{"id":32,"date":"2021-12-15T09:52:58","date_gmt":"2021-12-15T09:52:58","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien\/"},"modified":"2021-12-15T09:52:58","modified_gmt":"2021-12-15T09:52:58","slug":"weitere-lernmaterialien","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien\/","title":{"raw":"Weitere Lernmaterialien","rendered":"Weitere Lernmaterialien"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z1130e172d4c6\" class=\"sectionHead\"><span class=\"titlemark\">1.9 <\/span> <a id=\"x1-330009\"><\/a>Weitere Lernmaterialien<\/h3> <p class=\"noindent\">Wir wollen hier versuchen, Ihnen einen \u00dcberblick \u00fcber dieses Kapitel zu geben und auch weitere \u00dcbungsaufgaben zu den Themen des Kapitels zu sammeln. <a id=\"x1-33001r32\"><\/a> <\/p> <h4 id=\"z9955ba4b893f\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.1 <\/span> <a id=\"x1-340001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wir haben in diesem Kapitel viele Themen und unter anderem Tipps, Motivationen, einige Theorie und auch noch etwas Geschichte der Mathematik besprochen. Auf Grund dieser Vielfalt wollen wir kurz noch betonen, was Sie aus diesem Kapitel eigentlich f\u00fcr das Folgende mitnehmen m\u00fcssen: Die Themen aus den Abschnitten <a href=\"..\/..\/chapter\/logische-begriffe#x1-60003\">1.3<\/a>, <a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-110004\">1.4<\/a> und <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200006\">1.6<\/a> sind grundlegend und wir werden ohne Wiederholungen alle unsere weiteren Diskussionen auf diese Abschnitte aufbauen. Vor allem sollten Sie die folgenden Begriffe so lange \u00fcben, bis Sie diese ohne Zweifel im Ged\u00e4chnis haben. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Logische Operationen, insbesondere sollten Sie alle F\u00e4lle f\u00fcr das Oder, die Implikation und ihre Negation ohne das Nachbl\u00e4ttern der Wahrheitstabellen wissen. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Quantoren und deren Verhalten bei Kombination und Negation. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Mengenoperationen, de Morgan Gesetze. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Begriff der Funktion und elementare Eigenschaften wie Injektivit\u00e4t, Surjektivit\u00e4t und Bijektivit\u00e4t, aber auch die ungenaueren Begriffe \u201ewohldefiniert\u201c und \u201ekanonisch\u201c. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Verhalten dieser Eigenschaften unter Verkn\u00fcpfungen. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">\u00c4quivalenzrelationen, Partitionen, Quotientenraum<\/div><\/div> <p class=\"indent\">Bei logischen Aussagen werden wir, wie Sie vielleicht schon in Abschnitt <a href=\"..\/..\/chapter\/beweise#x1-220008\">1.8<\/a> gemerkt haben, die Notation in Zukunft etwas leichter halten. Unter anderem werden wir die Anf\u00fchrungszeichen <math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> bei logischen Ausdr\u00fccken weglassen und teilweise die Klammerung unterschlagen, wenn diese implizit klar ist. Zum Beispiel kann <math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> nur f\u00fcr den Ausdruck <math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> stehen, da ansonsten die Zahl <math display=\"inline\"><mi>n<\/mi><\/math> auf der rechten Seite von <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> nicht definiert ist. Im Zweifel empfehlen wir Ihnen aber lieber eine Klammer zu viel zu                                                                                                                                                                           schreiben. <\/p><p class=\"indent\">Die Diskussionen rund um Logik und Mengenlehre dieses Abschnitts k\u00f6nnte man mit Muskel\u00fcbungen f\u00fcr einen Schwimmunterricht fern von jeglicher Wasseroberfl\u00e4che vergleichen, denn wir haben logische Begriffe und Mengennotationen besprochen ohne konkrete Aussagen oder Mengen besprechen zu wollen. In der Tat sollten Sie sogar den Abschnitt <a href=\"..\/..\/chapter\/zahlenmengen#x1-190005\">1.5<\/a> (der als \u00dcbersicht gedacht war) wieder vergessen. Denn wir werden im n\u00e4chsten Kapitel logische Begriffe, Mengen und Funktionen verwenden um die reellen Zahlen axiomatisch einzuf\u00fchren. <\/p><p class=\"indent\">Abschnitt <a href=\"..\/..\/chapter\/beweise#x1-220008\">1.8<\/a> und die \u00dcbungsaufgaben dieses Abschnitts sollen Ihnen helfen Ihren eigenen Zugang zu Beweisen zu finden. Schwierige Fragen, die immer wieder auftauchen, sind, \u201eWas muss ich denn bei diesem Satz oder bei dieser Aufgabe eigentlich beweisen?\u201c und \u201eWelche Aussage kann ich als gegeben annehmen?\u201c. Dies ist in diesem Kapitel mitunter wirklich schwer zu beantworten (und wir haben hierzu im Laufe des Kapitels auch mehrmals unsere Meinung ge\u00e4ndert). Nach Einf\u00fchrung der Axiome im n\u00e4chsten Kapitel wird deutlich klarer sein, was wir beweisen m\u00fcssen: n\u00e4mlich ausser den Axiomen alles Weitere, wobei wir aber auf bereits bewiesene Aussagen zur\u00fcckgreifen d\u00fcrfen. Insbesondere d\u00fcrfen Sie in den w\u00f6chentlichen \u00dcbungsaufgaben die Aussagen der Vorlesung und ebenso die Aussagen des Skripts verwenden, aber keine \u00dcbungsaufgaben des Skripts und auch nur jene Seiten des Skripts, die bereits in der Vorlesung behandelt wurden. Obwohl die Axiome sehr einfach sein werden, werden wir im Laufe der Vorlesung viele komplizierte und auch \u00fcberraschende Aussagen beweisen k\u00f6nnen. <\/p><p class=\"indent\">Wir wollen hier noch einige Multiple-Choice-Fragen stellen, die Ihnen helfen sollten die Themen des Kapitels zu wiederholen. Es sind jeweils mehrere richtige Antworten m\u00f6glich. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zaab1e3014555\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen. Welche Aussagen sind (immer) wahr und welche sind (manchmal) falsch?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i) und (ii) verweisen wir auf die Diskussion nach<\/span> (<a href=\"..\/..\/chapter\/logische-begriffe#x1-8003r5\">1.5<\/a>)<span class=\"ecti-1095\">-<\/span>(<a href=\"..\/..\/chapter\/logische-begriffe#x1-8004r6\">1.6<\/a>)<span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rung<\/span> <span class=\"ecti-1095\">zu den Aussagen in (iii)-(vi) verweisen wir auf <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/logische-begriffe#x1-8011r11\"><span class=\"ecti-1095\">1.11<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za7d9b0fb131c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Abbildung und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Teilmengen. Welche der folgenden Aussagen sind immer wahr?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2287<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2287<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Als erstes bemerken wir, dass es sich bei dem Audruck<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">nicht um die inverse<\/span> <span class=\"ecti-1095\">Funktion von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(welche im Allgemeinen gar nicht existiert) handelt, sondern dass wir damit das Urbild der<\/span> <span class=\"ecti-1095\">angegebenen Menge bestimmen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (i) gilt per Definition: Jedes Element von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">wird unter<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">abgebildet, wodurch<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach Definition<\/span> <span class=\"ecti-1095\">vom Urbild von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Ein Gegenbeispiel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Die Aussage<\/span> <span class=\"ecti-1095\">gilt aber, falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich ein Gegenbeispiel finden. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">gilt die Aussage<\/span> <span class=\"ecti-1095\">nicht. Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">hingegen surjektiv ist, ist die Aussage wahr.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Per Definition ist (iv) wahr, weil jedes Element von<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">unter<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>B<\/mi><\/math> <span class=\"ecti-1095\">abgebildet wird.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z06b4b839b4f2\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Z<\/mi><\/math> <span class=\"ecti-1095\">Mengen und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">sowie <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">Funktionen. Gelten die folgenden Schl<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">sse allgemein?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist, dann ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist, dann ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Teilmenge <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i) finden wir ein Gegenbeispiel: Falls<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><span class=\"ecti-1095\">, so ist<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">surjektiv<\/span> <span class=\"ecti-1095\">obwohl <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht surjektiv ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Aussage (ii) ist richtig. Angenommen <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist surjektiv und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist<\/span> <span class=\"ecti-1095\">existiert ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">so<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <span class=\"ecti-1095\">jedoch <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Element von <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">beliebig<\/span> <span class=\"ecti-1095\">war, ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (iii) ist wahr. Wir nehmen an, dass <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist. Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">so dass<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Daraus folgt hingegen auch<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion ist, ist<\/span> <math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">wohldefiniert. Dies wird hier<\/span> <span class=\"ecti-1095\">implizit verwendet.<\/span><\/span><\/span><span class=\"ecti-1095\">, dass <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">als injektiv vorrausgesetzt<\/span> <span class=\"ecti-1095\">wurde, folgt nun <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Also ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iv) ist falsch. Ein Gegenbeispiel w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Aussage (v) ist wieder richtig. Angenommen<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">Teilmenge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Sei ausserdem<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und somit<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">nach Voraussetzung<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Also ist<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Dies beweise aber gerade<\/span> <span class=\"ecti-1095\">die Injektivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (vi) ist wahr. Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jede Teilmenge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">wissen wir<\/span> <span class=\"ecti-1095\">dann, dass <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">gilt.<\/span> <span class=\"ecti-1095\">Insbesondere ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nichtleer. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"ecti-1095\">per Definition <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math><span class=\"ecti-1095\">, was<\/span> <span class=\"ecti-1095\">die Surjektivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">beweist.<\/span><\/p><\/details>  <\/div> <a id=\"x1-34017r34\"><\/a> <h4 id=\"z099b6123eeee\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.2 <\/span> <a id=\"x1-350002\"><\/a>Fl\u00e4cheninhalt<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfd122d568e0d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Allgemeinere Bereiche unter der Parabel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter der Parabel<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei gegebene<\/span> <span class=\"ecti-1095\">reelle Zahlen mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">sind.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen Sie zur Vereinfachung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">an, womit die Situation der Aussage in Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\"><span class=\"ecti-1095\">1.1<\/span><\/a> <span class=\"ecti-1095\">sehr <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich wird. Der<\/span> <span class=\"ecti-1095\">Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und anschliessend auch der allgemeine Fall lassen sich auf diesen Spezialfall zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4616542b65de\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir eine Beweisvariante illustrieren, wie man auf Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4006r3\"><span class=\"ecti-1095\">1.3<\/span><\/a> <span class=\"ecti-1095\">schliessen kann ohne zuerst im Besitz der richtigen Formel zu sein. Wir schreiben dazu f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-35001r12\" \/><mstyle class=\"maketag\"><mtext>(1.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Gehen Sie nun wie folgt vor.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mi>a<\/mi><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie alle dabei verwendenten Rechenregeln benennen?<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifizieren Sie die Gleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf Lemma <\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4006r3\"><span class=\"ecti-1095\">1.3<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) verwenden Sie am besten (i) f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">setzen dies rechts in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-35001r12\">1.12<\/a>) <span class=\"ecti-1095\">ein. Verwenden Sie anschliessend die einfachere Summenformel in<\/span> <span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/beweise#x1-25001r85\"><span class=\"ecti-1095\">1.85<\/span><\/a> <span class=\"ecti-1095\">und eine elementare Gleichungsumformung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii).<\/span><\/p><\/details>  <\/div> <a id=\"x1-35005r35\"><\/a> <h4 id=\"z584dee1d4154\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.3 <\/span> <a id=\"x1-360003\"><\/a>Logik<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfec3a47e13de\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Draculas B\u00fccher)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In der Bibliothek des Grafen Dracula gibt es keine zwei B<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cher, deren Inhalt aus gleich<\/span> <span class=\"ecti-1095\">vielen W<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rtern besteht. Die Anzahl der B<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cher ist die Summe der Anzahl der W<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rter jedes<\/span> <span class=\"ecti-1095\">einzelnen Buches. Des Weiteren gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gen diese Aussagen, um den Inhalt mindestens eines<\/span> <span class=\"ecti-1095\">Buches aus Draculas Bibliothek genau zu beschreiben. Was steht in diesem Buch? Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span> <span class=\"ecti-1095\">enstammt dem Buch <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#Xamann-escher\"><span class=\"ecti-1095\">AE06<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd38469fb6ca7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Vier Aussagen in Pr\u00e4dikatenlogik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir sagen <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">falls<\/span> <span class=\"ecti-1095\">es ein <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gibt<\/span> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>d<\/mi><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beschreiben Sie die Bedeutung folgender Aussagen und bestimmen Sie, ob diese zutreffen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z94554da566b7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Allgemeinere Existenzquantoren)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir Quantoren f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Aussage, dass mehrere Elemente mit einer<\/span> <span class=\"ecti-1095\">Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">existieren, definieren.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie unter Verwendung des Existenzquantors einen neuen Quantor <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass die Aussage<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"maendquote\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">bedeutet, dass es mindestens zwei Elemente <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">haben.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verallgemeinern Sie (i) zu dem Quantor <\/span><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und verwenden Sie diese um auch den Quantor <\/span><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zu definieren, der besagen soll, dass es genau <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">besitzen. (Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen entweder informell Punkte verwenden, oder formal korrekter den<\/span> <span class=\"ecti-1095\">Funktionsbegriff, Eigenschaften von Funktionen, und die Menge <\/span><span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die genau <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Elemente hat.)<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie unter Verwendung des Existenzquantors, des Funktionsbegriffes, einer<\/span> <span class=\"ecti-1095\">Eigenschaft von Funktionen und der nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">einen neuen Quantor <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">der besagt, dass es unendlich viele Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">besitzen.<\/span><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> <span class=\"ecti-1095\">kann durch <\/span><math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> <span class=\"ecti-1095\">definiert werden. <\/span><\/p><\/details>  <\/div> <a id=\"x1-36004r36\"><\/a> <h4 id=\"z5420a98d47bd\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.4   <\/span> <a id=\"x1-370004\"><\/a>Funktionen und Relationen<\/h4> <div class=\"me meexample\"> <p class=\"indent\"> <\/p><h4 id=\"z5cfd4a37e7be\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Injektive Funktionen durch Fallunterscheidungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"> <span class=\"ecti-1095\">Zeigen Sie folgende Behauptung: Seien <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen und sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">eine Partition von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen es ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">eine injektive Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>P<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">gegeben und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">die eindeutige Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">nach Lemma <\/span><a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-16002r52\"><span class=\"ecti-1095\">1.52<\/span><\/a><span class=\"ecti-1095\">. Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann injektiv ist, falls die Mengen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>P<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">paarweise disjunkt sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"> <\/p><h4 id=\"z25a3c462b0c3\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine \u00c4quivalenzrelation auf dem kartesischen Produkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei nicht-leere<\/span> <span class=\"ecti-1095\">Mengen, sei <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Relation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Relation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Wir definieren<\/span> <span class=\"ecti-1095\">damit eine Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow> <mi>X<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow> <mi>Y<\/mi> <\/mrow><\/msub><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Zeigen Sie, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">genau dann eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist, wenn <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist. Gilt dies auch, wenn<\/span> <span class=\"ecti-1095\">eine der beiden Mengen <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">leer ist? Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung enstammt dem Buch <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#Xamann-escher\"><span class=\"ecti-1095\">AE06<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen Sie die drei definierenden Eigenschaften einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">(beziehungsweise<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>X<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5b028a988603\"> <span class=\"ecbx-1095\">Applet <\/span>(Nichtvertauschbarkeit der Verkn\u00fcpfung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"450px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/tcdwejfk\/width\/688\/height\/450\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten zwei Funktionen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nur durch den Graphen beschrieben ist und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">eine affine Funktion ist, die durch zwei Konstanten <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert wird. Durch Bewegen zweier Punkte am Graph von <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">lassen sich <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">definieren. Experimentieren Sie damit um sich an die geometrische Bedeutung von den Zahlen<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">zu errinnern, und beobachten Sie, wie sich die beiden Funktionen <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">im rechten Fenster unterschiedlich ver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndern. Es ist n<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">tzlich sich diese Ph<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nomene vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndig<\/span> <span class=\"ecti-1095\">zu erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, denn wir werden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlichen Verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfungen in unseren weiteren <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berlegungen<\/span> <span class=\"ecti-1095\">begegnen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1dcdda3c2f13\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Konstruktion der Menge der ganzen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen an, dass wir bereits die Menge der nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und damit auch die Menge der nicht-negativen ganzen Zahlen<\/span> <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u2294<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">mit allen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen<\/span> <span class=\"ecti-1095\">Operationen und Eigenschaften kennen und wollen daraus die ganzen Zahlen definieren. Dazu betrachten wir<\/span> <span class=\"ecti-1095\">eine Relation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math><\/span><span class=\"period\">:<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-37001r13\" \/><mstyle class=\"maketag\"><mtext>(1.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie unter der Annahme, dass die ganzen Zahlen schon bekannt sind, wieso wir<\/span> <span class=\"ecti-1095\">die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-37001r13\">1.13<\/a>)           <span class=\"ecti-1095\">betrachten wollen. <\/span><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Motiviert ist das ganze dadurch, dass man eine<\/span> <span class=\"ecti-1095\">ganze Zahl als Differenz von zwei nicht-negativen ganzen Zahlen auffassen kann. Diese<\/span> <span class=\"ecti-1095\">sind aber nicht eindeutig gegeben; deswegen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt man auf Tupeln von nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen<\/span> <span class=\"ecti-1095\">Zahlen obige <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ein. In der Tat hat man bei Definition<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-37001r13\">1.13<\/a>) <span class=\"ecti-1095\">eigentlich<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">im Hinterkopf, darf dies aber formal nicht verwenden, da die Subtraktion (noch) nicht<\/span> <span class=\"ecti-1095\">erlaubt ist. Wenn wir <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">sprechen, werden wir leicht schizophren an <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">denken, dies aber nicht verwenden.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Informell k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir uns das Tupel <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">als einen Vektor mit Anfangspunkt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und Endpunkt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">vorstellen, womit die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen allen Vektoren mit gleicher L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge und Richtung<\/span> <span class=\"ecti-1095\">entspricht. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Kontostand vor und <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Kontostand nach einer Transaktion darstellen und die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse entspricht<\/span> <span class=\"ecti-1095\">dann dem Nettogewinn\/-verlust.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definiert. <\/span><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math> <\/p><div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Symmetrie: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">denn <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">impliziert <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und daher auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">impliziert <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Durch Summieren dieser Gleichungen ergibt sich<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und durch Wegstreichen von <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(was eine der Eigenschaften von <\/span><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist und nicht die Subtraktion verwendet) erhalten wir<\/span> <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><span class=\"ecti-1095\">, also<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/div><\/div> <p class=\"noindent\"><\/p><\/details><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Der Quotient <\/span><math display=\"inline\"><msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math><span class=\"ecti-1095\">kann<\/span> <span class=\"ecti-1095\">als Definition von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">angesehen werden, wobei wir die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse auch als<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">schreiben. Insbesondere<\/span> <span class=\"ecti-1095\">identifizieren wir <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">schreiben <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch als <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folgende <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bungen erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wieso dies Sinn ergibt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildungen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">injektiv sind und disjunkte Bilder <\/span><math display=\"inline\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">haben, welche wir mit <\/span><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bezeichnen werden.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2294<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-37006r37\"><\/a> <h4 id=\"z45b65798acf8\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.5 <\/span> <a id=\"x1-380005\"><\/a>Beweismethoden<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z79d8878cd3cd\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl. Zeigen Sie, dass die Implikation<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;gerade&nbsp;<\/mtext><\/mstyle><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;ungerade<\/mtext><\/mstyle><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc986e192943d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(T\u00fcrme von Hanoi)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es seien <\/span><math display=\"inline\"><mn>3<\/mn><\/math> <span class=\"ecti-1095\">Ablagefl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chen gegeben. Angenommen auf der linken Ablagefl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che seien<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Bl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">cke f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine<\/span> <span class=\"ecti-1095\">nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">aufget<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rmt, wobei nie ein kleinerer Block unter einem gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sseren Block liegt.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=455&amp;url=Pictures\/Einfuehrung\/Induktionsbeispiele\/Hanoi\/hanoi1.pdf\"><img id=\"z670cd8bf7fde\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/Induktionsbeispiele\/Hanoi\/hanoi1.svg\" width=\"455\"><\/div>  <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass Sie den Turm von links nach rechts umschichten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen, ohne dass je ein kleinerer<\/span> <span class=\"ecti-1095\">Block unter einem gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sseren Block zu liegen kommt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za16d780107af\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie mittels vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahlen <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>5<\/mn><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>4<\/mn><mi>n<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Beginnen     Sie     die     Induktion     mit     dem     Induktionsanfang     bei<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>5<\/mn><\/math> <span class=\"ecti-1095\">(denn                                                                                                      f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><\/math> <span class=\"ecti-1095\">gilt die Aussage nicht).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zbe5557817843\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie     mittels     vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger     Induktion,     dass     die     Ungleichung<\/span> <math display=\"inline\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                           alle                           nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen                           Zahlen<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gilt. Beschreiben Sie die Variante des Induktionsbeweises, die sie hier verwenden.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Der       Beweis       des       Induktionsschrittes       ist       einfacher       falls<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">angenommen     werden     kann.     Aus     diesem     Grund     ist     es     besser     sowohl<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">als                                                                                                         auch<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">direkt            zu            <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen            um            im            Induktionsschritt<\/span> <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">dann                ohne                Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung                der                Allgemeinheit<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">zu verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3fce48ed5461\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir               f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rben               jeden               Punkt               im               Gitter<\/span> <math display=\"inline\"><msup><mrow><mi>\u2124<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit                                                  einer                                                  von<\/span> <math display=\"inline\"><mn>1<\/mn><mn>7<\/mn><\/math> <span class=\"ecti-1095\">verschiedenen   Farben   ein.   Zeigen   Sie,   dass   es   ein   achsenparalleles   Rechteck<\/span> <math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">in diesem Gitter gibt, dessen Ecken alle dieselbe Farbe besitzen.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Statt das ganze Gitter zu untersuchen, gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt es die Gitterpunkte in dem Rechteck<\/span> <math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mn>7<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">zu betrachten. Es gibt dann auf einer horizontalen Gerade mit ganzzahliger <\/span><math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">in dem Rechteck <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">jeweils eines von <\/span><math display=\"inline\"><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichen Farbmuster. Da es mehr horizontale Geraden (genau <\/span><math display=\"inline\"><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">als m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">gliche Farbmuster gibt, wiederholt sich eines der Farbmuster. Wir w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen diese beiden<\/span> <math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinaten<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r unser gesuchtes Rechteck <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In diesem Farbmuster gibt es horizontal <\/span><math display=\"inline\"><mn>1<\/mn><mn>8<\/mn><\/math> <span class=\"ecti-1095\">Positionen aber nur <\/span><math display=\"inline\"><mn>1<\/mn><mn>7<\/mn><\/math> <span class=\"ecti-1095\">Farben, womit sich eine der Farben wiederholt. Verwendet man diese <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Koordinaten,<\/span> <span class=\"ecti-1095\">so erhalten wir das gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Rechteck <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">bei dem alle Eckpunkte dieselbe Farbe besitzen. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0dc9d7a67ade\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><mi>S<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge von <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">mit Kardinalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass es Elemente <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>S<\/mi><\/math> <span class=\"ecti-1095\">gibt mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Es                                        empfiehlt                                        sich<\/span> <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">in Teilmengen zu partitionieren, die geometrische Progressionen darstellen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2ab8d4666c64\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>T<\/mi><\/math> <span class=\"ecti-1095\">eine endliche<\/span> <span class=\"ecti-1095\">Menge und seien <\/span><math display=\"inline\"><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">Teilmengen von <\/span><math display=\"inline\"><mi>T<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>k<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>T<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass es ein Element <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>T<\/mi><\/math> <span class=\"ecti-1095\">gibt, welches in mindestens <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">der Mengen <\/span><math display=\"inline\"><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">liegt.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen Sie indirekt an, dass jeder Punkt<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>T<\/mi><\/math> <span class=\"ecti-1095\">in h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens<\/span> <math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen enthalten<\/span> <span class=\"ecti-1095\">ist und zeigen Sie, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>T<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt dies in Worten oder geometrisch zu erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wir werden die Notation f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">formalen Beweis erst sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/p><\/details>  <\/div> <a id=\"x1-38001r38\"><\/a> <h4 id=\"z919b7a662800\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.6 <\/span> <a id=\"x1-390006\"><\/a>Geometrische Probleme<\/h4> <p class=\"noindent\">Wir empfehlen Ihnen folgende \u00dcbung zu l\u00f6sen, da der \u201eSatz von Pythagoras\u201c f\u00fcr uns sp\u00e4ter gewissermassen zu einer Definition werden wird. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf4eeeeac81f4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Satz von Pythagoras)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten ein rechtwinkliges Dreieck mit Katheten der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">und Hypotenuse<\/span> <span class=\"ecti-1095\">der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>c<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Hinweis: Betrachten Sie folgende Bilder<\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"329px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/XJ7K6R8e\/width\/688\/height\/329\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <span class=\"ecti-1095\">und gehen Sie dabei davon aus, dass gewisse<\/span> <span class=\"ecti-1095\">geometrische Begriffe wie Winkel, L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge und Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r elementare Bereiche und intuitiv<\/span> <span class=\"ecti-1095\">anschauliche Eigenschaften wie Invarianz der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter Verschiebung und Drehung bekannt<\/span> <span class=\"ecti-1095\">sind.<\/span> <\/p> <\/div> <a id=\"x1-39001r39\"><\/a> <h4 id=\"z4c0b4ec12664\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.7 <\/span> <a id=\"x1-400007\"><\/a>\u00dcbungen zu Primzahlen<\/h4> <p class=\"noindent\">Eine nat\u00fcrliche Zahl <math display=\"inline\"><mi>p<\/mi><\/math> gr\u00f6sser als <math display=\"inline\"><mn>1<\/mn><\/math> ist <span class=\"ecbx-1095\">irreduzibel<\/span>, falls sie nicht als Produkt von zwei kleineren nat\u00fcrlichen Zahlen geschrieben werden kann. Eine nat\u00fcrliche Zahl <math display=\"inline\"><mi>p<\/mi><\/math> gr\u00f6sser als <math display=\"inline\"><mn>1<\/mn><\/math> heisst eine <span class=\"ecbx-1095\">Primzahl<\/span>, falls ein Produkt <math display=\"inline\"><mi>a<\/mi><mi>b<\/mi><\/math> zweier nat\u00fcrlicher Zahlen <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> nur dann durch <math display=\"inline\"><mi>p<\/mi><\/math> teilbar ist, falls eine der beiden Zahlen durch <math display=\"inline\"><mi>p<\/mi><\/math> teilbar ist.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z266a3912d113\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Primzahlen sind irreduzibel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen               Sie,               dass               jede               Primzahl               in<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">auch irreduzibel ist.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich eine Primzahl <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">als Produkt zweier nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">schreiben (das heisst, <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi>b<\/mi><\/math><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">so teilt <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">das Produkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Diese beiden Begriffe sind in der Tat f\u00fcr die nat\u00fcrlichen Zahlen \u00e4quivalent (wir werden dies nochmals etwas genauer in Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-540004\">2.2.4<\/a> besprechen) und wir werden in diesem Abschnitt irreduzible Zahlen ebenso als Primzahlen bezeichnen. Es ist eine gute \u00dcbung im Folgenden genau zu erkl\u00e4ren welche der beiden Begriffe eigentlich verwendet wird. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z04c3af9b608b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Primfaktorzerlegung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie  mittels  vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger  Induktion,  dass  jede  nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche  Zahl  gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sser  als<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">als Produkt von Primzahlen geschrieben werden kann.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen hier eine Variante der vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigen Induktion verwenden und im<\/span> <span class=\"ecti-1095\">Induktionsschritt  annehmen,  dass  die  Aussage  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  alle  kleineren  Zahlen  bereits  bewiesen<\/span> <span class=\"ecti-1095\">wurde. Das einfachste Argument hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r verwendet eigentlich den Begriff der irreduziblen<\/span> <span class=\"ecti-1095\">Zahlen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za3f46d6beccc\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Unendlich viele Primzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass es unendliche viele Primzahlen gibt.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Das   einfachste   Argument   hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r   ist   das   Argument   von   Euklid:   Falls<\/span> <math display=\"inline\"><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">die einzigen             Primzahlen             w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren,             dann             w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u22ef<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">durch keine dieser  Primzahlen  teilbar  und  mittels  der  letzten  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung  f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nde  man  weitere<\/span> <span class=\"ecti-1095\">Primzahlen (genauer gesagt irreduzible Zahlen).<\/span><\/p><\/details>  <\/div> <a id=\"x1-40001r40\"><\/a> <h4 id=\"z3c9c03c2b21c\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.8 <\/span> <a id=\"x1-410008\"><\/a>Online Lernhilfen<\/h4> <p class=\"noindent\">Bei erster Verwendung dieses Skripts in der entsprechenden Vorlesung hat ein Student mehrere online-Tools zum Erlernen der Inhalte des Skripts programmiert. Leider haben sich aber die Inhalte seitdem etwas ver\u00e4ndert, ohne dass die Inhalte in den online-Tools angepasst wurden. Wir erw\u00e4hnen hier die <a href=\"https:\/\/janiks.me\/projects\/eth\/ana\/study\" target=\"_blank\" rel=\"noopener\">Webseite<\/a> einmalig. Sollte jemand dieses App aktualisieren oder ein alternatives App mit aktuellen Daten zur Verf\u00fcgung stellen wollen, so werden wir dieses gerne auch wieder am Ende von jedem Kapitel bewerben. <a id=\"x1-41001r41\"><\/a> <\/p> <h4 id=\"zd3cb8a14d53b\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.9 <\/span> <a id=\"x1-420009\"><\/a>SageMath<\/h4> <p class=\"noindent\">Schlussendlich wollen wir die auf Python basierende Programmiersprache SageMath erw\u00e4hnen. Diese ist f\u00fcr mathematische Experimente bestens geeignet und kann auch ohne einer aufwendigen Installation mittels <a href=\"https:\/\/sagecell.sagemath.org\" target=\"_blank\" rel=\"noopener\">SageMathCell<\/a> ben\u00fctzt werden. F\u00fcr aufwendigere oder auch rechenintensivere Programme empfiehlt sich allerdings eine lokale Installation von SageMath. <\/p><p class=\"indent\">Versuchen Sie doch folgende Zeilen in SageMath aus und experimentieren Sie etwas damit. <\/p> <div class=\"lstlisting\" id=\"listing-1\"><span class=\"label\"><a id=\"x1-42001r1\"><\/a><\/span><span class=\"ectt-1095\">table<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[[\"<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">\",\"<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">\",\"<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=&gt;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">\"]]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42002r2\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">+<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[[<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">,(<\/span><span class=\"ectt-1095\">not<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">or<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[<\/span><span class=\"ectt-1095\">true<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">false<\/span><span class=\"ectt-1095\">]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42003r3\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[<\/span><span class=\"ectt-1095\">true<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">false<\/span><span class=\"ectt-1095\">]]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Wir bemerken, dass einzelne Befehle normalerweise innerhalb einer Zeile stehen sollten, dass aber bei einer vorhanden offenen Klammer die anschliessende(n) Zeile(n) als Teil der ersten Zeile aufgefasst werden. Der Ausdruck in der ersten Zeile beschriftet die Kopfzeile und mit den beiden \u2019for\u2019-Konstruktionen werden alle M\u00f6glichkeiten durchgetestet. Insgesamt wird durch den Befehl <span class=\"ecbx-1095\">table <\/span>hier die Wahrheitstabelle der Implikation dargestellt. \u00c4ndern Sie obiges Beispiel und versuchen Sie zum Beispiel damit eine der Tautologie aus Abschnitt <a href=\"..\/..\/chapter\/logische-begriffe#x1-70001\">1.3.1<\/a> zu \u00fcberpr\u00fcfen. <\/p><p class=\"indent\">Hier einige Zeilen, die zeigen, wie wir in SageMath mit Mengen operieren k\u00f6nnen. Bei mehreren Befehlen hintereinander, die alle ein Ergebnis darstellen sollten, m\u00fcssen sie den Befehl <span class=\"ecbx-1095\">print<\/span> verwenden. <\/p>  <div class=\"lstlisting\" id=\"listing-2\"><span class=\"label\"><a id=\"x1-42004r1\"><\/a><\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">\"<\/span><span class=\"ectt-1095\">abcdabcd<\/span><span class=\"ectt-1095\">\"<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42005r2\"><\/a><\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[0,\"<\/span><span class=\"ectt-1095\">a<\/span><span class=\"ectt-1095\">\"]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42006r3\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=\",<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,\"<\/span><span class=\"ectt-1095\">und<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=\",<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42007r4\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Durchschnitt<\/span><span class=\"ectt-1095\">:\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">.<\/span><span class=\"ectt-1095\">intersection<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42008r5\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Vereingigung<\/span><span class=\"ectt-1095\">:\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">+<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42009r6\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42010r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42011r8\"><\/a><\/span><span class=\"ectt-1095\">def<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">Y<\/span><span class=\"ectt-1095\">):<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42012r9\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">return<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42013r10\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42014r11\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">:\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Zur Definition neuer Befehle mittels der obigen <span class=\"ecbx-1095\">def<\/span>-Konstruktion sollte man bemerken, dass die konsistente Einr\u00fcckung der folgenden Zeilen wichtig ist und man zur Erh\u00f6hung der \u00dcbersicht die Definition mit einer Leerzeile beenden sollte. <\/p><p class=\"indent\">Die folgende Routine testet die Injektivit\u00e4t der Einschr\u00e4nkung einer Funktion <math display=\"inline\"><mi>f<\/mi><\/math> auf einer endlichen Menge <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dazu wollen wir bemerken, dass <span class=\"ecbx-1095\">all <\/span>der Allquantor und <span class=\"ecbx-1095\">any <\/span>der Existenzquantor in SageMath ist. Des Weiteren, sollten Sie wissen, dass <span class=\"ecbx-1095\">= <\/span>wie in obigem Beispiel eine Anweisung ist, die einer Variable einen Wert zuweist, aber <span class=\"ecbx-1095\">== <\/span>die Frage nach Gleichheit und <span class=\"ecbx-1095\">!= <\/span>die Frage nach Ungleichheit darstellt. <\/p>  <div class=\"lstlisting\" id=\"listing-3\"><span class=\"label\"><a id=\"x1-42015r1\"><\/a><\/span><span class=\"ectt-1095\">def<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">TestInj<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">):<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42016r2\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">return<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">all<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">==<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">or<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">)!=<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">))<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42017r3\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42018r4\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42019r5\"><\/a><\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[0..10]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">#<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Liste<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">der<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">ganzen<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahlen<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">von<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">bis<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">10<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42020r6\"><\/a><\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">^2-2*<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">+1<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">#<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">die<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Funktion<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42021r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eingeschraenkt<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">auf<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">injektiv<\/span><span class=\"ectt-1095\">?\")<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42022r8\"><\/a><\/span><span class=\"ectt-1095\">TestInj<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Schreiben Sie anschliessend eine Routine <span class=\"ecti-1095\">TestWohl(F,X,Y)<\/span>, die \u00fcberpr\u00fcft, ob <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> gilt. Schreiben Sie eine Routine <span class=\"ecti-1095\">TestSurj(F,X,Y)<\/span>, die \u00fcberpr\u00fcft, ob <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <\/math> gilt. Schreiben Sie schlussendlich eine Routine <span class=\"ecti-1095\">TestGraph(G,X,Y)<\/span>, die f\u00fcr eine Menge <math display=\"inline\"><mi>G<\/mi><\/math> \u00fcberpr\u00fcft ob <math display=\"inline\"><mi>G<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo><\/math> der Graph einer Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> ist. <\/p><p class=\"indent\">Folgende Zeilen sollten auch zeigen, warum wir Ihnen SageMath als Programmiersprache f\u00fcr mathematische Experimente empfehlen. <\/p> <div class=\"lstlisting\" id=\"listing-4\"><span class=\"label\"><a id=\"x1-42023r1\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Wofuer<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">steht<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42024r2\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42025r3\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2^2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2^2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42026r4\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">die<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Wurzel<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">aus<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rational<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">sqrt<\/span><span class=\"ectt-1095\">(2)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42027r5\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42028r6\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Wofuer<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">steht<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42029r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">reelle<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br><span class=\"label\"><a id=\"x1-42030r8\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(\"<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?\",<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Wir \u00fcberlassen Ihnen die entsprechenden Internetnachforschungen, falls Sie mehr \u00fcber SageMath wissen wollen.                                                                                                                                                                                                                                                                                                                                                     <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; border-left: 0; border-right: 0; }\ncenter { margin-top:1em; margin-bottom:1em; }\ntd center { margin-top:0em; margin-bottom:0em; }\n.Canvas { position:relative; }\nmath { text-indent: 0em; }\nli p.indent { text-indent: 0em }\nli p:first-child{ margin-top:0em; }\nli p:last-child, li div:last-child { margin-bottom:0.5em; }\nli p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }\n.enumerate1 {list-style-type:decimal;}\n.enumerate2 {list-style-type:lower-alpha;}\n.enumerate3 {list-style-type:lower-roman;}\n.enumerate4 {list-style-type:upper-alpha;}\n.obeylines-h,.obeylines-v {white-space: nowrap; }\ndiv.obeylines-v p { margin-top:0; margin-bottom:0; }\n.overline{ text-decoration:overline; }\n.overline img{ border-top: 1px solid black; }\ntd.displaylines {text-align:center; white-space:nowrap;}\n.centerline {text-align:center;}\n.rightline {text-align:right;}\npre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }\n.fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.fbox {display:table}\ndiv.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.minipage{width:100%;}\ndiv.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}\ndiv.center {text-align: left;}\ndiv.flushright, div.flushright div.flushright {text-align: right;}\ndiv.flushright div {text-align: left;}\ndiv.flushleft {text-align: left;}\n.underline{ text-decoration:underline; }\n.underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }\n.framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\n.framebox-c {text-align:center;}\n.framebox-l {text-align:left;}\n.framebox-r {text-align:right;}\nspan.thank-mark{ vertical-align: super }\nspan.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td 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img {text-align:center;}\n.marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}\n.marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}\n.reversemarginpar{float:left;}\n.equation td{text-align:center; vertical-align:middle; }\ntd.eq-no{ width:5%; }\ntable.equation { width:100%; }\ndiv.math-display, div.par-math-display{text-align:center;}\nmtr.hline mtd{ border-bottom:black solid 1px; padding-top:2px; padding-bottom:0em; }\nmtr.hline mtd mo{ display:none }\nmath .texttt { font-family: monospace; }\nmath .textit { font-style: italic; }\nmath .textsl { font-style: oblique; }\nmath .textsf { font-family: sans-serif; }\nmath .textbf { font-weight: bold; }\nmo.MathClass-op + mi{margin-left:0.3em}\nmi + mo.MathClass-op{margin-left:0.3em}\n math mstyle[mathvariant=\"bold\"] { font-weight: bold; font-style: normal; }\n math mstyle[mathvariant=\"normal\"] { font-weight: normal; font-style: normal; }\n.partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}\n.index-item, .index-subitem, .index-subsubitem {display:block}\ndiv.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}\ndiv.caption span.id{font-weight: bold; white-space: nowrap; }\nh1.partHead{text-align: center}\np.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\np.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\n.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}\n.subparagraphHead, .likesubparagraphHead { font-weight: bold;}\n.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; margin-right:1em; text-align:justify;}\n.verse{white-space:nowrap; margin-left:2em}\ndiv.maketitle {text-align:center;}\nh2.titleHead{text-align:center;}\ndiv.maketitle{ margin-bottom: 2em; }\ndiv.author, div.date {text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z1130e172d4c6\" class=\"sectionHead\"><span class=\"titlemark\">1.9 <\/span> <a id=\"x1-330009\"><\/a>Weitere Lernmaterialien<\/h3> <p class=\"noindent\">Wir wollen hier versuchen, Ihnen einen \u00dcberblick \u00fcber dieses Kapitel zu geben und auch weitere \u00dcbungsaufgaben zu den Themen des Kapitels zu sammeln. <a id=\"x1-33001r32\"><\/a> <\/p> <h4 id=\"z9955ba4b893f\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.1 <\/span> <a id=\"x1-340001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wir haben in diesem Kapitel viele Themen und unter anderem Tipps, Motivationen, einige Theorie und auch noch etwas Geschichte der Mathematik besprochen. Auf Grund dieser Vielfalt wollen wir kurz noch betonen, was Sie aus diesem Kapitel eigentlich f\u00fcr das Folgende mitnehmen m\u00fcssen: Die Themen aus den Abschnitten <a href=\"..\/..\/chapter\/logische-begriffe#x1-60003\">1.3<\/a>, <a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-110004\">1.4<\/a> und <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200006\">1.6<\/a> sind grundlegend und wir werden ohne Wiederholungen alle unsere weiteren Diskussionen auf diese Abschnitte aufbauen. Vor allem sollten Sie die folgenden Begriffe so lange \u00fcben, bis Sie diese ohne Zweifel im Ged\u00e4chnis haben. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Logische Operationen, insbesondere sollten Sie alle F\u00e4lle f\u00fcr das Oder, die Implikation und ihre Negation ohne das Nachbl\u00e4ttern der Wahrheitstabellen wissen. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Quantoren und deren Verhalten bei Kombination und Negation. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Mengenoperationen, de Morgan Gesetze. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Begriff der Funktion und elementare Eigenschaften wie Injektivit\u00e4t, Surjektivit\u00e4t und Bijektivit\u00e4t, aber auch die ungenaueren Begriffe \u201ewohldefiniert\u201c und \u201ekanonisch\u201c. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Verhalten dieser Eigenschaften unter Verkn\u00fcpfungen. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">\u00c4quivalenzrelationen, Partitionen, Quotientenraum<\/div><\/div> <p class=\"indent\">Bei logischen Aussagen werden wir, wie Sie vielleicht schon in Abschnitt <a href=\"..\/..\/chapter\/beweise#x1-220008\">1.8<\/a> gemerkt haben, die Notation in Zukunft etwas leichter halten. Unter anderem werden wir die Anf\u00fchrungszeichen <math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> bei logischen Ausdr\u00fccken weglassen und teilweise die Klammerung unterschlagen, wenn diese implizit klar ist. Zum Beispiel kann <math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> nur f\u00fcr den Ausdruck <math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> stehen, da ansonsten die Zahl <math display=\"inline\"><mi>n<\/mi><\/math> auf der rechten Seite von <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mspace class=\"negthinspace\" width=\"-0.17em\" \/><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> nicht definiert ist. Im Zweifel empfehlen wir Ihnen aber lieber eine Klammer zu viel zu                                                                                                                                                                           schreiben. <\/p><p class=\"indent\">Die Diskussionen rund um Logik und Mengenlehre dieses Abschnitts k\u00f6nnte man mit Muskel\u00fcbungen f\u00fcr einen Schwimmunterricht fern von jeglicher Wasseroberfl\u00e4che vergleichen, denn wir haben logische Begriffe und Mengennotationen besprochen ohne konkrete Aussagen oder Mengen besprechen zu wollen. In der Tat sollten Sie sogar den Abschnitt <a href=\"..\/..\/chapter\/zahlenmengen#x1-190005\">1.5<\/a> (der als \u00dcbersicht gedacht war) wieder vergessen. Denn wir werden im n\u00e4chsten Kapitel logische Begriffe, Mengen und Funktionen verwenden um die reellen Zahlen axiomatisch einzuf\u00fchren. <\/p><p class=\"indent\">Abschnitt <a href=\"..\/..\/chapter\/beweise#x1-220008\">1.8<\/a> und die \u00dcbungsaufgaben dieses Abschnitts sollen Ihnen helfen Ihren eigenen Zugang zu Beweisen zu finden. Schwierige Fragen, die immer wieder auftauchen, sind, \u201eWas muss ich denn bei diesem Satz oder bei dieser Aufgabe eigentlich beweisen?\u201c und \u201eWelche Aussage kann ich als gegeben annehmen?\u201c. Dies ist in diesem Kapitel mitunter wirklich schwer zu beantworten (und wir haben hierzu im Laufe des Kapitels auch mehrmals unsere Meinung ge\u00e4ndert). Nach Einf\u00fchrung der Axiome im n\u00e4chsten Kapitel wird deutlich klarer sein, was wir beweisen m\u00fcssen: n\u00e4mlich ausser den Axiomen alles Weitere, wobei wir aber auf bereits bewiesene Aussagen zur\u00fcckgreifen d\u00fcrfen. Insbesondere d\u00fcrfen Sie in den w\u00f6chentlichen \u00dcbungsaufgaben die Aussagen der Vorlesung und ebenso die Aussagen des Skripts verwenden, aber keine \u00dcbungsaufgaben des Skripts und auch nur jene Seiten des Skripts, die bereits in der Vorlesung behandelt wurden. Obwohl die Axiome sehr einfach sein werden, werden wir im Laufe der Vorlesung viele komplizierte und auch \u00fcberraschende Aussagen beweisen k\u00f6nnen. <\/p><p class=\"indent\">Wir wollen hier noch einige Multiple-Choice-Fragen stellen, die Ihnen helfen sollten die Themen des Kapitels zu wiederholen. Es sind jeweils mehrere richtige Antworten m\u00f6glich. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zaab1e3014555\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen. Welche Aussagen sind (immer) wahr und welche sind (manchmal) falsch?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>B<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i) und (ii) verweisen wir auf die Diskussion nach<\/span> (<a href=\"..\/..\/chapter\/logische-begriffe#x1-8003r5\">1.5<\/a>)<span class=\"ecti-1095\">&#8211;<\/span>(<a href=\"..\/..\/chapter\/logische-begriffe#x1-8004r6\">1.6<\/a>)<span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rung<\/span> <span class=\"ecti-1095\">zu den Aussagen in (iii)-(vi) verweisen wir auf <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/logische-begriffe#x1-8011r11\"><span class=\"ecti-1095\">1.11<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za7d9b0fb131c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Abbildung und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Teilmengen. Welche der folgenden Aussagen sind immer wahr?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2287<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2287<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Als erstes bemerken wir, dass es sich bei dem Audruck<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">nicht um die inverse<\/span> <span class=\"ecti-1095\">Funktion von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(welche im Allgemeinen gar nicht existiert) handelt, sondern dass wir damit das Urbild der<\/span> <span class=\"ecti-1095\">angegebenen Menge bestimmen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (i) gilt per Definition: Jedes Element von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">wird unter<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">abgebildet, wodurch<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach Definition<\/span> <span class=\"ecti-1095\">vom Urbild von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Ein Gegenbeispiel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Die Aussage<\/span> <span class=\"ecti-1095\">gilt aber, falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich ein Gegenbeispiel finden. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">gilt die Aussage<\/span> <span class=\"ecti-1095\">nicht. Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">hingegen surjektiv ist, ist die Aussage wahr.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Per Definition ist (iv) wahr, weil jedes Element von<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">unter<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>B<\/mi><\/math> <span class=\"ecti-1095\">abgebildet wird.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z06b4b839b4f2\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Z<\/mi><\/math> <span class=\"ecti-1095\">Mengen und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">sowie <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">Funktionen. Gelten die folgenden Schl<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">sse allgemein?<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist, dann ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist, dann ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><details class=\"mcquest\"><summary class=\"mcquest\" style=\"color:#FF7F00\"><span class=\"ecti-1095\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <span class=\"ecti-1095\">Wenn <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Teilmenge <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i) finden wir ein Gegenbeispiel: Falls<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><span class=\"ecti-1095\">, so ist<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">surjektiv<\/span> <span class=\"ecti-1095\">obwohl <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht surjektiv ist.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Aussage (ii) ist richtig. Angenommen <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist surjektiv und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">surjektiv ist<\/span> <span class=\"ecti-1095\">existiert ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">so<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <span class=\"ecti-1095\">jedoch <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Element von <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">beliebig<\/span> <span class=\"ecti-1095\">war, ist <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">surjektiv.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (iii) ist wahr. Wir nehmen an, dass <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist. Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">so dass<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Daraus folgt hingegen auch<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion ist, ist<\/span> <math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">wohldefiniert. Dies wird hier<\/span> <span class=\"ecti-1095\">implizit verwendet.<\/span><\/span><\/span><span class=\"ecti-1095\">, dass <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">als injektiv vorrausgesetzt<\/span> <span class=\"ecti-1095\">wurde, folgt nun <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Also ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">injektiv.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iv) ist falsch. Ein Gegenbeispiel w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Aussage (v) ist wieder richtig. Angenommen<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">Teilmenge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Sei ausserdem<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und somit<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">nach Voraussetzung<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Also ist<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Dies beweise aber gerade<\/span> <span class=\"ecti-1095\">die Injektivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (vi) ist wahr. Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jede Teilmenge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">wissen wir<\/span> <span class=\"ecti-1095\">dann, dass <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">gilt.<\/span> <span class=\"ecti-1095\">Insbesondere ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nichtleer. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"ecti-1095\">per Definition <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math><span class=\"ecti-1095\">, was<\/span> <span class=\"ecti-1095\">die Surjektivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">beweist.<\/span><\/p><\/details>  <\/div> <a id=\"x1-34017r34\"><\/a> <h4 id=\"z099b6123eeee\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.2 <\/span> <a id=\"x1-350002\"><\/a>Fl\u00e4cheninhalt<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfd122d568e0d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Allgemeinere Bereiche unter der Parabel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter der Parabel<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwei gegebene<\/span> <span class=\"ecti-1095\">reelle Zahlen mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">sind.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen Sie zur Vereinfachung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">an, womit die Situation der Aussage in Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\"><span class=\"ecti-1095\">1.1<\/span><\/a> <span class=\"ecti-1095\">sehr <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich wird. Der<\/span> <span class=\"ecti-1095\">Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und anschliessend auch der allgemeine Fall lassen sich auf diesen Spezialfall zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4616542b65de\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir eine Beweisvariante illustrieren, wie man auf Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4006r3\"><span class=\"ecti-1095\">1.3<\/span><\/a> <span class=\"ecti-1095\">schliessen kann ohne zuerst im Besitz der richtigen Formel zu sein. Wir schreiben dazu f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-35001r12\" \/><mstyle class=\"maketag\"><mtext>(1.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Gehen Sie nun wie folgt vor.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mi>a<\/mi><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie alle dabei verwendenten Rechenregeln benennen?<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifizieren Sie die Gleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf Lemma <\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4006r3\"><span class=\"ecti-1095\">1.3<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) verwenden Sie am besten (i) f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">setzen dies rechts in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-35001r12\">1.12<\/a>) <span class=\"ecti-1095\">ein. Verwenden Sie anschliessend die einfachere Summenformel in<\/span> <span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/beweise#x1-25001r85\"><span class=\"ecti-1095\">1.85<\/span><\/a> <span class=\"ecti-1095\">und eine elementare Gleichungsumformung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii).<\/span><\/p><\/details>  <\/div> <a id=\"x1-35005r35\"><\/a> <h4 id=\"z584dee1d4154\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.3 <\/span> <a id=\"x1-360003\"><\/a>Logik<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfec3a47e13de\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Draculas B\u00fccher)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In der Bibliothek des Grafen Dracula gibt es keine zwei B<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cher, deren Inhalt aus gleich<\/span> <span class=\"ecti-1095\">vielen W<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rtern besteht. Die Anzahl der B<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cher ist die Summe der Anzahl der W<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rter jedes<\/span> <span class=\"ecti-1095\">einzelnen Buches. Des Weiteren gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gen diese Aussagen, um den Inhalt mindestens eines<\/span> <span class=\"ecti-1095\">Buches aus Draculas Bibliothek genau zu beschreiben. Was steht in diesem Buch? Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span> <span class=\"ecti-1095\">enstammt dem Buch <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#Xamann-escher\"><span class=\"ecti-1095\">AE06<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd38469fb6ca7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Vier Aussagen in Pr\u00e4dikatenlogik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir sagen <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">falls<\/span> <span class=\"ecti-1095\">es ein <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gibt<\/span> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>d<\/mi><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Beschreiben Sie die Bedeutung folgender Aussagen und bestimmen Sie, ob diese zutreffen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><\/math> <span class=\"ecti-1095\">teilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z94554da566b7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Allgemeinere Existenzquantoren)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir Quantoren f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Aussage, dass mehrere Elemente mit einer<\/span> <span class=\"ecti-1095\">Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">existieren, definieren.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie unter Verwendung des Existenzquantors einen neuen Quantor <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass die Aussage<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"maendquote\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">bedeutet, dass es mindestens zwei Elemente <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">haben.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verallgemeinern Sie (i) zu dem Quantor <\/span><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und verwenden Sie diese um auch den Quantor <\/span><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zu definieren, der besagen soll, dass es genau <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">besitzen. (Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen entweder informell Punkte verwenden, oder formal korrekter den<\/span> <span class=\"ecti-1095\">Funktionsbegriff, Eigenschaften von Funktionen, und die Menge <\/span><span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die genau <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Elemente hat.)<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie unter Verwendung des Existenzquantors, des Funktionsbegriffes, einer<\/span> <span class=\"ecti-1095\">Eigenschaft von Funktionen und der nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">einen neuen Quantor <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">der besagt, dass es unendlich viele Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gibt, die die Eigenschaft <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">besitzen.<\/span><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><msup><mrow><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> <span class=\"ecti-1095\">kann durch <\/span><math display=\"inline\"><mstyle class=\"text\"><mtext>\u201e<\/mtext><\/mstyle><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mstyle class=\"text\"><mtext>\u201c<\/mtext><\/mstyle><\/math> <span class=\"ecti-1095\">definiert werden. <\/span><\/p><\/details>  <\/div> <a id=\"x1-36004r36\"><\/a> <h4 id=\"z5420a98d47bd\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.4   <\/span> <a id=\"x1-370004\"><\/a>Funktionen und Relationen<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5cfd4a37e7be\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Injektive Funktionen durch Fallunterscheidungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"> <span class=\"ecti-1095\">Zeigen Sie folgende Behauptung: Seien <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen und sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">eine Partition von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen es ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">eine injektive Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>P<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">gegeben und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">die eindeutige Funktion mit <\/span><math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>P<\/mi> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">nach Lemma <\/span><a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-16002r52\"><span class=\"ecti-1095\">1.52<\/span><\/a><span class=\"ecti-1095\">. Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann injektiv ist, falls die Mengen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>P<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">paarweise disjunkt sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z25a3c462b0c3\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine \u00c4quivalenzrelation auf dem kartesischen Produkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei nicht-leere<\/span> <span class=\"ecti-1095\">Mengen, sei <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Relation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Relation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Wir definieren<\/span> <span class=\"ecti-1095\">damit eine Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow> <mi>X<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow> <mi>Y<\/mi> <\/mrow><\/msub><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Zeigen Sie, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">genau dann eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist, wenn <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist. Gilt dies auch, wenn<\/span> <span class=\"ecti-1095\">eine der beiden Mengen <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">leer ist? Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung enstammt dem Buch <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#Xamann-escher\"><span class=\"ecti-1095\">AE06<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span><span class=\"ecti-1095\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen Sie die drei definierenden Eigenschaften einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">(beziehungsweise<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>X<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5b028a988603\"> <span class=\"ecbx-1095\">Applet <\/span>(Nichtvertauschbarkeit der Verkn\u00fcpfung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"450px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/tcdwejfk\/width\/688\/height\/450\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten zwei Funktionen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nur durch den Graphen beschrieben ist und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">eine affine Funktion ist, die durch zwei Konstanten <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert wird. Durch Bewegen zweier Punkte am Graph von <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">lassen sich <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">definieren. Experimentieren Sie damit um sich an die geometrische Bedeutung von den Zahlen<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">zu errinnern, und beobachten Sie, wie sich die beiden Funktionen <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">im rechten Fenster unterschiedlich ver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndern. Es ist n<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">tzlich sich diese Ph<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nomene vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndig<\/span> <span class=\"ecti-1095\">zu erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, denn wir werden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlichen Verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfungen in unseren weiteren <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berlegungen<\/span> <span class=\"ecti-1095\">begegnen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1dcdda3c2f13\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Konstruktion der Menge der ganzen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen an, dass wir bereits die Menge der nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und damit auch die Menge der nicht-negativen ganzen Zahlen<\/span> <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u2294<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">mit allen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen<\/span> <span class=\"ecti-1095\">Operationen und Eigenschaften kennen und wollen daraus die ganzen Zahlen definieren. Dazu betrachten wir<\/span> <span class=\"ecti-1095\">eine Relation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math><\/span><span class=\"period\">:<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-37001r13\" \/><mstyle class=\"maketag\"><mtext>(1.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie unter der Annahme, dass die ganzen Zahlen schon bekannt sind, wieso wir<\/span> <span class=\"ecti-1095\">die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-37001r13\">1.13<\/a>)           <span class=\"ecti-1095\">betrachten wollen. <\/span><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Motiviert ist das ganze dadurch, dass man eine<\/span> <span class=\"ecti-1095\">ganze Zahl als Differenz von zwei nicht-negativen ganzen Zahlen auffassen kann. Diese<\/span> <span class=\"ecti-1095\">sind aber nicht eindeutig gegeben; deswegen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt man auf Tupeln von nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen<\/span> <span class=\"ecti-1095\">Zahlen obige <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ein. In der Tat hat man bei Definition<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-37001r13\">1.13<\/a>) <span class=\"ecti-1095\">eigentlich<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">im Hinterkopf, darf dies aber formal nicht verwenden, da die Subtraktion (noch) nicht<\/span> <span class=\"ecti-1095\">erlaubt ist. Wenn wir <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">sprechen, werden wir leicht schizophren an <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">denken, dies aber nicht verwenden.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Informell k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir uns das Tupel <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">als einen Vektor mit Anfangspunkt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und Endpunkt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">vorstellen, womit die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen allen Vektoren mit gleicher L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge und Richtung<\/span> <span class=\"ecti-1095\">entspricht. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Kontostand vor und <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Kontostand nach einer Transaktion darstellen und die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse entspricht<\/span> <span class=\"ecti-1095\">dann dem Nettogewinn\/-verlust.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definiert. <\/span><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Es gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/math> <\/p><div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Symmetrie: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">denn <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">impliziert <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und daher auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">impliziert <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Durch Summieren dieser Gleichungen ergibt sich<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und durch Wegstreichen von <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(was eine der Eigenschaften von <\/span><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist und nicht die Subtraktion verwendet) erhalten wir<\/span> <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><span class=\"ecti-1095\">, also<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/div><\/div> <div class=\"wp-nocaption \"><\/div><\/details><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Der Quotient <\/span><math display=\"inline\"><msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math><span class=\"ecti-1095\">kann<\/span> <span class=\"ecti-1095\">als Definition von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">angesehen werden, wobei wir die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse auch als<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">schreiben. Insbesondere<\/span> <span class=\"ecti-1095\">identifizieren wir <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">schreiben <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch als <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folgende <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bungen erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wieso dies Sinn ergibt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildungen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">injektiv sind und disjunkte Bilder <\/span><math display=\"inline\"><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">haben, welche wir mit <\/span><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bezeichnen werden.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2294<\/mo> <msub><mrow><mi>\u03b9<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-37006r37\"><\/a> <h4 id=\"z45b65798acf8\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.5 <\/span> <a id=\"x1-380005\"><\/a>Beweismethoden<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z79d8878cd3cd\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl. Zeigen Sie, dass die Implikation<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mn>7<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mn>3<\/mn><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;gerade&nbsp;<\/mtext><\/mstyle><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;ungerade<\/mtext><\/mstyle><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc986e192943d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(T\u00fcrme von Hanoi)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es seien <\/span><math display=\"inline\"><mn>3<\/mn><\/math> <span class=\"ecti-1095\">Ablagefl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chen gegeben. Angenommen auf der linken Ablagefl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che seien<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Bl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">cke f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine<\/span> <span class=\"ecti-1095\">nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">aufget<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rmt, wobei nie ein kleinerer Block unter einem gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sseren Block liegt.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=455&amp;url=Pictures\/Einfuehrung\/Induktionsbeispiele\/Hanoi\/hanoi1.pdf\"><img decoding=\"async\" id=\"z670cd8bf7fde\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/Induktionsbeispiele\/Hanoi\/hanoi1.svg\" width=\"455\" \/><\/div>  <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass Sie den Turm von links nach rechts umschichten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen, ohne dass je ein kleinerer<\/span> <span class=\"ecti-1095\">Block unter einem gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sseren Block zu liegen kommt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za16d780107af\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie mittels vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahlen <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>5<\/mn><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>4<\/mn><mi>n<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Beginnen     Sie     die     Induktion     mit     dem     Induktionsanfang     bei<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>5<\/mn><\/math> <span class=\"ecti-1095\">(denn                                                                                                      f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><\/math> <span class=\"ecti-1095\">gilt die Aussage nicht).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbe5557817843\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie     mittels     vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger     Induktion,     dass     die     Ungleichung<\/span> <math display=\"inline\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>3<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                           alle                           nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen                           Zahlen<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gilt. Beschreiben Sie die Variante des Induktionsbeweises, die sie hier verwenden.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Der       Beweis       des       Induktionsschrittes       ist       einfacher       falls<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">angenommen     werden     kann.     Aus     diesem     Grund     ist     es     besser     sowohl<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">als                                                                                                         auch<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">direkt            zu            <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen            um            im            Induktionsschritt<\/span> <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">dann                ohne                Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung                der                Allgemeinheit<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">zu verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3fce48ed5461\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir               f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rben               jeden               Punkt               im               Gitter<\/span> <math display=\"inline\"><msup><mrow><mi>\u2124<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit                                                  einer                                                  von<\/span> <math display=\"inline\"><mn>1<\/mn><mn>7<\/mn><\/math> <span class=\"ecti-1095\">verschiedenen   Farben   ein.   Zeigen   Sie,   dass   es   ein   achsenparalleles   Rechteck<\/span> <math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">in diesem Gitter gibt, dessen Ecken alle dieselbe Farbe besitzen.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Statt das ganze Gitter zu untersuchen, gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt es die Gitterpunkte in dem Rechteck<\/span> <math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mn>7<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">zu betrachten. Es gibt dann auf einer horizontalen Gerade mit ganzzahliger <\/span><math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">in dem Rechteck <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">jeweils eines von <\/span><math display=\"inline\"><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichen Farbmuster. Da es mehr horizontale Geraden (genau <\/span><math display=\"inline\"><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>1<\/mn><mn>8<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">als m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">gliche Farbmuster gibt, wiederholt sich eines der Farbmuster. Wir w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen diese beiden<\/span> <math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinaten<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r unser gesuchtes Rechteck <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In diesem Farbmuster gibt es horizontal <\/span><math display=\"inline\"><mn>1<\/mn><mn>8<\/mn><\/math> <span class=\"ecti-1095\">Positionen aber nur <\/span><math display=\"inline\"><mn>1<\/mn><mn>7<\/mn><\/math> <span class=\"ecti-1095\">Farben, womit sich eine der Farben wiederholt. Verwendet man diese <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Koordinaten,<\/span> <span class=\"ecti-1095\">so erhalten wir das gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Rechteck <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">bei dem alle Eckpunkte dieselbe Farbe besitzen. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0dc9d7a67ade\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><mi>S<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge von <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">mit Kardinalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass es Elemente <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>S<\/mi><\/math> <span class=\"ecti-1095\">gibt mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Es                                        empfiehlt                                        sich<\/span> <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">in Teilmengen zu partitionieren, die geometrische Progressionen darstellen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2ab8d4666c64\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>T<\/mi><\/math> <span class=\"ecti-1095\">eine endliche<\/span> <span class=\"ecti-1095\">Menge und seien <\/span><math display=\"inline\"><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">Teilmengen von <\/span><math display=\"inline\"><mi>T<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>k<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>T<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass es ein Element <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>T<\/mi><\/math> <span class=\"ecti-1095\">gibt, welches in mindestens <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">der Mengen <\/span><math display=\"inline\"><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">liegt.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen Sie indirekt an, dass jeder Punkt<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>T<\/mi><\/math> <span class=\"ecti-1095\">in h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens<\/span> <math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen enthalten<\/span> <span class=\"ecti-1095\">ist und zeigen Sie, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>T<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt dies in Worten oder geometrisch zu erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren, wir werden die Notation f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">formalen Beweis erst sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/p><\/details>  <\/div> <a id=\"x1-38001r38\"><\/a> <h4 id=\"z919b7a662800\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.6 <\/span> <a id=\"x1-390006\"><\/a>Geometrische Probleme<\/h4> <p class=\"noindent\">Wir empfehlen Ihnen folgende \u00dcbung zu l\u00f6sen, da der \u201eSatz von Pythagoras\u201c f\u00fcr uns sp\u00e4ter gewissermassen zu einer Definition werden wird. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf4eeeeac81f4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Satz von Pythagoras)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten ein rechtwinkliges Dreieck mit Katheten der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">und Hypotenuse<\/span> <span class=\"ecti-1095\">der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>c<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Hinweis: Betrachten Sie folgende Bilder<\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"329px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/XJ7K6R8e\/width\/688\/height\/329\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <span class=\"ecti-1095\">und gehen Sie dabei davon aus, dass gewisse<\/span> <span class=\"ecti-1095\">geometrische Begriffe wie Winkel, L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge und Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r elementare Bereiche und intuitiv<\/span> <span class=\"ecti-1095\">anschauliche Eigenschaften wie Invarianz der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter Verschiebung und Drehung bekannt<\/span> <span class=\"ecti-1095\">sind.<\/span> <\/p> <\/div> <a id=\"x1-39001r39\"><\/a> <h4 id=\"z4c0b4ec12664\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.7 <\/span> <a id=\"x1-400007\"><\/a>\u00dcbungen zu Primzahlen<\/h4> <p class=\"noindent\">Eine nat\u00fcrliche Zahl <math display=\"inline\"><mi>p<\/mi><\/math> gr\u00f6sser als <math display=\"inline\"><mn>1<\/mn><\/math> ist <span class=\"ecbx-1095\">irreduzibel<\/span>, falls sie nicht als Produkt von zwei kleineren nat\u00fcrlichen Zahlen geschrieben werden kann. Eine nat\u00fcrliche Zahl <math display=\"inline\"><mi>p<\/mi><\/math> gr\u00f6sser als <math display=\"inline\"><mn>1<\/mn><\/math> heisst eine <span class=\"ecbx-1095\">Primzahl<\/span>, falls ein Produkt <math display=\"inline\"><mi>a<\/mi><mi>b<\/mi><\/math> zweier nat\u00fcrlicher Zahlen <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> nur dann durch <math display=\"inline\"><mi>p<\/mi><\/math> teilbar ist, falls eine der beiden Zahlen durch <math display=\"inline\"><mi>p<\/mi><\/math> teilbar ist.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z266a3912d113\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Primzahlen sind irreduzibel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen               Sie,               dass               jede               Primzahl               in<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">auch irreduzibel ist.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich eine Primzahl <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">als Produkt zweier nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">schreiben (das heisst, <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><mi>b<\/mi><\/math><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">so teilt <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">das Produkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Diese beiden Begriffe sind in der Tat f\u00fcr die nat\u00fcrlichen Zahlen \u00e4quivalent (wir werden dies nochmals etwas genauer in Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-540004\">2.2.4<\/a> besprechen) und wir werden in diesem Abschnitt irreduzible Zahlen ebenso als Primzahlen bezeichnen. Es ist eine gute \u00dcbung im Folgenden genau zu erkl\u00e4ren welche der beiden Begriffe eigentlich verwendet wird. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z04c3af9b608b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Primfaktorzerlegung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie  mittels  vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger  Induktion,  dass  jede  nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche  Zahl  gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sser  als<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">als Produkt von Primzahlen geschrieben werden kann.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen hier eine Variante der vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigen Induktion verwenden und im<\/span> <span class=\"ecti-1095\">Induktionsschritt  annehmen,  dass  die  Aussage  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  alle  kleineren  Zahlen  bereits  bewiesen<\/span> <span class=\"ecti-1095\">wurde. Das einfachste Argument hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r verwendet eigentlich den Begriff der irreduziblen<\/span> <span class=\"ecti-1095\">Zahlen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za3f46d6beccc\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Unendlich viele Primzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass es unendliche viele Primzahlen gibt.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Das   einfachste   Argument   hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r   ist   das   Argument   von   Euklid:   Falls<\/span> <math display=\"inline\"><msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">die einzigen             Primzahlen             w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren,             dann             w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u22ef<\/mo> <msub><mrow><mi>p<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">durch keine dieser  Primzahlen  teilbar  und  mittels  der  letzten  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung  f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nde  man  weitere<\/span> <span class=\"ecti-1095\">Primzahlen (genauer gesagt irreduzible Zahlen).<\/span><\/p><\/details>  <\/div> <a id=\"x1-40001r40\"><\/a> <h4 id=\"z3c9c03c2b21c\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.8 <\/span> <a id=\"x1-410008\"><\/a>Online Lernhilfen<\/h4> <p class=\"noindent\">Bei erster Verwendung dieses Skripts in der entsprechenden Vorlesung hat ein Student mehrere online-Tools zum Erlernen der Inhalte des Skripts programmiert. Leider haben sich aber die Inhalte seitdem etwas ver\u00e4ndert, ohne dass die Inhalte in den online-Tools angepasst wurden. Wir erw\u00e4hnen hier die <a href=\"https:\/\/janiks.me\/projects\/eth\/ana\/study\" target=\"_blank\" rel=\"noopener\">Webseite<\/a> einmalig. Sollte jemand dieses App aktualisieren oder ein alternatives App mit aktuellen Daten zur Verf\u00fcgung stellen wollen, so werden wir dieses gerne auch wieder am Ende von jedem Kapitel bewerben. <a id=\"x1-41001r41\"><\/a> <\/p> <h4 id=\"zd3cb8a14d53b\" class=\"subsectionHead\"><span class=\"titlemark\">1.9.9 <\/span> <a id=\"x1-420009\"><\/a>SageMath<\/h4> <p class=\"noindent\">Schlussendlich wollen wir die auf Python basierende Programmiersprache SageMath erw\u00e4hnen. Diese ist f\u00fcr mathematische Experimente bestens geeignet und kann auch ohne einer aufwendigen Installation mittels <a href=\"https:\/\/sagecell.sagemath.org\" target=\"_blank\" rel=\"noopener\">SageMathCell<\/a> ben\u00fctzt werden. F\u00fcr aufwendigere oder auch rechenintensivere Programme empfiehlt sich allerdings eine lokale Installation von SageMath. <\/p><p class=\"indent\">Versuchen Sie doch folgende Zeilen in SageMath aus und experimentieren Sie etwas damit. <\/p> <div class=\"lstlisting\" id=\"listing-1\"><span class=\"label\"><a id=\"x1-42001r1\"><\/a><\/span><span class=\"ectt-1095\">table<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[[&#171;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&#171;,&#187;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&#171;,&#187;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=&gt;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&#171;]]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42002r2\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">+<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[[<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">,(<\/span><span class=\"ectt-1095\">not<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">or<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[<\/span><span class=\"ectt-1095\">true<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">false<\/span><span class=\"ectt-1095\">]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42003r3\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[<\/span><span class=\"ectt-1095\">true<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">false<\/span><span class=\"ectt-1095\">]]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Wir bemerken, dass einzelne Befehle normalerweise innerhalb einer Zeile stehen sollten, dass aber bei einer vorhanden offenen Klammer die anschliessende(n) Zeile(n) als Teil der ersten Zeile aufgefasst werden. Der Ausdruck in der ersten Zeile beschriftet die Kopfzeile und mit den beiden \u2019for\u2019-Konstruktionen werden alle M\u00f6glichkeiten durchgetestet. Insgesamt wird durch den Befehl <span class=\"ecbx-1095\">table <\/span>hier die Wahrheitstabelle der Implikation dargestellt. \u00c4ndern Sie obiges Beispiel und versuchen Sie zum Beispiel damit eine der Tautologie aus Abschnitt <a href=\"..\/..\/chapter\/logische-begriffe#x1-70001\">1.3.1<\/a> zu \u00fcberpr\u00fcfen. <\/p><p class=\"indent\">Hier einige Zeilen, die zeigen, wie wir in SageMath mit Mengen operieren k\u00f6nnen. Bei mehreren Befehlen hintereinander, die alle ein Ergebnis darstellen sollten, m\u00fcssen sie den Befehl <span class=\"ecbx-1095\">print<\/span> verwenden. <\/p>  <div class=\"lstlisting\" id=\"listing-2\"><span class=\"label\"><a id=\"x1-42004r1\"><\/a><\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&#171;<\/span><span class=\"ectt-1095\">abcdabcd<\/span><span class=\"ectt-1095\">&#171;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42005r2\"><\/a><\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[0,&#187;<\/span><span class=\"ectt-1095\">a<\/span><span class=\"ectt-1095\">&#171;]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42006r3\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=&#187;,<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,&#187;<\/span><span class=\"ectt-1095\">und<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=&#187;,<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42007r4\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Durchschnitt<\/span><span class=\"ectt-1095\">:&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">.<\/span><span class=\"ectt-1095\">intersection<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42008r5\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Vereingigung<\/span><span class=\"ectt-1095\">:&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">+<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42009r6\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42010r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42011r8\"><\/a><\/span><span class=\"ectt-1095\">def<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">Y<\/span><span class=\"ectt-1095\">):<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42012r9\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">return<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Set<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42013r10\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42014r11\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">:&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Produktmenge<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">A<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">B<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Zur Definition neuer Befehle mittels der obigen <span class=\"ecbx-1095\">def<\/span>-Konstruktion sollte man bemerken, dass die konsistente Einr\u00fcckung der folgenden Zeilen wichtig ist und man zur Erh\u00f6hung der \u00dcbersicht die Definition mit einer Leerzeile beenden sollte. <\/p><p class=\"indent\">Die folgende Routine testet die Injektivit\u00e4t der Einschr\u00e4nkung einer Funktion <math display=\"inline\"><mi>f<\/mi><\/math> auf einer endlichen Menge <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dazu wollen wir bemerken, dass <span class=\"ecbx-1095\">all <\/span>der Allquantor und <span class=\"ecbx-1095\">any <\/span>der Existenzquantor in SageMath ist. Des Weiteren, sollten Sie wissen, dass <span class=\"ecbx-1095\">= <\/span>wie in obigem Beispiel eine Anweisung ist, die einer Variable einen Wert zuweist, aber <span class=\"ecbx-1095\">== <\/span>die Frage nach Gleichheit und <span class=\"ecbx-1095\">!= <\/span>die Frage nach Ungleichheit darstellt. <\/p>  <div class=\"lstlisting\" id=\"listing-3\"><span class=\"label\"><a id=\"x1-42015r1\"><\/a><\/span><span class=\"ectt-1095\">def<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">TestInj<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">):<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42016r2\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">return<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">all<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">==<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">or<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">)!=<\/span><span class=\"ectt-1095\">F<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">))<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42017r3\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">x<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">for<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">y<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42018r4\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42019r5\"><\/a><\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">[0..10]<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">#<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Liste<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">der<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">ganzen<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahlen<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">von<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">0<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">bis<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">10<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42020r6\"><\/a><\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">=<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">^2-2*<\/span><span class=\"ectt-1095\">n<\/span><span class=\"ectt-1095\">+1<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">#<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">die<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Funktion<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42021r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eingeschraenkt<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">auf<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">injektiv<\/span><span class=\"ectt-1095\">?&#187;)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42022r8\"><\/a><\/span><span class=\"ectt-1095\">TestInj<\/span><span class=\"ectt-1095\">(<\/span><span class=\"ectt-1095\">g<\/span><span class=\"ectt-1095\">,<\/span><span class=\"ectt-1095\">X<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Schreiben Sie anschliessend eine Routine <span class=\"ecti-1095\">TestWohl(F,X,Y)<\/span>, die \u00fcberpr\u00fcft, ob <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> gilt. Schreiben Sie eine Routine <span class=\"ecti-1095\">TestSurj(F,X,Y)<\/span>, die \u00fcberpr\u00fcft, ob <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <\/math> gilt. Schreiben Sie schlussendlich eine Routine <span class=\"ecti-1095\">TestGraph(G,X,Y)<\/span>, die f\u00fcr eine Menge <math display=\"inline\"><mi>G<\/mi><\/math> \u00fcberpr\u00fcft ob <math display=\"inline\"><mi>G<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-close\">)<\/mo><\/math> der Graph einer Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> ist. <\/p><p class=\"indent\">Folgende Zeilen sollten auch zeigen, warum wir Ihnen SageMath als Programmiersprache f\u00fcr mathematische Experimente empfehlen. <\/p> <div class=\"lstlisting\" id=\"listing-4\"><span class=\"label\"><a id=\"x1-42023r1\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Wofuer<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">steht<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42024r2\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42025r3\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2^2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2^2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42026r4\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">die<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Wurzel<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">aus<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">2<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rational<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">sqrt<\/span><span class=\"ectt-1095\">(2)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42027r5\"><\/a><\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42028r6\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Wofuer<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">steht<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42029r7\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">reelle<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">RR<\/span><span class=\"ectt-1095\">)<\/span><span class=\"ectt-1095\">&nbsp;<\/span><br \/><span class=\"label\"><a id=\"x1-42030r8\"><\/a><\/span><span class=\"ectt-1095\">print<\/span><span class=\"ectt-1095\">(&#171;<\/span><span class=\"ectt-1095\">Ist<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">eine<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">rationale<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">Zahl<\/span><span class=\"ectt-1095\">?&#187;,<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">pi<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">in<\/span><span class=\"ectt-1095\">&nbsp;<\/span><span class=\"ectt-1095\">QQ<\/span><span class=\"ectt-1095\">)<\/span> <\/div> <p class=\"indent\">Wir \u00fcberlassen Ihnen die entsprechenden Internetnachforschungen, falls Sie mehr \u00fcber SageMath wissen wollen.                                                                                                                                                                                                                                                                                                                                                     <\/p> 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