{"id":40,"date":"2021-12-15T09:53:02","date_gmt":"2021-12-15T09:53:02","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-2\/"},"modified":"2021-12-15T09:53:02","modified_gmt":"2021-12-15T09:53:02","slug":"weitere-lernmaterialien-2","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-2\/","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=\"z4022e73eefa1\" class=\"sectionHead\"><span class=\"titlemark\">2.7 <\/span> <a id=\"x1-730007\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-73001r72\"><\/a> <h4 id=\"z49e881b75709\" class=\"subsectionHead\"><span class=\"titlemark\">2.7.1 <\/span> <a id=\"x1-740001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Die Themen dieses Kapitels stellen den Anfang unserer Entwicklung der Analysis dar und sind aus diesem Grunde f\u00fcr das Folgende fundamental. Wie bereits erw\u00e4hnt werden wir die \u00fcblichen Eigenschaften der reellen, nat\u00fcrlichen, ganzen, rationalen und komplexen Zahlen (inklusive der Konjugation komplexer Zahlen) im Folgenden ohne Verweise verwenden. Es ist auch nicht notwendig, die Beweise der elementaren Aussagen in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-440001\">2.1<\/a> auswendig zu lernen. Manche der Beweise in Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-500002\">2.2<\/a> sind auch etwas zu formal, als dass sie f\u00fcr das Folgende von grosser Bedeutung sein werden. F\u00fcr ein fundiertes Verst\u00e4ndnis der Induktion sind die besprochenen Varianten der Induktion samt Beweise wichtig und auch die Beweise der algebraischen und geometrischen Aussagen stellen eine gute \u00dcbung dar. In Abschnitt <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-580004\">2.4<\/a> haben wir einige Ihnen wahrscheinlich bekannte Definition ausgesprochen, doch werden auch die Ihnen wahrscheinlich neuen Begriffe \u201eoffen\u201c und \u201eabgeschlossen\u201c zunehmend an Bedeutung gewinnen. <\/p><p class=\"indent\">Die Kernthemen dieses Kapitels sind hingegen in folgender Liste enthalten. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Vollst\u00e4ndigkeitsaxiom in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Existenz und Eigenschaften des Supremums und Infimums in Abschnitt <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-620005\">2.5<\/a> (inbesondere beispielsweise die Unterscheidung von Maximum und Supremum). <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Korollare der Vollst\u00e4ndigkeit in Abschnitt <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-670006\">2.6<\/a>: Das Archimedische Prinzip (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\">2.68<\/a>), die Existenz von H\u00e4ufungspunkten f\u00fcr beschr\u00e4nkte unendliche Mengen (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\">2.75<\/a>), das Intervallschachtelungsprinzip (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70001r77\">2.77<\/a>), und die \u00dcberabz\u00e4hlbarkeit von <math display=\"inline\"><mi>\u211d<\/mi><\/math> in Korollar&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-71001r81\">2.81<\/a>.<\/div><\/div> <p class=\"noindent\">Diese Themen und deren Beweismethoden sind von zentraler Bedeutung f\u00fcr das Folgende und Sie werden weitere Vorlesungsstunden besser verstehen, wenn Sie diese Kernthemen bereits im Ged\u00e4chnis und auf Abruf bereit haben. <\/p><p class=\"indent\">Im Laufe dieses Kapitels haben wir auch bereits einige grundlegende Funktionen eingef\u00fchrt, welche wir ohne Verweis und mit den \u00fcblichen Eigenschaften in Zukunft wieder ben\u00f6tigen werden. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die K\u00f6rperoperationen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">:<\/span> Addition, Subtraktion, Multiplikation, Division. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Quadrieren <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Quadratwurzel <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/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\">Der Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Vorzeichenfunktion <math display=\"inline\"><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der ganzzahlige Anteil <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">\u230a<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">\u230b<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der Nachkommaanteil <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/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> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Maximum <math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><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\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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> und das Minimum <math display=\"inline\"><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><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\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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> zweier reeller Zahlen <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ergeben sich als Spezialf\u00e4lle von Maximum und Minimum der Menge <math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> (welche auf Grund einer Fallunterscheidung basierend auf die Trichotomie reeller Zahlen immer existieren).<\/div><\/div> <p class=\"indent\">Sollten Sie noch nicht mit dem Anlegen einer pers\u00f6nlichen Zusammenfassung aller wichtigen Inhalte der Vorlesung begonnen haben, dann legen wir Ihnen nahe dies jetzt in Angriff zu nehmen. Die Inhalte aus Kapitel <a href=\"..\/..\/part\/einfuehrung#x1-30001\">1<\/a> sollten schnell wiederholt und zusammengefasst sein. Doch in diesem Kapitel haben wir bereits unsere ersten grundlegenden S\u00e4tze der reellen Analysis und deren Beweise kennengelernt. Deswegen wird eine pers\u00f6nlich erstellte Zusammenfassung nun wahrscheinlich schon einige Seiten lang sein. Welche Form und Detailreiche eine derartige Zusammenfassung oder Mindmap haben sollte, ist Geschmackssache und Ihnen \u00fcberlassen. Zum Beispiel k\u00f6nnte f\u00fcr den Beweis der Existenz eines H\u00e4ufungspunktes einer beschr\u00e4nkten unendlichen Menge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> (Satz <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\">2.75<\/a>) folgende Zusammenfassung aussreichen: \u201eWir definieren <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und zeigen, dass <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>X<\/mi><\/math> ein H\u00e4ufungspunkt der Menge <math display=\"inline\"><mi>A<\/mi><\/math> ist.\u201c Vielleicht reicht Ihnen dies bereits als Anfangspunkt um den Beweis zu vervollst\u00e4ndigen, oder Sie erg\u00e4nzen die Zusammenfassung noch um ein bis zwei S\u00e4tze.                                                                                                                                                                           <\/p><p class=\"indent\">Wir stellen nochmals einige Multiple-Choice-Fragen, die Ihnen zur Wiederholung des Kapitels helfen sollten. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z62d7c53cc1b5\"> <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>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sind die folgenden Aussagen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zur Aussage, dass<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist?<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">!<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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\">In (i) wird eine eindeutige Existenz eines Punktes nahe an<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">verlangt. Aber f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">einer<\/span> <span class=\"ecti-1095\">Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">gibt es in der<\/span> <span class=\"ecti-1095\">Tat f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">sogar unendlich<\/span> <span class=\"ecti-1095\">viele Punkte von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die zur <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">-Umgebung<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (ii) ist genau die Formulierung der Definition eines H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunktes in<\/span> <span class=\"ecti-1095\">Pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">dikatenlogik.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In (iii) schr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nken wir die Definition auf alle<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gend kleinen<\/span><span class=\"ecti-1095\">\u201c<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein. Dies ist zur Definition<\/span> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent. Denn falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir die eingeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Behauptung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">anwenden<\/span> <span class=\"ecti-1095\">und ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">finden. Also <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert diese Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung die Bedeutung der Aussage nicht.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung auf alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert allerdings den Begriff auf drastische Weise. In der Tat hat zum Beispiel<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">keinen einzigen H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt,<\/span> <span class=\"ecti-1095\">aber jedes beliebige <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">die Aussage in (iv).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za0cc5e3a5743\"> <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>X<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Menge mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>X<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist<\/span><span class=\"ecti-1095\">\u2026<\/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;<span class=\"ecti-1095\">\u2026eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026eine lineare Ordnungsrelation.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026eine Ordnungsrelation, die nicht linear ist.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026keins der Obigen.<\/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\">Die Aussage (i) ist falsch. Da <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">nichtleer ist, gelten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Relationen <\/span><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-rel\">\u2284<\/mo> <mi>\u2205<\/mi><\/math><span class=\"ecti-1095\">. Also<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">nicht symmetrisch.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (ii) ist nicht richtig. Seien <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mi>y<\/mi><\/math><span class=\"ecti-1095\">. Diese Elemente<\/span> <span class=\"ecti-1095\">existieren, da <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>X<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt weder <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2286<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">noch <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">ist eine Ordnungsrelation, denn sie ist reflexiv, da 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> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">stets<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt; sie ist<\/span> <span class=\"ecti-1095\">transitiv, da 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-punc\">,<\/mo><mi>C<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">aus <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">auch<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">folgt; und sie ist<\/span> <span class=\"ecti-1095\">antisymmetrisch, da 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 mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">schon <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">gilt. Sie ist nicht linear nach (ii), also ist (iii) richtig und somit muss (iv) falsch sein.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z66126eede26d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sind die folgenden Mengen (mit der <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Addition und Multiplikation) Beispiele f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, die angeordnet werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen?<\/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; <math display=\"inline\"><mi>\u2115<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u2124<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u211a<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u211d<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u2102<\/mi><\/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\">Die Menge <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist kein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, da zum Beispiel <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">kein<\/span> <span class=\"ecti-1095\">additives Inverses besitzt (da <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">\u2209<\/mo><mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist kein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper.<\/span> <span class=\"ecti-1095\">Beispielsweise hat <\/span><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">kein<\/span> <span class=\"ecti-1095\">multiplikatives Inverse in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Sowohl <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als auch <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">sind angeordnete K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper. Wir verweisen dazu auf die Abschnitte <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-460002\"><span class=\"ecti-1095\">2.1.2<\/span><\/a> <span class=\"ecti-1095\">und <\/span><a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-530003\"><span class=\"ecti-1095\">2.2.3<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In jedem angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper gelten <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung (s)), also <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung (q)). Weiters ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung<\/span> <span class=\"ecti-1095\">(r)). In <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt aber<\/span> <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">. Dies erg<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be nun einen<\/span> <span class=\"ecti-1095\">Widerspruch, wenn <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit einer geeigneten Ordnung zu einem angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper gemacht werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z884ec46357e4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es bezeichne <\/span><math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die imagin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Einheit. Welche der folgenden Formeln sind richtig?<\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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\">Durch Ausrechnen erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt man die richtigen L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sungen. Dabei ist es hilfreich f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i)-(ii) zuerst<\/span> <math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">(<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">(iii)-(iv) zuerst <\/span><math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">(<\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auszurechnen. SageMath kann dies nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlich auch sehr schnell berechnen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z926f4d336508\"> <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>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">nichtleere, nach oben<\/span> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Welche der folgenden Aussagen gelten im Allgemeinen?<\/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;<span class=\"ecti-1095\">Gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so folgt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mi>B<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Existiert das Maximum der Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so existiert das Maximum von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/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\">Die erste Aussage ist richtig, denn aufgrund der Inklusion<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math> <span class=\"ecti-1095\">eine obere Schranke<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math><span class=\"ecti-1095\">. Das Supremum<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist als kleinste<\/span> <span class=\"ecti-1095\">obere Schranke von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">somit h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu (ii) finden wir ein Gegenbeispiel: Seien <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/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>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">es gibt 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> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">kein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) verweisen wir auf Proposition <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\"><span class=\"ecti-1095\">2.63<\/span><\/a> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die vierte Aussage ist falsch. Ein Gegenbeispiel:<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/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\">. Dann gilt<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">. Die Aussage<\/span> <span class=\"ecti-1095\">gilt aber, wenn <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-65003r66\"><span class=\"ecti-1095\">2.66<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu (v): Das Maximum <\/span><math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist, wenn<\/span> <span class=\"ecti-1095\">es existiert, eine obere Schranke von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und es kann keine kleinere obere Schranke geben, da nach Definition eines Maximums<\/span> <math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt. Also ist<\/span> <math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi><\/math> <span class=\"ecti-1095\">die kleinste obere<\/span> <span class=\"ecti-1095\">Schranke von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und dies ist die definierende Eigenschaft des Supremums.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (vi) ist richtig. Da das Supremum von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">per Definition eine obere<\/span> <span class=\"ecti-1095\">Schranke von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist, erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <span class=\"ecti-1095\">es im Fall <\/span><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">die definierende<\/span> <span class=\"ecti-1095\">Eigenschaft eines Maximums von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-74024r74\"><\/a> <h4 id=\"za4df50aedd3e\" class=\"subsectionHead\"><span class=\"titlemark\">2.7.2 <\/span> <a id=\"x1-750002\"><\/a>Weitere \u00dcbungsaufgaben<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z529d659527c6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Parallelogrammidentit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Gleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/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> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7c1c9205e48e\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Mittelsenkrechte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">zwei verschiedene Punkte. Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren und beweisen Sie, wieso die Teilmenge <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine Gerade ist. Eine Gerade ist dabei eine Teilmenge der Form <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>t<\/mi><mi>v<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/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>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7f7baf97d9d6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(K\u00f6rper mit zwei Elementen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                    Sie,                    dass                    die                    Menge<\/span> <math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit den in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-45040r7\"><span class=\"ecti-1095\">2.7<\/span><\/a> <span class=\"ecti-1095\">definierten Operationen einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit zwei Elementen bildet. Wieso<\/span> <span class=\"ecti-1095\">gibt es keinen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit nur einem Element?<\/span> <\/p> <\/div> <p class=\"indent\">In den n\u00e4chsten beiden \u00dcbungen konstruieren wir f\u00fcr eine Primzahl <math display=\"inline\"><mi>p<\/mi><\/math> den K\u00f6rper mit <math display=\"inline\"><mi>p<\/mi><\/math> Elementen. In der Praxis (insbesondere in der Informatik) finden diese viele Anwendungen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2fbe7d78005a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kongruente Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">sagen, dass <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">kongruent modulo <\/span><math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">sind, falls <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">teilbar ist. In diesem<\/span> <span class=\"ecti-1095\">Fall schreiben wir auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2261<\/mo> <mi>b<\/mi><mspace width=\"0.3em\" \/><mi class=\"MathClass-op\">mod<\/mi><mo> <\/mo><mspace width=\"0.3em\" \/><mi>q<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2261<\/mo> <mi>b<\/mi><mspace width=\"0.3em\" \/><mi class=\"MathClass-op\">mod<\/mi><mo> <\/mo><mspace width=\"0.3em\" \/><mi>q<\/mi><\/math> <span class=\"ecti-1095\">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>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definiert.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Den Quotienten bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich dieser <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation bezeichnet man meist als<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">und die<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse von <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist durch <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">gegeben. Genau wie die Zahlenmengen, die wir bereits kennen, verf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt die Menge<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <span class=\"ecti-1095\">zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzliche Struktur wie Addition und Multiplikation.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/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\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><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\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><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\">wohldefiniert sind.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifzieren Sie mit Division mit Rest, dass<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">genau<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">Elemente hat.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8e68cc01e322\"> <span class=\"ecbx-1095\">Applet <\/span>(Darstellung des Quotienten modulo Kongruenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"378px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/u5ranukq\/width\/688\/height\/378\/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 stellen in diesem Applet den Quotienten <\/span><math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">(f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r verschiedene Werte von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">dar. Es macht Sinn sich die Punkte <\/span><math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">entlang eines Kreises vorzustellen, doch hat dies formal (vorerst) keine Bedeutung.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd832a01ff2b0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(K\u00f6rper von Primzahlordnung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine Primzahl<\/span> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ausgestattet<\/span> <span class=\"ecti-1095\">mit Addition <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Multiplikation <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo><\/math> <span class=\"ecti-1095\">aus der vorherigen <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung. Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung zeigen, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper ist.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">allen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperaxiomen bis auf<\/span> (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450166\">6<\/a>) <span class=\"ecti-1095\">gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt, wobei das Nullelement durch <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und das Einselement durch <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass jedes Element <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine multiplikative Inverse besitzt. Betrachten Sie dazu die Multiplikation mit diesem<\/span> <span class=\"ecti-1095\">Element auf <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen Sie zuerst, dass diese injektiv (und damit auch surjektiv) ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass es keine Ordnung auf <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gibt, die <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zu einem angeordnetem K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper macht.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Wir bemerken auch, dass sich f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Primzahlpotenz wie zum Beispiel<\/span> <math display=\"inline\"><mn>4<\/mn><\/math> <span class=\"ecti-1095\">oder<\/span> <math display=\"inline\"><mn>9<\/mn><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper definieren l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst; siehe n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chstes Kapitel.<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen Sie auch <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-54002r32\"><span class=\"ecti-1095\">2.32<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfc6516f42dab\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden Sie bei den folgenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">jeweils, ob sie offen, abgeschlossen oder weder noch sind.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Zahlenmengen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2102<\/mi><\/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\">Die Teilmenge der komplexen Zahlen mit Absolutbetrag Eins.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Das Rechteck <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">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-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z061da2f5b472\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Topologie auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <\/math><span class=\"ecti-1095\">die Menge der<\/span> <span class=\"ecti-1095\">offenen Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass folgende Eigenschaften erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sind.<\/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>\u2205<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>U<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>U<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>U<\/mi><\/mrow><mrow><mi>i<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Kollektion <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>U<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><\/mrow><\/msub><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">In Worten ausgedr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckt sind also endliche Schnitte und beliebige Vereinigungen von<\/span> <span class=\"ecti-1095\">offenen Mengen offen. Die analoge Aussage gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die offenen Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Was<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r abgeschlossene Mengen?<\/span> <\/p> <\/div> <p class=\"indent\">In Abschnitt <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-680001\">2.6.1<\/a> haben wir bereits beschrieben, was Dichtheit der rationalen Zahlen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> bedeutet. Allgemeiner sagt man, dass eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecbx-1095\">dicht <\/span>ist, wenn f\u00fcr jedes offene, nicht-leere Intervall <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> der Schnitt <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>A<\/mi><\/math> nicht-leer ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze57c8d867d80\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Charakterisierung von Dichtheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass folgende Aussagen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber eine Teilmenge<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist dicht.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Menge der H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Jede abgeschlossene Menge, die <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt, ist gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zba2e3177ba57\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Dichtheit der irrationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Menge <\/span><math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">der irrationalen Zahlen dicht liegt in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/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\">Verschieben Sie die Menge der rationalen Zahlen um eine irrationale Zahl.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z39d1a5a529bb\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte folgender Teilmengen von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2223<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2223<\/mo><mi>r<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z88faccc4681a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Supremum als H\u00e4ufungspunkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine von        oben        beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte        Teilmenge.        Zeigen        Sie,        dass<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ein Maximum           besitzt           oder           das           Supremum           von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ein                          H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt                          der                          Menge<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">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\">Falls<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">kombinieren Sie am besten die Aussage in Satz <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">mit Definition <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69001r73\"><span class=\"ecti-1095\">2.73<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z207aba883704\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(\u00dcberabz\u00e4hlbare Mengen haben H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar    (aber    m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glicherweise    unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt).    Zeigen    Sie,    dass    dann<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">einen H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt besitzt.<\/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\">Betrachten                    Sie                    die                    Durschschnitte<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und ob diese endlich oder unendlich sind.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7c554aeb086c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall <\/span><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit rationalen Endpunkten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">wie in obigem Satz, so dass <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt. Schliessen Sie daraus, dass das Intervallschachtelungsprinzip in <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt ist. (Hierbei ist ein Intervall in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">definiert als der Durchschnitt von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">mit einem reellen Intervall mit rationalen Endpunkten.)<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z51e158bd913c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Das Vollst\u00e4ndigkeitsaxiom und das Supremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie,  dass  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">zum  Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom  (Axiom<\/span>  (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>)<span class=\"ecti-1095\">)  <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent  ist.<\/span> <span class=\"ecti-1095\">Genauer formuliert: zeigen Sie, dass die Axiome eines angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers (das w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren<\/span> <span class=\"ecti-1095\">Axiome<\/span> (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450011\">1<\/a>) <span class=\"ecti-1095\">\u2013<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4600615\">15<\/a>)<span class=\"ecti-1095\">)  gemeinsam  mit  der  Aussage  in  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">das  Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom<\/span> <span class=\"ecti-1095\">(Axiom<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) <span class=\"ecti-1095\">) implizieren.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5f6cfe75fc74\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine weitere Formen des Vollst\u00e4ndigkeitsaxioms)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie  in  Analogie  zu  obiger  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung,  dass  unter  Annahme  der  Axiome  eines<\/span> <span class=\"ecti-1095\">angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers<\/span>   (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450011\">1<\/a>)<span class=\"ecti-1095\">\u2013<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4600615\">15<\/a>)  <span class=\"ecti-1095\">das  Intervallschachtelungsprinzip  zusammen  mit  dem<\/span> <span class=\"ecti-1095\">Archimedischen Prinzip <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zum Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1fc74ce42006\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen       Sie,       dass       jede       nichtleere       offene       Teilmenge       von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar 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\">Verifizieren       Sie       der       Einfachheit       halber       zuerst,       dass<\/span> <math display=\"inline\"><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\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9f5e56df327a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Multiplikation mit 3 auf der Cantor-Menge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>3<\/mn><mi>x<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/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\">wohldefiniert ist. Intuitiv sagt uns die Abbildung<\/span> <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">also,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">aus zwei H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lften besteht, die jeweils aussehen wie kontrahierte Kopien von<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wieso?)<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zafba726f65d1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Rechtecksschachtelungprinzip in <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir bezeichnen eine Menge der Form<\/span> <\/p><table id=\"z3ed9dae8b21c\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">als ein abgeschlossenes beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes Rechteck. Beweisen Sie folgendes Rechtecksschachtelungsprinzip<\/span> <span class=\"ecti-1095\">in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">:<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein abgeschlossenes beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes Rechteck so dass <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2287<\/mo> <msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/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>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist der abz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbare Durchschnitt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht-leer.<\/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\">Verwenden Sie zuerst das Intervallschachtelungsprinzip f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Projektionen der<\/span> <span class=\"ecti-1095\">Rechtecke auf die reelle Achse.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1307cf03b8fc\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(H\u00e4ufungspunkte in <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann heisst <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von der Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">falls es zu jedem <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gibt mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine unendliche und beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte (das heisst, es existiert <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>M<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">Teilmenge. Zeigen Sie, dass ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt der Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">existiert.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Eine kurze Anleitung: Auf Grund der Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktheit der Menge<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">existiert<\/span> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so<\/span> <span class=\"ecti-1095\">dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis zuerst obiges Rechtecksschachtelungsprinzip beweisen und<\/span> <span class=\"ecti-1095\">dann verwenden. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie den Beweis von Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\"><span class=\"ecti-1095\">2.75<\/span><\/a> <span class=\"ecti-1095\">adaptieren: definieren<\/span> <span class=\"ecti-1095\">Sie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>X<\/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\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-punc\">:<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>Y<\/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\">und zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd10d04a026ce\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Gibt es eine Kollektion <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">von Teilmengen von <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit der Eigenschaft <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u228a<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u211d<\/mi><\/mrow><\/msub><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"endquote\">?<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Kryptischer    Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Ja,    es    gibt    derartige    Mengen.    Verwenden    Sie,    dass<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chtig sind.<\/span><\/p><\/details>  <\/div> <p class=\"indent\"> <\/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; 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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=\"z4022e73eefa1\" class=\"sectionHead\"><span class=\"titlemark\">2.7 <\/span> <a id=\"x1-730007\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-73001r72\"><\/a> <h4 id=\"z49e881b75709\" class=\"subsectionHead\"><span class=\"titlemark\">2.7.1 <\/span> <a id=\"x1-740001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Die Themen dieses Kapitels stellen den Anfang unserer Entwicklung der Analysis dar und sind aus diesem Grunde f\u00fcr das Folgende fundamental. Wie bereits erw\u00e4hnt werden wir die \u00fcblichen Eigenschaften der reellen, nat\u00fcrlichen, ganzen, rationalen und komplexen Zahlen (inklusive der Konjugation komplexer Zahlen) im Folgenden ohne Verweise verwenden. Es ist auch nicht notwendig, die Beweise der elementaren Aussagen in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-440001\">2.1<\/a> auswendig zu lernen. Manche der Beweise in Abschnitt <a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-500002\">2.2<\/a> sind auch etwas zu formal, als dass sie f\u00fcr das Folgende von grosser Bedeutung sein werden. F\u00fcr ein fundiertes Verst\u00e4ndnis der Induktion sind die besprochenen Varianten der Induktion samt Beweise wichtig und auch die Beweise der algebraischen und geometrischen Aussagen stellen eine gute \u00dcbung dar. In Abschnitt <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-580004\">2.4<\/a> haben wir einige Ihnen wahrscheinlich bekannte Definition ausgesprochen, doch werden auch die Ihnen wahrscheinlich neuen Begriffe \u201eoffen\u201c und \u201eabgeschlossen\u201c zunehmend an Bedeutung gewinnen. <\/p><p class=\"indent\">Die Kernthemen dieses Kapitels sind hingegen in folgender Liste enthalten. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Vollst\u00e4ndigkeitsaxiom in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-470003\">2.1.3<\/a>. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Existenz und Eigenschaften des Supremums und Infimums in Abschnitt <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-620005\">2.5<\/a> (inbesondere beispielsweise die Unterscheidung von Maximum und Supremum). <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Korollare der Vollst\u00e4ndigkeit in Abschnitt <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-670006\">2.6<\/a>: Das Archimedische Prinzip (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\">2.68<\/a>), die Existenz von H\u00e4ufungspunkten f\u00fcr beschr\u00e4nkte unendliche Mengen (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\">2.75<\/a>), das Intervallschachtelungsprinzip (Satz&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70001r77\">2.77<\/a>), und die \u00dcberabz\u00e4hlbarkeit von <math display=\"inline\"><mi>\u211d<\/mi><\/math> in Korollar&nbsp;<a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-71001r81\">2.81<\/a>.<\/div><\/div> <p class=\"noindent\">Diese Themen und deren Beweismethoden sind von zentraler Bedeutung f\u00fcr das Folgende und Sie werden weitere Vorlesungsstunden besser verstehen, wenn Sie diese Kernthemen bereits im Ged\u00e4chnis und auf Abruf bereit haben. <\/p><p class=\"indent\">Im Laufe dieses Kapitels haben wir auch bereits einige grundlegende Funktionen eingef\u00fchrt, welche wir ohne Verweis und mit den \u00fcblichen Eigenschaften in Zukunft wieder ben\u00f6tigen werden. <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die K\u00f6rperoperationen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">:<\/span> Addition, Subtraktion, Multiplikation, Division. <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Quadrieren <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Quadratwurzel <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/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\">Der Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Die Vorzeichenfunktion <math display=\"inline\"><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der ganzzahlige Anteil <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">\u230a<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">\u230b<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Der Nachkommaanteil <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/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> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Das Maximum <math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><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\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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> und das Minimum <math display=\"inline\"><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><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\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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> zweier reeller Zahlen <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ergeben sich als Spezialf\u00e4lle von Maximum und Minimum der Menge <math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> (welche auf Grund einer Fallunterscheidung basierend auf die Trichotomie reeller Zahlen immer existieren).<\/div><\/div> <p class=\"indent\">Sollten Sie noch nicht mit dem Anlegen einer pers\u00f6nlichen Zusammenfassung aller wichtigen Inhalte der Vorlesung begonnen haben, dann legen wir Ihnen nahe dies jetzt in Angriff zu nehmen. Die Inhalte aus Kapitel <a href=\"..\/..\/part\/einfuehrung#x1-30001\">1<\/a> sollten schnell wiederholt und zusammengefasst sein. Doch in diesem Kapitel haben wir bereits unsere ersten grundlegenden S\u00e4tze der reellen Analysis und deren Beweise kennengelernt. Deswegen wird eine pers\u00f6nlich erstellte Zusammenfassung nun wahrscheinlich schon einige Seiten lang sein. Welche Form und Detailreiche eine derartige Zusammenfassung oder Mindmap haben sollte, ist Geschmackssache und Ihnen \u00fcberlassen. Zum Beispiel k\u00f6nnte f\u00fcr den Beweis der Existenz eines H\u00e4ufungspunktes einer beschr\u00e4nkten unendlichen Menge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> (Satz <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\">2.75<\/a>) folgende Zusammenfassung aussreichen: \u201eWir definieren <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und zeigen, dass <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>X<\/mi><\/math> ein H\u00e4ufungspunkt der Menge <math display=\"inline\"><mi>A<\/mi><\/math> ist.\u201c Vielleicht reicht Ihnen dies bereits als Anfangspunkt um den Beweis zu vervollst\u00e4ndigen, oder Sie erg\u00e4nzen die Zusammenfassung noch um ein bis zwei S\u00e4tze.                                                                                                                                                                           <\/p><p class=\"indent\">Wir stellen nochmals einige Multiple-Choice-Fragen, die Ihnen zur Wiederholung des Kapitels helfen sollten. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z62d7c53cc1b5\"> <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>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sind die folgenden Aussagen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zur Aussage, dass<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist?<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">!<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/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\">In (i) wird eine eindeutige Existenz eines Punktes nahe an<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">verlangt. Aber f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">einer<\/span> <span class=\"ecti-1095\">Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">gibt es in der<\/span> <span class=\"ecti-1095\">Tat f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">sogar unendlich<\/span> <span class=\"ecti-1095\">viele Punkte von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die zur <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">-Umgebung<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (ii) ist genau die Formulierung der Definition eines H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunktes in<\/span> <span class=\"ecti-1095\">Pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">dikatenlogik.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In (iii) schr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nken wir die Definition auf alle<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gend kleinen<\/span><span class=\"ecti-1095\">\u201c<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein. Dies ist zur Definition<\/span> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent. Denn falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir die eingeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Behauptung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">anwenden<\/span> <span class=\"ecti-1095\">und ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> <span class=\"ecti-1095\">finden. Also <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert diese Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung die Bedeutung der Aussage nicht.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung auf alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert allerdings den Begriff auf drastische Weise. In der Tat hat zum Beispiel<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">keinen einzigen H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt,<\/span> <span class=\"ecti-1095\">aber jedes beliebige <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">die Aussage in (iv).<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za0cc5e3a5743\"> <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>X<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Menge mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>X<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist<\/span><span class=\"ecti-1095\">\u2026<\/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;<span class=\"ecti-1095\">\u2026eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026eine lineare Ordnungsrelation.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026eine Ordnungsrelation, die nicht linear ist.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">\u2026keins der Obigen.<\/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\">Die Aussage (i) ist falsch. Da <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">nichtleer ist, gelten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Relationen <\/span><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-rel\">\u2284<\/mo> <mi>\u2205<\/mi><\/math><span class=\"ecti-1095\">. Also<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">nicht symmetrisch.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (ii) ist nicht richtig. Seien <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mi>y<\/mi><\/math><span class=\"ecti-1095\">. Diese Elemente<\/span> <span class=\"ecti-1095\">existieren, da <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>X<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt weder <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2286<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">noch <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">ist eine Ordnungsrelation, denn sie ist reflexiv, da 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> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">stets<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt; sie ist<\/span> <span class=\"ecti-1095\">transitiv, da 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-punc\">,<\/mo><mi>C<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">aus <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">auch<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>C<\/mi><\/math> <span class=\"ecti-1095\">folgt; und sie ist<\/span> <span class=\"ecti-1095\">antisymmetrisch, da 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 mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">schon <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">gilt. Sie ist nicht linear nach (ii), also ist (iii) richtig und somit muss (iv) falsch sein.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z66126eede26d\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sind die folgenden Mengen (mit der <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Addition und Multiplikation) Beispiele f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, die angeordnet werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen?<\/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; <math display=\"inline\"><mi>\u2115<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u2124<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u211a<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u211d<\/mi><\/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\">(J\/N)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp; <math display=\"inline\"><mi>\u2102<\/mi><\/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\">Die Menge <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist kein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, da zum Beispiel <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">kein<\/span> <span class=\"ecti-1095\">additives Inverses besitzt (da <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">\u2209<\/mo><mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist kein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper.<\/span> <span class=\"ecti-1095\">Beispielsweise hat <\/span><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">kein<\/span> <span class=\"ecti-1095\">multiplikatives Inverse in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Sowohl <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als auch <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">sind angeordnete K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper. Wir verweisen dazu auf die Abschnitte <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-460002\"><span class=\"ecti-1095\">2.1.2<\/span><\/a> <span class=\"ecti-1095\">und <\/span><a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-530003\"><span class=\"ecti-1095\">2.2.3<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In jedem angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper gelten <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung (s)), also <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung (q)). Weiters ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(Folgerung<\/span> <span class=\"ecti-1095\">(r)). In <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt aber<\/span> <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">. Dies erg<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be nun einen<\/span> <span class=\"ecti-1095\">Widerspruch, wenn <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit einer geeigneten Ordnung zu einem angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper gemacht werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z884ec46357e4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Es bezeichne <\/span><math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die imagin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Einheit. Welche der folgenden Formeln sind richtig?<\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">( <\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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\">Durch Ausrechnen erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt man die richtigen L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sungen. Dabei ist es hilfreich f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (i)-(ii) zuerst<\/span> <math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">(<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">(iii)-(iv) zuerst <\/span><math display=\"inline\"><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.61em\">(<\/mo><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msqrt><mrow><mn>3<\/mn><\/mrow><\/msqrt><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.61em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auszurechnen. SageMath kann dies nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlich auch sehr schnell berechnen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z926f4d336508\"> <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>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">nichtleere, nach oben<\/span> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Welche der folgenden Aussagen gelten im Allgemeinen?<\/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;<span class=\"ecti-1095\">Gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so folgt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so gibt es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\ud83d\udeab&nbsp;<\/span><\/details>&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mi>B<\/mi> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mi>b<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Existiert das Maximum der Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/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\">(W\/F)<\/span>&nbsp;<\/summary><span style=\"vertical-align: middle\">\u2705&nbsp;<\/span><\/details>&nbsp;<span class=\"ecti-1095\">Ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so existiert das Maximum von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/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\">Die erste Aussage ist richtig, denn aufgrund der Inklusion<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math> <span class=\"ecti-1095\">eine obere Schranke<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math><span class=\"ecti-1095\">. Das Supremum<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist als kleinste<\/span> <span class=\"ecti-1095\">obere Schranke von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">somit h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu (ii) finden wir ein Gegenbeispiel: Seien <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/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>2<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>B<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">es gibt 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> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>B<\/mi><\/math> <span class=\"ecti-1095\">kein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) verweisen wir auf Proposition <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\"><span class=\"ecti-1095\">2.63<\/span><\/a> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die vierte Aussage ist falsch. Ein Gegenbeispiel:<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/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\">. Dann gilt<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><span class=\"ecti-1095\">. Die Aussage<\/span> <span class=\"ecti-1095\">gilt aber, wenn <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>B<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-65003r66\"><span class=\"ecti-1095\">2.66<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu (v): Das Maximum <\/span><math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist, wenn<\/span> <span class=\"ecti-1095\">es existiert, eine obere Schranke von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und es kann keine kleinere obere Schranke geben, da nach Definition eines Maximums<\/span> <math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gilt. Also ist<\/span> <math display=\"inline\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mi>A<\/mi><\/math> <span class=\"ecti-1095\">die kleinste obere<\/span> <span class=\"ecti-1095\">Schranke von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und dies ist die definierende Eigenschaft des Supremums.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Auch (vi) ist richtig. Da das Supremum von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">per Definition eine obere<\/span> <span class=\"ecti-1095\">Schranke von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist, erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <span class=\"ecti-1095\">es im Fall <\/span><math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">die definierende<\/span> <span class=\"ecti-1095\">Eigenschaft eines Maximums von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-74024r74\"><\/a> <h4 id=\"za4df50aedd3e\" class=\"subsectionHead\"><span class=\"titlemark\">2.7.2 <\/span> <a id=\"x1-750002\"><\/a>Weitere \u00dcbungsaufgaben<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z529d659527c6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Parallelogrammidentit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Gleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/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> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7c1c9205e48e\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Mittelsenkrechte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">zwei verschiedene Punkte. Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren und beweisen Sie, wieso die Teilmenge <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine Gerade ist. Eine Gerade ist dabei eine Teilmenge der Form <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>t<\/mi><mi>v<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/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>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7f7baf97d9d6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(K\u00f6rper mit zwei Elementen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                    Sie,                    dass                    die                    Menge<\/span> <math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit den in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-45040r7\"><span class=\"ecti-1095\">2.7<\/span><\/a> <span class=\"ecti-1095\">definierten Operationen einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit zwei Elementen bildet. Wieso<\/span> <span class=\"ecti-1095\">gibt es keinen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit nur einem Element?<\/span> <\/p> <\/div> <p class=\"indent\">In den n\u00e4chsten beiden \u00dcbungen konstruieren wir f\u00fcr eine Primzahl <math display=\"inline\"><mi>p<\/mi><\/math> den K\u00f6rper mit <math display=\"inline\"><mi>p<\/mi><\/math> Elementen. In der Praxis (insbesondere in der Informatik) finden diese viele Anwendungen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2fbe7d78005a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kongruente Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">sagen, dass <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">kongruent modulo <\/span><math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">sind, falls <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">teilbar ist. In diesem<\/span> <span class=\"ecti-1095\">Fall schreiben wir auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2261<\/mo> <mi>b<\/mi><mspace width=\"0.3em\" \/><mi class=\"MathClass-op\">mod<\/mi><mo> <\/mo><mspace width=\"0.3em\" \/><mi>q<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2261<\/mo> <mi>b<\/mi><mspace width=\"0.3em\" \/><mi class=\"MathClass-op\">mod<\/mi><mo> <\/mo><mspace width=\"0.3em\" \/><mi>q<\/mi><\/math> <span class=\"ecti-1095\">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>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definiert.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Den Quotienten bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich dieser <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation bezeichnet man meist als<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">und die<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse von <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist durch <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">gegeben. Genau wie die Zahlenmengen, die wir bereits kennen, verf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt die Menge<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <span class=\"ecti-1095\">zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzliche Struktur wie Addition und Multiplikation.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/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\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><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\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>\u2124<\/mi><\/mstyle><mtext \/><mstyle class=\"math\"><mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> \/<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mstyle><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><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\">wohldefiniert sind.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifzieren Sie mit Division mit Rest, dass<\/span> <math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">genau<\/span> <math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">Elemente hat.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8e68cc01e322\"> <span class=\"ecbx-1095\">Applet <\/span>(Darstellung des Quotienten modulo Kongruenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"378px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/u5ranukq\/width\/688\/height\/378\/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 stellen in diesem Applet den Quotienten <\/span><math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">(f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r verschiedene Werte von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">dar. Es macht Sinn sich die Punkte <\/span><math display=\"inline\"><mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>q<\/mi><mi>\u2124<\/mi><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">entlang eines Kreises vorzustellen, doch hat dies formal (vorerst) keine Bedeutung.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd832a01ff2b0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(K\u00f6rper von Primzahlordnung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine Primzahl<\/span> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ausgestattet<\/span> <span class=\"ecti-1095\">mit Addition <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Multiplikation <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo><\/math> <span class=\"ecti-1095\">aus der vorherigen <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung. Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung zeigen, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper ist.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">allen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperaxiomen bis auf<\/span> (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450166\">6<\/a>) <span class=\"ecti-1095\">gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt, wobei das Nullelement durch <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und das Einselement durch <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass jedes Element <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>p<\/mi><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine multiplikative Inverse besitzt. Betrachten Sie dazu die Multiplikation mit diesem<\/span> <span class=\"ecti-1095\">Element auf <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen Sie zuerst, dass diese injektiv (und damit auch surjektiv) ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass es keine Ordnung auf <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gibt, die <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zu einem angeordnetem K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper macht.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Wir bemerken auch, dass sich f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Primzahlpotenz wie zum Beispiel<\/span> <math display=\"inline\"><mn>4<\/mn><\/math> <span class=\"ecti-1095\">oder<\/span> <math display=\"inline\"><mn>9<\/mn><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper definieren l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst; siehe n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chstes Kapitel.<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (iii) d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen Sie auch <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-natuerlichen-zahlen#x1-54002r32\"><span class=\"ecti-1095\">2.32<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfc6516f42dab\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Entscheiden Sie bei den folgenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">jeweils, ob sie offen, abgeschlossen oder weder noch sind.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Zahlenmengen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u2102<\/mi><\/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\">Die Teilmenge der komplexen Zahlen mit Absolutbetrag Eins.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Das Rechteck <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">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-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z061da2f5b472\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Topologie auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <\/math><span class=\"ecti-1095\">die Menge der<\/span> <span class=\"ecti-1095\">offenen Teilmengen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass folgende Eigenschaften erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sind.<\/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>\u2205<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>U<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>U<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>U<\/mi><\/mrow><mrow><mi>i<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Kollektion <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>U<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><\/mrow><\/msub><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">In Worten ausgedr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckt sind also endliche Schnitte und beliebige Vereinigungen von<\/span> <span class=\"ecti-1095\">offenen Mengen offen. Die analoge Aussage gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die offenen Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">. Was<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r abgeschlossene Mengen?<\/span> <\/p> <\/div> <p class=\"indent\">In Abschnitt <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-680001\">2.6.1<\/a> haben wir bereits beschrieben, was Dichtheit der rationalen Zahlen in <math display=\"inline\"><mi>\u211d<\/mi><\/math> bedeutet. Allgemeiner sagt man, dass eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecbx-1095\">dicht <\/span>ist, wenn f\u00fcr jedes offene, nicht-leere Intervall <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> der Schnitt <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>A<\/mi><\/math> nicht-leer ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze57c8d867d80\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Charakterisierung von Dichtheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass folgende Aussagen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber eine Teilmenge<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist dicht.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Menge der H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte von <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ist gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Jede abgeschlossene Menge, die <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt, ist gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zba2e3177ba57\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Dichtheit der irrationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Menge <\/span><math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">der irrationalen Zahlen dicht liegt in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/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\">Verschieben Sie die Menge der rationalen Zahlen um eine irrationale Zahl.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z39d1a5a529bb\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkte folgender Teilmengen von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2223<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2223<\/mo><mi>r<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z88faccc4681a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Supremum als H\u00e4ufungspunkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine von        oben        beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte        Teilmenge.        Zeigen        Sie,        dass<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ein Maximum           besitzt           oder           das           Supremum           von<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">ein                          H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt                          der                          Menge<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">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\">Falls<\/span> <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>A<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">kombinieren Sie am besten die Aussage in Satz <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">mit Definition <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69001r73\"><span class=\"ecti-1095\">2.73<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z207aba883704\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(\u00dcberabz\u00e4hlbare Mengen haben H\u00e4ufungspunkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar    (aber    m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glicherweise    unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt).    Zeigen    Sie,    dass    dann<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">einen H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt besitzt.<\/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\">Betrachten                    Sie                    die                    Durschschnitte<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und ob diese endlich oder unendlich sind.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7c554aeb086c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall <\/span><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit rationalen Endpunkten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">wie in obigem Satz, so dass <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msqrt><mrow><mn>2<\/mn><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">gilt. Schliessen Sie daraus, dass das Intervallschachtelungsprinzip in <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt ist. (Hierbei ist ein Intervall in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">definiert als der Durchschnitt von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">mit einem reellen Intervall mit rationalen Endpunkten.)<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z51e158bd913c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Das Vollst\u00e4ndigkeitsaxiom und das Supremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie,  dass  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">zum  Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom  (Axiom<\/span>  (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>)<span class=\"ecti-1095\">)  <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent  ist.<\/span> <span class=\"ecti-1095\">Genauer formuliert: zeigen Sie, dass die Axiome eines angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers (das w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren<\/span> <span class=\"ecti-1095\">Axiome<\/span> (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450011\">1<\/a>) <span class=\"ecti-1095\">\u2013<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4600615\">15<\/a>)<span class=\"ecti-1095\">)  gemeinsam  mit  der  Aussage  in  Satz  <\/span><a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\"><span class=\"ecti-1095\">2.59<\/span><\/a> <span class=\"ecti-1095\">das  Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom<\/span> <span class=\"ecti-1095\">(Axiom<\/span><span class=\"ecti-1095\">&nbsp;<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4700116\">16<\/a>) <span class=\"ecti-1095\">) implizieren.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5f6cfe75fc74\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eine weitere Formen des Vollst\u00e4ndigkeitsaxioms)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie  in  Analogie  zu  obiger  <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung,  dass  unter  Annahme  der  Axiome  eines<\/span> <span class=\"ecti-1095\">angeordneten K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers<\/span>   (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450011\">1<\/a>)<span class=\"ecti-1095\">\u2013<\/span>(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4600615\">15<\/a>)  <span class=\"ecti-1095\">das  Intervallschachtelungsprinzip  zusammen  mit  dem<\/span> <span class=\"ecti-1095\">Archimedischen Prinzip <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent zum Vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndigkeitsaxiom sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1fc74ce42006\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen       Sie,       dass       jede       nichtleere       offene       Teilmenge       von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar 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\">Verifizieren       Sie       der       Einfachheit       halber       zuerst,       dass<\/span> <math display=\"inline\"><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\">\u00fc<\/span><span class=\"ecti-1095\">berabz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar ist.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9f5e56df327a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Multiplikation mit 3 auf der Cantor-Menge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>3<\/mn><mi>x<\/mi> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>3<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/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\">wohldefiniert ist. Intuitiv sagt uns die Abbildung<\/span> <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">also,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">aus zwei H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lften besteht, die jeweils aussehen wie kontrahierte Kopien von<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wieso?)<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zafba726f65d1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Rechtecksschachtelungprinzip in <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir bezeichnen eine Menge der Form<\/span> <\/p><table id=\"z3ed9dae8b21c\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">als ein abgeschlossenes beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes Rechteck. Beweisen Sie folgendes Rechtecksschachtelungsprinzip<\/span> <span class=\"ecti-1095\">in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">:<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein abgeschlossenes beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes Rechteck so dass <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2287<\/mo> <msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/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>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist der abz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbare Durchschnitt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u22c2<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht-leer.<\/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\">Verwenden Sie zuerst das Intervallschachtelungsprinzip f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Projektionen der<\/span> <span class=\"ecti-1095\">Rechtecke auf die reelle Achse.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1307cf03b8fc\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(H\u00e4ufungspunkte in <math display=\"inline\"><mi>\u2102<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann heisst <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von der Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">falls es zu jedem <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> <span class=\"ecti-1095\">gibt mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine unendliche und beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte (das heisst, es existiert <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>M<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">Teilmenge. Zeigen Sie, dass ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt der Menge <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">existiert.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Eine kurze Anleitung: Auf Grund der Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktheit der Menge<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">existiert<\/span> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">so<\/span> <span class=\"ecti-1095\">dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis zuerst obiges Rechtecksschachtelungsprinzip beweisen und<\/span> <span class=\"ecti-1095\">dann verwenden. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie den Beweis von Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-69003r75\"><span class=\"ecti-1095\">2.75<\/span><\/a> <span class=\"ecti-1095\">adaptieren: definieren<\/span> <span class=\"ecti-1095\">Sie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>X<\/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\"><mi>Y<\/mi> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-punc\">:<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><mrow><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><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>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><\/mtd> <mtd class=\"align-even\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi>Y<\/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\">und zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd10d04a026ce\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Gibt es eine Kollektion <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">von Teilmengen von <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit der Eigenschaft <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u228a<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u211d<\/mi><\/mrow><\/msub><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"endquote\">?<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Kryptischer    Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Ja,    es    gibt    derartige    Mengen.    Verwenden    Sie,    dass<\/span> <math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chtig sind.<\/span><\/p><\/details>  <\/div> <div class=\"wp-nocaption \"><\/div> \n","protected":false},"author":1089,"menu_order":7,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-40","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/40","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/40\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=40"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=40"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=40"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}