{"id":29,"date":"2021-12-15T09:52:57","date_gmt":"2021-12-15T09:52:57","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/aequivalenzrelationen\/"},"modified":"2021-12-15T09:52:57","modified_gmt":"2021-12-15T09:52:57","slug":"aequivalenzrelationen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/aequivalenzrelationen\/","title":{"raw":"\u00c4quivalenzrelationen","rendered":"\u00c4quivalenzrelationen"},"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=\"z38b870205ca7\" class=\"sectionHead\"><span class=\"titlemark\">1.6 <\/span> <a id=\"x1-200006\"><\/a>\u00c4quivalenzrelationen<\/h3> <p class=\"noindent\">In diesem Abschnitt besprechen wir Relationen auf Mengen. Mit dem Begriff der \u00c4quivalenzrelation werden wir im Gegensatz zum vorherigen Abschnitt in der Lage sein, zum Beispiel die Menge der rationalen Zahlen formal korrekt aus den ganzen Zahlen (und mit etwas mehr Arbeit auch aus den nat\u00fcrlichen Zahlen) zu konstruieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zb8237b94e49f\"> <a id=\"x1-20001r58\"><\/a> <span class=\"ecbx-1095\">Definition 1.58 <\/span>(Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><mi>Y<\/mi> <\/math> Mengen. Eine <span class=\"ecbx-1095\">Relation <\/span>auf <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> ist eine Teilmenge <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Wir schreiben auch <math display=\"inline\"><mi>x<\/mi><mi mathvariant=\"bold-script\">\u211b<\/mi><mi>y<\/mi><\/math> falls <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><\/math> und verwenden oft Symbole<a id=\"dx1-20002\"><\/a> wie <math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u226a<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2261<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> f\u00fcr Relationen. Falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <\/math> ist, dann sprechen wir auch von einer Relation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wenn <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> (resp. <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo> <\/math><\/span><span class=\"period\">,<\/span> \u2026) eine Relation ist, dann schreiben wir auch \u201e<span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo><mi>y<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> (resp. \u201e <span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2247<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span>, \u2026) f\u00fcr \u201e <span class=\"maendquote\"><math display=\"inline\"><mi class=\"MathClass-op\">\u00ac<\/mi><mo> <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span> (resp. \u201e <span class=\"maendquote\"><math display=\"inline\"><mi class=\"MathClass-op\">\u00ac<\/mi><mo> <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span>, \u2026). <\/p> <\/div> <p class=\"indent\">Der Begriff der Relation umfasst viele verschiedene Beispiele, die in verschiedene Typen von Relation eingeteilt werden k\u00f6nnen. Um diese Allgemenheit zu erm\u00f6glichen, ist obige Definition eher abstrakt formuliert. Wir werden aber f\u00fcr den allgemeinen Begriff keinerlei theoretischen \u00dcberlegungen anstellen. Stattdessen wollen wir nur kurz einige bekannte Beispiele von Relation besprechen und anschliessend einen wichtigen speziellen Typ von Relationen genauer untersuchen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z569dce0ec59e\"> <a id=\"x1-20003r59\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.59 <\/span>(Beispiele von Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zum Beispiel sind <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">Relationen auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die wir mittels der Addition auf <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">folgendermassen definieren k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnten: Wir schreiben <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">falls es ein <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">gibt, und wir schreiben <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ebenso k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">auch als Relationen auf<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auffassen. In der Tat<\/span> <span class=\"ecti-1095\">werden wir die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">mittels unserem Axiomensytem der reellen Zahlen im n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chsten Kapitel einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren und dabei<\/span> <span class=\"ecti-1095\">nochmals ihre wichtigsten Eigenschaften besprechen. Im Sinne der obigen Definition sollten Sie<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">mit der Teilmenge<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20006r12\"><span class=\"ecti-1095\">1.12<\/span><\/a> <span class=\"ecti-1095\">identifizieren. Denn <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">genau dann wenn der Punkt auf oder links von der Diagonale liegt.<\/span> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=270&amp;url=Pictures\/Einfuehrung\/relation\/leq-relation.pdf\"><img id=\"z8ee58dede116\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/leq-relation.svg\" width=\"270\"><\/div> <a id=\"x1-20006r12\"><\/a> <a id=\"x1-20007\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.12: <\/span><span class=\"content\">Dies stellt die <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math>-Relation               als Teilmenge von <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><\/math>                 dar.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Manchmal verwendet man auch eine Relation<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2248<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken soll, dass zwei Zahlen in etwa gleich sind. Um diese Relation<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Obwohl wir hier<\/span> <span class=\"ecti-1095\">ein schwammiges<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Ungef<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hr-Gleich<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">betrachten, ben<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">tigen wir eine pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">zise Definition um<\/span> <span class=\"ecti-1095\">dar<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber sprechen zu k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span><\/span><\/span> <span class=\"ecti-1095\">zu definieren, w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen wir eine fix gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlte positive Konstante<\/span> <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Mit dieser defineren<\/span> <span class=\"ecti-1095\">wir die Relation <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">(wobei<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">den Abstand<\/span> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">bestimmt). Als<\/span> <span class=\"ecti-1095\">Teilmenge von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist diese Relation ein Streifen rund um die Diagonale (siehe Figur<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20009r13\"><span class=\"ecti-1095\">1.13<\/span><\/a><span class=\"ecti-1095\">).<\/span> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=301&amp;url=Pictures\/Einfuehrung\/relation\/approx-relation.pdf\"><img id=\"za27e81625359\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/approx-relation.svg\" width=\"301\"><\/div> <a id=\"x1-20009r13\"><\/a> <a id=\"x1-20010\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.13:             <\/span><span class=\"content\">Dies             stellt             die             Relation               <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math>                als                                                                             Teilmenge               <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><\/math>                 dar,                                                                                 wobei               <math display=\"inline\"><mi>\u03b4<\/mi><\/math>             die vertikale (oder horizontale) halbe Breite des Streifens rund um die               Diagonale ist.                                                                            &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine beliebige Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math><span class=\"ecti-1095\">als eine Relation<\/span> <span class=\"ecti-1095\">auf der Potenzmenge <\/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\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">betrachten. Dieses Beispiel ist viel schwieriger zu visualisieren. Aber in dem Spezialfall der Menge<\/span> <math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit paarweise<\/span> <span class=\"ecti-1095\">verschiedenen Elemente <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>c<\/mi><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte man die<\/span> <span class=\"ecti-1095\">Inklusionsrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">wie in der Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20012r14\"><span class=\"ecti-1095\">1.14<\/span><\/a> <span class=\"ecti-1095\">visualisieren.<\/span> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=513&amp;url=Pictures\/Einfuehrung\/relation\/incl-relation.pdf\"><img id=\"z7492065bb30c\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/incl-relation.svg\" width=\"513\"><\/div> <a id=\"x1-20012r14\"><\/a> <a id=\"x1-20013\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.14: <\/span><span class=\"content\">Hier wird die Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math>       auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               dargestellt: Ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>B<\/mi><\/math>             oder      existiert      eine      Pfeil-Kette      von      einer      Menge               <math display=\"inline\"><mi>A<\/mi><\/math>             links zu einer anderen Menge <math display=\"inline\"><mi>B<\/mi><\/math>             rechts (ohne Pfeile in gegengesetzter Richtung zu verwenden), so gilt               <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/span> &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/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\">auch wie in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20014r15\"><span class=\"ecti-1095\">1.15<\/span><\/a> <span class=\"ecti-1095\">beschreiben.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=484&amp;url=Pictures\/Einfuehrung\/relation\/incl-relation-as-square.pdf\"><img id=\"z15b2677c24c9\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/incl-relation-as-square.svg\" width=\"484\"><\/div> <a id=\"x1-20014r15\"><\/a> <a id=\"x1-20015\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.15:         <\/span><span class=\"content\">Dies         stellt         ebenso         die         Relation               <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math>       auf               <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               dar.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z60165fbeeff9\"> <a id=\"x1-20016r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.60 <\/span>(Eine bekannte Relation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In einem gewissen Sinne haben wir bereits eine gewisse wichtige Klasse von Relationen betrachtet. Seien<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math><span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Relation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die folgende Eigenschaft erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt:<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">!<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><mi>y<\/mi><\/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\">Wie nennen wir eine solche Relation gemeinsam mit<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">? Welches Symbol verwenden<\/span> <span class=\"ecti-1095\">wir statt dem Symbol <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math> <span class=\"ecti-1095\">in diesem Zusammenhang?<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Eine derartige Relationen entspricht einer Abbildung von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist der Graph und<\/span> <span class=\"ecti-1095\">wir schreiben meist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u21a6<\/mo><\/math> <span class=\"ecti-1095\">statt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr den Rest dieses Abschnittes wollen wir uns aber vorwiegend mit folgendem wichtigen Typ von Relationen besch\u00e4ftigen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zcc9e0c4d584c\"> <a id=\"x1-20017r61\"><\/a> <span class=\"ecbx-1095\">Definition 1.61 <\/span>(\u00c4quivalenzrelationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> auf <math display=\"inline\"><mi>X<\/mi><\/math> ist eine <span class=\"ecbx-1095\">\u00c4<\/span><span class=\"ecbx-1095\">quivalenzrelation<\/span>, falls folgende drei Eigenschaften erf\u00fcllt sind: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Reflexivit\u00e4t: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Symmetrie: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Transitivit\u00e4t: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <p class=\"indent\">\u00c4quivalenzrelationen sind oft durch eine Gleichheit in gewissen Aspekten definiert und sollten als eine Form von einer Gleichheit angesehen werden. In der Tat bezieht sich der Ausdruck \u201edas Gleiche\u201c in der deutschen Sprache auf eine Art \u201e\u00c4quivalenzrelation\u201c (die je nach Zusammenhang verschieden sein kann), wohingegen der Ausdruck \u201edasselbe\u201c nur f\u00fcr \u201eein und dasselbe\u201c Objekt verwendet werden sollte. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf4ee98ed6714\"> <a id=\"x1-20018r62\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.62 <\/span>(Beispiele von \u00c4quivalenzrelationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Das einfachste Beispiel einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf einer beliebigen Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist die Gleichheit, also die Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>X<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ein weiteres allgemeines Beispiel ist die sogenannte triviale <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>X<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der je zwei Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir betrachten ein Beispiel in der ebenen (euklidschen) Geometrie. Sei<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Geraden in der Ebene. Zu zwei Geraden<\/span> <math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">schreiben wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;sind&nbsp;parallel<\/mtext><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies definiert eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">(wieso?).<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><a id=\"x1-200224\"><\/a><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">die Relation, die durch Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a> <span class=\"ecti-1095\">definiert wird. Wir behaupten, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">darstellt.<\/span> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=365&amp;url=Pictures\/Einfuehrung\/relation\/number-relation.pdf\"><img id=\"z026bff2f0505\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/number-relation.svg\" width=\"365\"><\/div> <a id=\"x1-20023r16\"><\/a> <a id=\"x1-20024\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.16:           <\/span><span class=\"content\">Wir           definieren           eine           Relation               <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math>       als                                    Teilmenge                                    von               <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math>             und behaupten, dass diese sogar eine \u00c4quivalenzrelation auf der Menge               <math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math>       darstellt.                                                                                  &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat entspricht der Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">der Aussage, dass<\/span> <span class=\"ecti-1095\">die Diagonale <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">in der Relation enthalten ist. Ebenso entspricht der Symmetrie von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">der Aussage, dass die Teilmenge bei Spiegelung um die Diagonale (bei Vertauschung der beiden<\/span> <span class=\"ecti-1095\">Koordinaten) unver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert bleibt. Beides ist in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t hat keine<\/span> <span class=\"ecti-1095\">derartige einfache visuelle Interpretation. Stattdessen sehen wir in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Ebenso<\/span> <span class=\"ecti-1095\">gilt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mn>6<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mn>6<\/mn><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Und<\/span> <span class=\"ecti-1095\">schlussendlich gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>6<\/mn> <mo class=\"MathClass-rel\">\u223c<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Aus dieser Information ergibt sich nun die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t durch Fallunterscheidung: Angenommen<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">so gilt dies<\/span> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>z<\/mi><\/math><span class=\"ecti-1095\">, womit damit<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">so gilt dies<\/span> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Falls aber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math> <span class=\"ecti-1095\">und damit<\/span> <span class=\"ecti-1095\">ebenso <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen werden in anderen Wissenschaften oder auch im Alltag h<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufig verwendet.<\/span> <span class=\"ecti-1095\">Beispielsweise kann man auf der Menge aller Lebewesen eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definieren, in<\/span> <span class=\"ecti-1095\">dem man zwei Lebewesen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rt, wenn sie zur selben Art (oder Gattung oder<\/span> <span class=\"ecti-1095\">Familie) geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5c05ec20b472\"> <a id=\"x1-20026r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.63 <\/span>(Zwei weitere Beispiele)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bungen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zwei weitere Beispiele von <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen besprechen. Sei<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Menge.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Quetschen einer Teilmenge) Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge. Zeigen Sie, dass die Relation gegeben durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub><mi>y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/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\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definiert.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenz <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber eine Abbildung) Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Abbildung in eine<\/span> <span class=\"ecti-1095\">weitere Menge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">definieren eine Relation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definiert.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7f30e516bae7\"> <a id=\"x1-20029r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.64 <\/span>(Ein falscher Beweis)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser Aufgabe behaupten wir f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lschlicherweise, dass jede symmetrische und transitive Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf einer<\/span> <span class=\"ecti-1095\">Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">auch reflexiv ist (d.h. eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist). Finden Sie den Fehler in folgendem<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Beweis<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">:<\/span> <\/p><blockquote class=\"quote\"> <p class=\"noindent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ein Element. Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wegen Symmetrie der Relation gilt also auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folglich gilt unter Verwendung der Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t der Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was zu zeigen war.<\/span><\/p><\/blockquote> <p class=\"noindent\"><span class=\"ecti-1095\">Finden Sie ein Beispiel einer Relation, die symmetrisch und transitiv, aber nicht reflexiv<\/span> <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\">Die Aussage ist auf jeden Fall falsch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">leere Relation<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">, welche<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2241<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7d7899f12eb0\"> <a id=\"x1-20030r65\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.65 <\/span>(Beispiele allgemeiner Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie eine Relation (auf <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">oder anderen Mengen), die von den Eigenschaften einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/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\">nur die Symmetrie,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">nur die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">die Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t und Symmetrie, aber nicht die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">die Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t und Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t, aber nicht die Symmetrie<\/span><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/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\">Bestimmen Sie die Eigenschaften der Relationen in Beispiel <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20003r59\"><span class=\"ecti-1095\">1.59<\/span><\/a> <span class=\"ecti-1095\">und auch von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2260<\/mo><\/math><span class=\"ecti-1095\">. Insbesondere ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes fest gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlte <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Relation <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">keine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">da <\/span><math display=\"inline\"><mn>0<\/mn> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u03b4<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mn>2<\/mn><mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">aber<\/span> <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><msub><mrow><mo class=\"MathClass-rel\">\u2249<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mn>2<\/mn><mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wie schon erw\u00e4hnt ist eine \u00c4quivalenzrelation gewissermassen eine Form von Gleichheit. Dies l\u00e4sst sich auch formalisieren: <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z3b4b1d100629\"> <a id=\"x1-20031r66\"><\/a> <span class=\"ecbx-1095\">Definition 1.66 <\/span>(\u00c4quivalenzklassen und die Quotientenmenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf einer Menge <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann wird f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> die Menge                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/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> <p class=\"noindent\">die <span class=\"ecbx-1095\">\u00c4<\/span><span class=\"ecbx-1095\">quivalenzklasse<\/span><a id=\"dx1-20032\"><\/a> von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. Weiters ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>X<\/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\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/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> <p class=\"noindent\">der <span class=\"ecbx-1095\">Quotient <\/span>(oder die <span class=\"ecbx-1095\">Quotientenmenge<\/span>)<a id=\"dx1-20033\"><\/a> von <math display=\"inline\"><mi>X<\/mi><\/math> modulo <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> Ein Element <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> wird auch <span class=\"ecbx-1095\">Repr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">sentant<\/span> seiner \u00c4quivalenzklasse <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> genannt. <\/p> <\/div> <p class=\"indent\">Anschaulich gesprochen geben wir \u00e4quivalente Elemente von <math display=\"inline\"><mi>X<\/mi><\/math> in ein und denselben Topf und nicht \u00e4quivalente Elemente in verschiedene T\u00f6pfe. In diesem Bild besteht die \u00c4quivalenzklasse eines Elements aus allen Elementen, die im gleichen Topf sind. Der Quotient modulo <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> wiederum ist die Menge der T\u00f6pfe. In Beispiel <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20018r62\">1.62<\/a>(<a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200224\">iv<\/a>) hatten wir bereits implizit die drei \u00c4quivalenzklassen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> in unserer Diskussion verwendet. <\/p><p class=\"indent\">Die Begriffe der \u00c4quivalenzrelation und der Partition in Definition <a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-16001r51\">1.51<\/a> sind auf folgende Weise eng verwandt.                                                                                                                                                                           <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd40376e790d6\"> <a id=\"x1-20034r67\"><\/a> <span class=\"ecbx-1095\">Proposition 1.67 <\/span>(Korrespondenz zwischen \u00c4quivalenzrelationen und Partitionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge. Dann entsprechen <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und Partitionen<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">einander im folgenden Sinne: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine gegebene <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist die<\/span> <span class=\"ecti-1095\">Menge<\/span><a id=\"dx1-20035\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/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> <p class=\"noindent\"><span class=\"ecti-1095\">eine Partition von <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Umgekehrt<\/span> <span class=\"ecti-1095\">definiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Partition <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math><a id=\"dx1-20036\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><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\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Des Weiteren sind die Konstruktion der Partition aus der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation und die<\/span> <span class=\"ecti-1095\">Konstruktion der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation aus der Partition zueinander invers: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Partition<\/span> <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge, <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> wie in Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a> definiert. Selbstverst\u00e4ndlich gilt nach den Definitionen eigentlich <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten jedoch zwei verschiedene Symbole mitf\u00fchren, da wir <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> jeweils verschieden interpretieren m\u00f6chten. <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> werden wir stets als eine Kollektion von Teilmengen von <math display=\"inline\"><mi>X<\/mi><\/math> auffassen; <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> hingegen werden wir als einen neuen Raum erachten, wo die Punkte durch Identifikation (\u201eAneinanderkleben\u201c) von gewissen Punkten in <math display=\"inline\"><mi>X<\/mi><\/math> entstanden sind. (Die Punkte in <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> sind die Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> die in <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> enthalten sind). <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0168eec249db\"> <a id=\"x1-20037r68\"><\/a> <span class=\"ecbx-1095\">Applet 1.68 <\/span>(Eine \u00c4quivalenzrelation und deren Quotient)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"412px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/n2vtpjrw\/width\/688\/height\/412\/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\">Links wird eine Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">partitioniert, was einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">entspricht. Rechts betrachten wir die Menge der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen, also den Quotienten von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">bz<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Menge links k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte eine abstrakte Menge darstellen oder den Einheitskreis. Im letzteren<\/span> <span class=\"ecti-1095\">Fall muss klar definiert sein, zu welcher Menge die Punkte der Kanten, bei denen der Kreis<\/span> <span class=\"ecti-1095\">unterteilt wird, geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren. Wir haben dies im Beispiel mit Farben angedeutet.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za1abeecf416a\"> <a id=\"x1-20038r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.69 <\/span>(Zwei Quotienten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Charakterisieren Sie die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen der beiden in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20026r63\"><span class=\"ecti-1095\">1.63<\/span><\/a> <span class=\"ecti-1095\">(ii) definierten <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie jeweils, dass <\/span><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(wie in obiger Proposition definiert) eine Partition ist (ohne auf die noch nicht-bewiesene<\/span> <span class=\"ecti-1095\">Proposition zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckzugreifen) und beschreiben Sie <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">intuitiv.<\/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\">In            (i)            werden            alle            Punkte            in<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">miteinander                 identifiziert                 und                 man                 kann<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">als<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">auffassen.                    In                    diesem                    Sinne                    wird<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">zu einem         Punkt         gequetscht.         In         (ii)         k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen         wir<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in Beziehung    bringen    und    auf    diese    Weise    eine    kanonische    Bijektion    von<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definieren. <\/span><\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr den Beweis der Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a> ziehen wir folgende Behauptung vor: <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z2f4fc8da73eb\"> <span class=\"ecbx-1095\">Behauptung.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Dann sind folgende<\/span> <span class=\"ecti-1095\">drei Aussagen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\">(paarweise) <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/dd><\/dl> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis der Behauptung.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir beweisen die Implikationen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> woraus folgt, dass alle drei Aussagen \u00e4quivalent sind. Seien <span class=\"maperiod\"><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><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir nehmen also zuerst an, dass <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> gilt wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Dann ist <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> und also <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wegen der Transitivit\u00e4t. Die andere Inklusion folgt analog (durch Vertauschen von <math display=\"inline\"><mi>x<\/mi><\/math> und <math display=\"inline\"><mi>y<\/mi><\/math>) und es gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so folgt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> wie in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>                                                                                                                                                                           wegen Reflexivit\u00e4t. <\/p><p class=\"indent\">Gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so existiert ein <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Aus Symmetrie und Transitivit\u00e4t folgt daher <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Sei <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><mi>P<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>X<\/mi><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> da <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Paarweise Disjunktheit der Elemente von <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> gilt dank der Behauptung und es folgt, dass <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> eine Partition ist. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> eine Partition von <math display=\"inline\"><mi>X<\/mi><\/math> und sei die Relation <math display=\"inline\"><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/math> wie in der Proposition definiert. Es ist f\u00fcr jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> auch <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> da es wegen <math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>P<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><\/math> ein <math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">;<\/span> dies zeigt Reflexivit\u00e4t. Falls <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><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><span class=\"period\">,<\/span> dann folgt <math display=\"inline\"><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>x<\/mi><\/math> direkt aus der Definition, das heisst <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> ist symmetrisch. Angenommen es gilt <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein Partitionselement <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> mit <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und daher <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> nach den Eigenschaften der Partition. Es folgt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>z<\/mi><\/math> und die Transitivit\u00e4t                                                                                                                                                                           der Relation <span class=\"maperiod\"><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Daher ist <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> eine \u00c4quivalenzrelation. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> ist die \u00c4quivalenzklasse bez\u00fcglich <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mo> \u22c3<\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-bin\">\u2227<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>P<\/mi> <\/mrow><\/munder><mi>P<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber die Elemente von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> paarweise disjunkt sind, kann <math display=\"inline\"><mi>x<\/mi><\/math> nur in einem Element enthalten sein. Insbesondere folgt, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> das eindeutig bestimmte Element der Partition <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> ist, das <math display=\"inline\"><mi>x<\/mi><\/math> enth\u00e4lt, und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">,<\/span> so gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">,<\/span> also <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Ist umgekehrt <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> die entsprechende Partition, dann gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach der Definition von <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><\/math> und der Behauptung. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Die folgende \u00dcbung zeigt, dass man sich jede \u00c4quivalenzrelation bildlich wie eine disjunkte Vereinigung von Quadraten vorstellen kann, die wie in Beispiel <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20018r62\">1.62<\/a>(<a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200224\">iv<\/a>) entlang der Diagonale angeordnet in <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math> liegen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z087392517b26\"> <a id=\"x1-20042r70\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.70.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>X<\/mi><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben ist. Zeigen Sie auch, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/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\">entweder <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">oder <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2205<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Man verwendet Quotienten modulo \u00c4quivalenzrelationen in der Mathematik oft f\u00fcr die Konstruktion von gewissen R\u00e4umen und auch von neuen Zahlenmengen. Wir betrachten zu letzterem ein einfaches und grundlegendes Beispiel: Wir konstruieren die rationalen Zahlen aus den ganzen Zahlen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za061ee32cf1e\"> <a id=\"x1-20043r71\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.71 <\/span>(Konstruktion der rationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen       an,       dass       wir       bereits       die       ganzen       Zahlen<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und die              Addition              und              Multiplikation              auf<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit allen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen  Eigenschaften  kennen.  Wir  wollen  damit  die  rationalen  Zahlen<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">definieren,                           wobei                           ein                           Element<\/span> <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als                          <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse                          des                          Tupels<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definiert sein wird.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu diesem Zwecke betrachten wir die Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Diese<\/span> <span class=\"ecti-1095\">Definition r<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt daher, dass wir eben die rationale Zahlen als Br<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">che von ganzen Zahlen auffassen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten.<\/span> <span class=\"ecti-1095\">Dabei m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen wir allerdings solche identifizieren, die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">nach K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzen<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">gleich sind; zum Beispiel sollte gelten<\/span> <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><mn>0<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>5<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Allerdings wollen<\/span> <span class=\"ecti-1095\">wir hier davon ausgehen, dass wir die rationalen Zahlen noch nicht kennen, weswegen wir anstatt Gleichungen der<\/span> <span class=\"ecti-1095\">Form <\/span><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">Gleichungen der<\/span> <span class=\"ecti-1095\">Form <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit Ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken<\/span> <span class=\"ecti-1095\">innerhalb von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">betrachten<\/span> <span class=\"ecti-1095\">(im Beispiel <\/span><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>5<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>6<\/mn><\/math><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Nun verifzieren wir, dass obige Relation tats<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chlich eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist. Seien dazu<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/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\">Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Symmetrie: Angenommen es gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist per Definition also <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und daher auch <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ergibt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Durch Multiplikation der ersten Gleichungen mit<\/span> <math display=\"inline\"><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und der zweiten<\/span> <span class=\"ecti-1095\">Gleichung mit <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erhalten wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht Null ist, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir in obiger Gleichung<\/span> <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Wegstreichen (was eine<\/span> <span class=\"ecti-1095\">der Eigenschaften von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">nicht<\/span> <span class=\"ecti-1095\">verwendet), womit sich <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ergibt.<\/span><\/p><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Der Quotient <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math><span class=\"ecti-1095\">kann nun als Definition<\/span> <span class=\"ecti-1095\">der rationalen Zahlen <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">angesehen<\/span> <span class=\"ecti-1095\">werden. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">schreibt man wie <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blich <\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Damit gilt nun die Gleichung<\/span> <\/p><table id=\"zc0dbc865d5b8\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mfrac><mrow><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat bezeichnen nach Definition beide Seiten <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen und die Gleichung gilt genau<\/span> <span class=\"ecti-1095\">wenn<\/span> <\/p> <table id=\"z2c3be99be83d\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">oder <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalenterweise <\/span><math display=\"inline\"><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt ist. Da letzteres gilt, erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen damit die so definierten rationalen Zahlen die <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen<\/span> <span class=\"ecti-1095\">Erweiterungs- und K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzungsregeln.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Des Weiteren l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">auffassen. In der Tat ist die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <\/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\">injektiv, denn die Gleichung <\/span><math display=\"inline\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">per Definition<\/span> <span class=\"ecti-1095\">genau dann erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt, wenn <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">ist. Wir identifizieren <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit dem Bild obiger Abbildung und schreiben insbesondere<\/span> <math display=\"inline\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen kurz zu allgemeinen Quotienten zur\u00fcckkehren, also sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> H\u00e4ufig will man eine Funktion auf <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> unter Verwendung der Elemente von <math display=\"inline\"><mi>X<\/mi><\/math> (also der Repr\u00e4sentanten der \u00c4quivalenzklassen) definieren. Zum Beispiel m\u00f6chten wir in obigem Beispiel in der Lage sein, zus\u00e4tzliche Abbildungen auf <math display=\"inline\"><mi>\u211a<\/mi><\/math> zu definieren (unter anderem die Addition und die Multiplikation). <\/p><p class=\"indent\">Konkreter, wenn <math display=\"inline\"><mi>Y<\/mi> <\/math> eine weitere Menge ist und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion ist, wollen wir m\u00f6glicherweise durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>X<\/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\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine Funktion definieren. Dies ist aber nur dann m\u00f6glich, wenn <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> (also <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math>) auch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> impliziert. In diesem Fall ist <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> unabh\u00e4ngig von der Wahl des Repr\u00e4sentanten <math display=\"inline\"><mi>x<\/mi><\/math> der \u00c4quivalenzklasse <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Also definiert dies in der Tat eine Funktion <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">,<\/span> die jedem Element <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> des Definitionsbereichs <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> ein eindeutig bestimmtes Element <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zuordnet. Wie bereits erw\u00e4hnt, sagen wir zur Betonung dieser (f\u00fcr Funktionen notwendiger) Eigenschaft, dass <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> <span class=\"ecbx-1095\">wohldefiniert <\/span>ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z11e432c05f84\"> <a id=\"x1-20044r72\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.72 <\/span>(Addition und Multiplikation auf den rationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir definieren nun zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzliche Strukturen auf<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Abbildungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/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 sind. Verifizieren Sie des Weiteren die Rechenregeln<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen in dieser Aufgabe zwar alle <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechenregeln und Eigenschaften von<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">verwenden, aber<\/span> <span class=\"ecti-1095\">nicht jene von <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">(da wir letztere ja definieren wollen).<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir zeigen am Beispiel der Addition, dass die Aufl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung bekannte Rechengesetze verwendet.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Dann gilt nach<\/span> <span class=\"ecti-1095\">Definition <\/span><math display=\"inline\"><mi>m<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>n<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>p<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir verwenden nun Erweitern (siehe Beispiel <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20043r71\"><span class=\"ecti-1095\">1.71<\/span><\/a><span class=\"ecti-1095\">), Umformungen des Z<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlers (welcher in<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">liegt und wo die <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechengesetze angenommen werden), K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzen, und erhalten<\/span> <span class=\"ecti-1095\">dadurch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>p<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><mi>q<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><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\">Dies zeigt, dass die definierte Abbildung <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">in der Tat wohldefiniert ist. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb3f2a2646496\"> <a id=\"x1-20045r73\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.73.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Angenommen wir wollen die reellen Zahlen<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <span class=\"ecti-1095\">Dezimalbruchentwicklungen definieren<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Es gibt auch mehrere andere M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten, siehe auch<\/span> <span class=\"ecti-1095\">Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"#x1-2960002\"><span class=\"ecti-1095\">A.2<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/span><\/span><span class=\"ecti-1095\">. Ebenso wollen wir auch noch, dass alle <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechenregeln erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sein sollten.<\/span> <span class=\"ecti-1095\">Dann muss man eine reelle Zahl als eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse von Dezimalbr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chen definieren. Der<\/span> <span class=\"ecti-1095\">Grund daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ist, dass auf Grund <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blicher Rechenmethoden<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und ebenso<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2217<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">da kein <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bertrag notwendig ist. Aber eigentlich sollte<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2217<\/mo> <mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">sein. Damit also unsere Rechenmethoden Sinn machen, muss also<\/span> <math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <\/math> <span class=\"ecti-1095\">identifiziert werden. Anders formuliert, muss also eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf der Menge der Dezimalbruchentwicklungen definiert werden, so dass<\/span> <math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math><span class=\"ecti-1095\">. Die reelle Zahl<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist dann eigentlich<\/span> <span class=\"ecti-1095\">die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-20046r20\"><\/a> \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.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z38b870205ca7\" class=\"sectionHead\"><span class=\"titlemark\">1.6 <\/span> <a id=\"x1-200006\"><\/a>\u00c4quivalenzrelationen<\/h3> <p class=\"noindent\">In diesem Abschnitt besprechen wir Relationen auf Mengen. Mit dem Begriff der \u00c4quivalenzrelation werden wir im Gegensatz zum vorherigen Abschnitt in der Lage sein, zum Beispiel die Menge der rationalen Zahlen formal korrekt aus den ganzen Zahlen (und mit etwas mehr Arbeit auch aus den nat\u00fcrlichen Zahlen) zu konstruieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb8237b94e49f\"> <a id=\"x1-20001r58\"><\/a> <span class=\"ecbx-1095\">Definition 1.58 <\/span>(Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><mi>Y<\/mi> <\/math> Mengen. Eine <span class=\"ecbx-1095\">Relation <\/span>auf <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> ist eine Teilmenge <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Wir schreiben auch <math display=\"inline\"><mi>x<\/mi><mi mathvariant=\"bold-script\">\u211b<\/mi><mi>y<\/mi><\/math> falls <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><\/math> und verwenden oft Symbole<a id=\"dx1-20002\"><\/a> wie <math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u226a<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u2261<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> f\u00fcr Relationen. Falls <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>Y<\/mi> <\/math> ist, dann sprechen wir auch von einer Relation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wenn <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> (resp. <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo> <\/math><\/span><span class=\"period\">,<\/span> \u2026) eine Relation ist, dann schreiben wir auch \u201e<span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo><mi>y<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> (resp. \u201e <span class=\"maendquote\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2247<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span>, \u2026) f\u00fcr \u201e <span class=\"maendquote\"><math display=\"inline\"><mi class=\"MathClass-op\">\u00ac<\/mi><mo> <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span> (resp. \u201e <span class=\"maendquote\"><math display=\"inline\"><mi class=\"MathClass-op\">\u00ac<\/mi><mo> <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">\u201c<\/span>, \u2026). <\/p> <\/div> <p class=\"indent\">Der Begriff der Relation umfasst viele verschiedene Beispiele, die in verschiedene Typen von Relation eingeteilt werden k\u00f6nnen. Um diese Allgemenheit zu erm\u00f6glichen, ist obige Definition eher abstrakt formuliert. Wir werden aber f\u00fcr den allgemeinen Begriff keinerlei theoretischen \u00dcberlegungen anstellen. Stattdessen wollen wir nur kurz einige bekannte Beispiele von Relation besprechen und anschliessend einen wichtigen speziellen Typ von Relationen genauer untersuchen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z569dce0ec59e\"> <a id=\"x1-20003r59\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.59 <\/span>(Beispiele von Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zum Beispiel sind <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">Relationen auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die wir mittels der Addition auf <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">folgendermassen definieren k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnten: Wir schreiben <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">falls es ein <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">gibt, und wir schreiben <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ebenso k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">&lt;<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">auch als Relationen auf<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auffassen. In der Tat<\/span> <span class=\"ecti-1095\">werden wir die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">mittels unserem Axiomensytem der reellen Zahlen im n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chsten Kapitel einf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren und dabei<\/span> <span class=\"ecti-1095\">nochmals ihre wichtigsten Eigenschaften besprechen. Im Sinne der obigen Definition sollten Sie<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">mit der Teilmenge<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20006r12\"><span class=\"ecti-1095\">1.12<\/span><\/a> <span class=\"ecti-1095\">identifizieren. Denn <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">genau dann wenn der Punkt auf oder links von der Diagonale liegt.<\/span> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=270&amp;url=Pictures\/Einfuehrung\/relation\/leq-relation.pdf\"><img decoding=\"async\" id=\"z8ee58dede116\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/leq-relation.svg\" width=\"270\" \/><\/div> <a id=\"x1-20006r12\"><\/a> <a id=\"x1-20007\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.12: <\/span><span class=\"content\">Dies stellt die <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math>-Relation               als Teilmenge von <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><\/math>                 dar.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Manchmal verwendet man auch eine Relation<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2248<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken soll, dass zwei Zahlen in etwa gleich sind. Um diese Relation<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Obwohl wir hier<\/span> <span class=\"ecti-1095\">ein schwammiges<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Ungef<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hr-Gleich<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">betrachten, ben<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">tigen wir eine pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">zise Definition um<\/span> <span class=\"ecti-1095\">dar<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber sprechen zu k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span><\/span><\/span> <span class=\"ecti-1095\">zu definieren, w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen wir eine fix gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlte positive Konstante<\/span> <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Mit dieser defineren<\/span> <span class=\"ecti-1095\">wir die Relation <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">(wobei<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">den Abstand<\/span> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">bestimmt). Als<\/span> <span class=\"ecti-1095\">Teilmenge von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist diese Relation ein Streifen rund um die Diagonale (siehe Figur<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20009r13\"><span class=\"ecti-1095\">1.13<\/span><\/a><span class=\"ecti-1095\">).<\/span> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=301&amp;url=Pictures\/Einfuehrung\/relation\/approx-relation.pdf\"><img decoding=\"async\" id=\"za27e81625359\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/approx-relation.svg\" width=\"301\" \/><\/div> <a id=\"x1-20009r13\"><\/a> <a id=\"x1-20010\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.13:             <\/span><span class=\"content\">Dies             stellt             die             Relation               <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math>                als                                                                             Teilmenge               <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211d<\/mi><\/math>                 dar,                                                                                 wobei               <math display=\"inline\"><mi>\u03b4<\/mi><\/math>             die vertikale (oder horizontale) halbe Breite des Streifens rund um die               Diagonale ist.                                                                            &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine beliebige Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math><span class=\"ecti-1095\">als eine Relation<\/span> <span class=\"ecti-1095\">auf der Potenzmenge <\/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\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">betrachten. Dieses Beispiel ist viel schwieriger zu visualisieren. Aber in dem Spezialfall der Menge<\/span> <math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">mit paarweise<\/span> <span class=\"ecti-1095\">verschiedenen Elemente <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>c<\/mi><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte man die<\/span> <span class=\"ecti-1095\">Inklusionsrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">wie in der Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20012r14\"><span class=\"ecti-1095\">1.14<\/span><\/a> <span class=\"ecti-1095\">visualisieren.<\/span> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=513&amp;url=Pictures\/Einfuehrung\/relation\/incl-relation.pdf\"><img decoding=\"async\" id=\"z7492065bb30c\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/incl-relation.svg\" width=\"513\" \/><\/div> <a id=\"x1-20012r14\"><\/a> <a id=\"x1-20013\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.14: <\/span><span class=\"content\">Hier wird die Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math>       auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               dargestellt: Ist <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>B<\/mi><\/math>             oder      existiert      eine      Pfeil-Kette      von      einer      Menge               <math display=\"inline\"><mi>A<\/mi><\/math>             links zu einer anderen Menge <math display=\"inline\"><mi>B<\/mi><\/math>             rechts (ohne Pfeile in gegengesetzter Richtung zu verwenden), so gilt               <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math><\/span><span class=\"period\">.<\/span> &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/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\">auch wie in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20014r15\"><span class=\"ecti-1095\">1.15<\/span><\/a> <span class=\"ecti-1095\">beschreiben.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=484&amp;url=Pictures\/Einfuehrung\/relation\/incl-relation-as-square.pdf\"><img decoding=\"async\" id=\"z15b2677c24c9\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/incl-relation-as-square.svg\" width=\"484\" \/><\/div> <a id=\"x1-20014r15\"><\/a> <a id=\"x1-20015\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.15:         <\/span><span class=\"content\">Dies         stellt         ebenso         die         Relation               <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2286<\/mo><\/math>       auf               <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               dar.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z60165fbeeff9\"> <a id=\"x1-20016r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.60 <\/span>(Eine bekannte Relation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In einem gewissen Sinne haben wir bereits eine gewisse wichtige Klasse von Relationen betrachtet. Seien<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">Mengen und<\/span> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math><span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Relation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die folgende Eigenschaft erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt:<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">!<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><mi>y<\/mi><\/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\">Wie nennen wir eine solche Relation gemeinsam mit<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">? Welches Symbol verwenden<\/span> <span class=\"ecti-1095\">wir statt dem Symbol <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math> <span class=\"ecti-1095\">in diesem Zusammenhang?<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Eine derartige Relationen entspricht einer Abbildung von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>Y<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">ist der Graph und<\/span> <span class=\"ecti-1095\">wir schreiben meist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u21a6<\/mo><\/math> <span class=\"ecti-1095\">statt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca2<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr den Rest dieses Abschnittes wollen wir uns aber vorwiegend mit folgendem wichtigen Typ von Relationen besch\u00e4ftigen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zcc9e0c4d584c\"> <a id=\"x1-20017r61\"><\/a> <span class=\"ecbx-1095\">Definition 1.61 <\/span>(\u00c4quivalenzrelationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> auf <math display=\"inline\"><mi>X<\/mi><\/math> ist eine <span class=\"ecbx-1095\">\u00c4<\/span><span class=\"ecbx-1095\">quivalenzrelation<\/span>, falls folgende drei Eigenschaften erf\u00fcllt sind: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Reflexivit\u00e4t: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Symmetrie: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Transitivit\u00e4t: <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <p class=\"indent\">\u00c4quivalenzrelationen sind oft durch eine Gleichheit in gewissen Aspekten definiert und sollten als eine Form von einer Gleichheit angesehen werden. In der Tat bezieht sich der Ausdruck \u201edas Gleiche\u201c in der deutschen Sprache auf eine Art \u201e\u00c4quivalenzrelation\u201c (die je nach Zusammenhang verschieden sein kann), wohingegen der Ausdruck \u201edasselbe\u201c nur f\u00fcr \u201eein und dasselbe\u201c Objekt verwendet werden sollte. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf4ee98ed6714\"> <a id=\"x1-20018r62\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.62 <\/span>(Beispiele von \u00c4quivalenzrelationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Das einfachste Beispiel einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf einer beliebigen Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist die Gleichheit, also die Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>X<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ein weiteres allgemeines Beispiel ist die sogenannte triviale <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>X<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der je zwei Elemente in <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent sind.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir betrachten ein Beispiel in der ebenen (euklidschen) Geometrie. Sei<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Geraden in der Ebene. Zu zwei Geraden<\/span> <math display=\"inline\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">schreiben wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>G<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;sind&nbsp;parallel<\/mtext><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies definiert eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">(wieso?).<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><a id=\"x1-200224\"><\/a><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">die Relation, die durch Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a> <span class=\"ecti-1095\">definiert wird. Wir behaupten, dass<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">darstellt.<\/span> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=365&amp;url=Pictures\/Einfuehrung\/relation\/number-relation.pdf\"><img decoding=\"async\" id=\"z026bff2f0505\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Einfuehrung\/relation\/number-relation.svg\" width=\"365\" \/><\/div> <a id=\"x1-20023r16\"><\/a> <a id=\"x1-20024\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;1.16:           <\/span><span class=\"content\">Wir           definieren           eine           Relation               <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math>       als                                    Teilmenge                                    von               <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math>             und behaupten, dass diese sogar eine \u00c4quivalenzrelation auf der Menge               <math display=\"inline\"><mi>X<\/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-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math>       darstellt.                                                                                  &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat entspricht der Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">der Aussage, dass<\/span> <span class=\"ecti-1095\">die Diagonale <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">in der Relation enthalten ist. Ebenso entspricht der Symmetrie von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">der Aussage, dass die Teilmenge bei Spiegelung um die Diagonale (bei Vertauschung der beiden<\/span> <span class=\"ecti-1095\">Koordinaten) unver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert bleibt. Beides ist in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t hat keine<\/span> <span class=\"ecti-1095\">derartige einfache visuelle Interpretation. Stattdessen sehen wir in Figur <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20023r16\"><span class=\"ecti-1095\">1.16<\/span><\/a><span class=\"ecti-1095\">, dass<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Ebenso<\/span> <span class=\"ecti-1095\">gilt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">aber<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2241<\/mo> <mn>6<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mn>6<\/mn><mo class=\"MathClass-rel\">\u2241<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><span class=\"ecti-1095\">. Und<\/span> <span class=\"ecti-1095\">schlussendlich gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>6<\/mn> <mo class=\"MathClass-rel\">\u223c<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Aus dieser Information ergibt sich nun die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t durch Fallunterscheidung: Angenommen<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">so gilt dies<\/span> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>z<\/mi><\/math><span class=\"ecti-1095\">, womit damit<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">so gilt dies<\/span> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Falls aber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>6<\/mn><\/math> <span class=\"ecti-1095\">und damit<\/span> <span class=\"ecti-1095\">ebenso <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen werden in anderen Wissenschaften oder auch im Alltag h<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufig verwendet.<\/span> <span class=\"ecti-1095\">Beispielsweise kann man auf der Menge aller Lebewesen eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation definieren, in<\/span> <span class=\"ecti-1095\">dem man zwei Lebewesen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rt, wenn sie zur selben Art (oder Gattung oder<\/span> <span class=\"ecti-1095\">Familie) geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5c05ec20b472\"> <a id=\"x1-20026r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.63 <\/span>(Zwei weitere Beispiele)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bungen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zwei weitere Beispiele von <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen besprechen. Sei<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Menge.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Quetschen einer Teilmenge) Sei <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge. Zeigen Sie, dass die Relation gegeben durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>A<\/mi><\/mrow><\/msub><mi>y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/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\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definiert.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenz <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber eine Abbildung) Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Abbildung in eine<\/span> <span class=\"ecti-1095\">weitere Menge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>Y<\/mi> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">definieren eine Relation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definiert.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7f30e516bae7\"> <a id=\"x1-20029r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.64 <\/span>(Ein falscher Beweis)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser Aufgabe behaupten wir f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lschlicherweise, dass jede symmetrische und transitive Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf einer<\/span> <span class=\"ecti-1095\">Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">auch reflexiv ist (d.h. eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist). Finden Sie den Fehler in folgendem<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Beweis<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">:<\/span> <\/p><blockquote class=\"quote\"> <p class=\"noindent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ein Element. Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wegen Symmetrie der Relation gilt also auch <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Folglich gilt unter Verwendung der Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t der Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2227<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was zu zeigen war.<\/span><\/p><\/blockquote> <p class=\"noindent\"><span class=\"ecti-1095\">Finden Sie ein Beispiel einer Relation, die symmetrisch und transitiv, aber nicht reflexiv<\/span> <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\">Die Aussage ist auf jeden Fall falsch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">leere Relation<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">, welche<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2241<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7d7899f12eb0\"> <a id=\"x1-20030r65\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.65 <\/span>(Beispiele allgemeiner Relationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie eine Relation (auf <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">oder anderen Mengen), die von den Eigenschaften einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/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\">nur die Symmetrie,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">nur die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">die Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t und Symmetrie, aber nicht die Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">die Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t und Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t, aber nicht die Symmetrie<\/span><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/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\">Bestimmen Sie die Eigenschaften der Relationen in Beispiel <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20003r59\"><span class=\"ecti-1095\">1.59<\/span><\/a> <span class=\"ecti-1095\">und auch von<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2260<\/mo><\/math><span class=\"ecti-1095\">. Insbesondere ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes fest gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlte <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Relation <\/span><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">keine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">da <\/span><math display=\"inline\"><mn>0<\/mn> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>\u03b4<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u2248<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mn>2<\/mn><mi>\u03b4<\/mi><\/math> <span class=\"ecti-1095\">aber<\/span> <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><msub><mrow><mo class=\"MathClass-rel\">\u2249<\/mo> <\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mn>2<\/mn><mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wie schon erw\u00e4hnt ist eine \u00c4quivalenzrelation gewissermassen eine Form von Gleichheit. Dies l\u00e4sst sich auch formalisieren: <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3b4b1d100629\"> <a id=\"x1-20031r66\"><\/a> <span class=\"ecbx-1095\">Definition 1.66 <\/span>(\u00c4quivalenzklassen und die Quotientenmenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf einer Menge <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann wird f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> die Menge                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/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> <p class=\"noindent\">die <span class=\"ecbx-1095\">\u00c4<\/span><span class=\"ecbx-1095\">quivalenzklasse<\/span><a id=\"dx1-20032\"><\/a> von <math display=\"inline\"><mi>x<\/mi><\/math> genannt. Weiters ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>X<\/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\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/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> <p class=\"noindent\">der <span class=\"ecbx-1095\">Quotient <\/span>(oder die <span class=\"ecbx-1095\">Quotientenmenge<\/span>)<a id=\"dx1-20033\"><\/a> von <math display=\"inline\"><mi>X<\/mi><\/math> modulo <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> Ein Element <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> wird auch <span class=\"ecbx-1095\">Repr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">sentant<\/span> seiner \u00c4quivalenzklasse <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> genannt. <\/p> <\/div> <p class=\"indent\">Anschaulich gesprochen geben wir \u00e4quivalente Elemente von <math display=\"inline\"><mi>X<\/mi><\/math> in ein und denselben Topf und nicht \u00e4quivalente Elemente in verschiedene T\u00f6pfe. In diesem Bild besteht die \u00c4quivalenzklasse eines Elements aus allen Elementen, die im gleichen Topf sind. Der Quotient modulo <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> wiederum ist die Menge der T\u00f6pfe. In Beispiel <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20018r62\">1.62<\/a>(<a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200224\">iv<\/a>) hatten wir bereits implizit die drei \u00c4quivalenzklassen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>4<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>5<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>6<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> in unserer Diskussion verwendet. <\/p><p class=\"indent\">Die Begriffe der \u00c4quivalenzrelation und der Partition in Definition <a href=\"..\/..\/chapter\/mengenlehre-und-abbildungen#x1-16001r51\">1.51<\/a> sind auf folgende Weise eng verwandt.                                                                                                                                                                           <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd40376e790d6\"> <a id=\"x1-20034r67\"><\/a> <span class=\"ecbx-1095\">Proposition 1.67 <\/span>(Korrespondenz zwischen \u00c4quivalenzrelationen und Partitionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge. Dann entsprechen <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelationen auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und Partitionen<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">einander im folgenden Sinne: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine gegebene <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist die<\/span> <span class=\"ecti-1095\">Menge<\/span><a id=\"dx1-20035\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/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> <p class=\"noindent\"><span class=\"ecti-1095\">eine Partition von <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Umgekehrt<\/span> <span class=\"ecti-1095\">definiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Partition <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>X<\/mi><\/math><a id=\"dx1-20036\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi> <mo class=\"MathClass-bin\">\u2227<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><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\">eine<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Des Weiteren sind die Konstruktion der Partition aus der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation und die<\/span> <span class=\"ecti-1095\">Konstruktion der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation aus der Partition zueinander invers: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Partition<\/span> <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede<\/span> <span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge, <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> wie in Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a> definiert. Selbstverst\u00e4ndlich gilt nach den Definitionen eigentlich <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten jedoch zwei verschiedene Symbole mitf\u00fchren, da wir <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> und <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> jeweils verschieden interpretieren m\u00f6chten. <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> werden wir stets als eine Kollektion von Teilmengen von <math display=\"inline\"><mi>X<\/mi><\/math> auffassen; <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> hingegen werden wir als einen neuen Raum erachten, wo die Punkte durch Identifikation (\u201eAneinanderkleben\u201c) von gewissen Punkten in <math display=\"inline\"><mi>X<\/mi><\/math> entstanden sind. (Die Punkte in <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> sind die Teilmengen von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> die in <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> enthalten sind). <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0168eec249db\"> <a id=\"x1-20037r68\"><\/a> <span class=\"ecbx-1095\">Applet 1.68 <\/span>(Eine \u00c4quivalenzrelation und deren Quotient)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"412px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/n2vtpjrw\/width\/688\/height\/412\/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\">Links wird eine Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">partitioniert, was einer <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">entspricht. Rechts betrachten wir die Menge der <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen, also den Quotienten von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">bz<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Menge links k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte eine abstrakte Menge darstellen oder den Einheitskreis. Im letzteren<\/span> <span class=\"ecti-1095\">Fall muss klar definiert sein, zu welcher Menge die Punkte der Kanten, bei denen der Kreis<\/span> <span class=\"ecti-1095\">unterteilt wird, geh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren. Wir haben dies im Beispiel mit Farben angedeutet.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za1abeecf416a\"> <a id=\"x1-20038r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.69 <\/span>(Zwei Quotienten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Charakterisieren Sie die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen der beiden in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20026r63\"><span class=\"ecti-1095\">1.63<\/span><\/a> <span class=\"ecti-1095\">(ii) definierten <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie jeweils, dass <\/span><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(wie in obiger Proposition definiert) eine Partition ist (ohne auf die noch nicht-bewiesene<\/span> <span class=\"ecti-1095\">Proposition zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckzugreifen) und beschreiben Sie <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">intuitiv.<\/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\">In            (i)            werden            alle            Punkte            in<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">miteinander                 identifiziert                 und                 man                 kann<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">als<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>A<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">auffassen.                    In                    diesem                    Sinne                    wird<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">zu einem         Punkt         gequetscht.         In         (ii)         k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen         wir<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">{<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in Beziehung    bringen    und    auf    diese    Weise    eine    kanonische    Bijektion    von<\/span> <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definieren. <\/span><\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr den Beweis der Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a> ziehen wir folgende Behauptung vor: <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2f4fc8da73eb\"> <span class=\"ecbx-1095\">Behauptung.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math><span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Dann sind folgende<\/span> <span class=\"ecti-1095\">drei Aussagen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/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\">(paarweise) <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis der Behauptung.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir beweisen die Implikationen <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> woraus folgt, dass alle drei Aussagen \u00e4quivalent sind. Seien <span class=\"maperiod\"><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><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir nehmen also zuerst an, dass <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> gilt wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Sei <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Dann ist <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> und also <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wegen der Transitivit\u00e4t. Die andere Inklusion folgt analog (durch Vertauschen von <math display=\"inline\"><mi>x<\/mi><\/math> und <math display=\"inline\"><mi>y<\/mi><\/math>) und es gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so folgt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> wie in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>                                                                                                                                                                           wegen Reflexivit\u00e4t. <\/p><p class=\"indent\">Gilt <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so existiert ein <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>x<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Aus Symmetrie und Transitivit\u00e4t folgt daher <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/math> wie in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Proposition <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20034r67\">1.67<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Sei <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><mi>P<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>X<\/mi><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> da <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Paarweise Disjunktheit der Elemente von <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> gilt dank der Behauptung und es folgt, dass <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> eine Partition ist. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> eine Partition von <math display=\"inline\"><mi>X<\/mi><\/math> und sei die Relation <math display=\"inline\"><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/math> wie in der Proposition definiert. Es ist f\u00fcr jedes <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> auch <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> da es wegen <math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>P<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>X<\/mi><\/math> ein <math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">;<\/span> dies zeigt Reflexivit\u00e4t. Falls <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><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><span class=\"period\">,<\/span> dann folgt <math display=\"inline\"><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>x<\/mi><\/math> direkt aus der Definition, das heisst <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> ist symmetrisch. Angenommen es gilt <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>y<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein Partitionselement <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> mit <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und daher <math display=\"inline\"><msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> nach den Eigenschaften der Partition. Es folgt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>P<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <mi>z<\/mi><\/math> und die Transitivit\u00e4t                                                                                                                                                                           der Relation <span class=\"maperiod\"><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Daher ist <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> eine \u00c4quivalenzrelation. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> ist die \u00c4quivalenzklasse bez\u00fcglich <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/math> gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mo> \u22c3<\/mo> <\/mrow><mrow><mi>P<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><mo class=\"MathClass-bin\">\u2227<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>P<\/mi> <\/mrow><\/munder><mi>P<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber die Elemente von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> paarweise disjunkt sind, kann <math display=\"inline\"><mi>x<\/mi><\/math> nur in einem Element enthalten sein. Insbesondere folgt, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> das eindeutig bestimmte Element der Partition <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> ist, das <math display=\"inline\"><mi>x<\/mi><\/math> enth\u00e4lt, und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <math display=\"inline\"><mi>P<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">,<\/span> so gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>P<\/mi><\/math><\/span><span class=\"period\">,<\/span> also <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Ist umgekehrt <math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <math display=\"inline\"><mi>X<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <\/math> die entsprechende Partition, dann gilt f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u223c<\/mo> <mi>y<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach der Definition von <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><mrow><msub><mrow><mi mathvariant=\"bold-script\">\ud835\udcab<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mrow><\/msub><\/math> und der Behauptung. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Die folgende \u00dcbung zeigt, dass man sich jede \u00c4quivalenzrelation bildlich wie eine disjunkte Vereinigung von Quadraten vorstellen kann, die wie in Beispiel <a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20018r62\">1.62<\/a>(<a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-200224\">iv<\/a>) entlang der Diagonale angeordnet in <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math> liegen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z087392517b26\"> <a id=\"x1-20042r70\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.70.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Relation <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\"> \u22c3<\/mi><mo> <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>X<\/mi><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben ist. Zeigen Sie auch, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/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\">entweder <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">oder <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2205<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Man verwendet Quotienten modulo \u00c4quivalenzrelationen in der Mathematik oft f\u00fcr die Konstruktion von gewissen R\u00e4umen und auch von neuen Zahlenmengen. Wir betrachten zu letzterem ein einfaches und grundlegendes Beispiel: Wir konstruieren die rationalen Zahlen aus den ganzen Zahlen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za061ee32cf1e\"> <a id=\"x1-20043r71\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.71 <\/span>(Konstruktion der rationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen       an,       dass       wir       bereits       die       ganzen       Zahlen<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und die              Addition              und              Multiplikation              auf<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit allen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen  Eigenschaften  kennen.  Wir  wollen  damit  die  rationalen  Zahlen<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">definieren,                           wobei                           ein                           Element<\/span> <math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">als                          <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse                          des                          Tupels<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definiert sein wird.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Zu diesem Zwecke betrachten wir die Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Diese<\/span> <span class=\"ecti-1095\">Definition r<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt daher, dass wir eben die rationale Zahlen als Br<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">che von ganzen Zahlen auffassen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten.<\/span> <span class=\"ecti-1095\">Dabei m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen wir allerdings solche identifizieren, die<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">nach K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzen<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">gleich sind; zum Beispiel sollte gelten<\/span> <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><mn>0<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>5<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Allerdings wollen<\/span> <span class=\"ecti-1095\">wir hier davon ausgehen, dass wir die rationalen Zahlen noch nicht kennen, weswegen wir anstatt Gleichungen der<\/span> <span class=\"ecti-1095\">Form <\/span><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">Gleichungen der<\/span> <span class=\"ecti-1095\">Form <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit Ausdr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken<\/span> <span class=\"ecti-1095\">innerhalb von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">betrachten<\/span> <span class=\"ecti-1095\">(im Beispiel <\/span><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>5<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>6<\/mn><\/math><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Nun verifzieren wir, dass obige Relation tats<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chlich eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist. Seien dazu<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/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\">Reflexivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Symmetrie: Angenommen es gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist per Definition also <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und daher auch <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Transitivit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t: <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ergibt <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Durch Multiplikation der ersten Gleichungen mit<\/span> <math display=\"inline\"><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und der zweiten<\/span> <span class=\"ecti-1095\">Gleichung mit <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erhalten wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht Null ist, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir in obiger Gleichung<\/span> <math display=\"inline\"><msub><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Wegstreichen (was eine<\/span> <span class=\"ecti-1095\">der Eigenschaften von <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">nicht<\/span> <span class=\"ecti-1095\">verwendet), womit sich <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und damit <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>q<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ergibt.<\/span><\/p><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Der Quotient <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math><span class=\"ecti-1095\">kann nun als Definition<\/span> <span class=\"ecti-1095\">der rationalen Zahlen <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">angesehen<\/span> <span class=\"ecti-1095\">werden. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">schreibt man wie <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blich <\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Damit gilt nun die Gleichung<\/span> <\/p><table id=\"zc0dbc865d5b8\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mfrac><mrow><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat bezeichnen nach Definition beide Seiten <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklassen und die Gleichung gilt genau<\/span> <span class=\"ecti-1095\">wenn<\/span> <\/p> <table id=\"z2c3be99be83d\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">oder <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalenterweise <\/span><math display=\"inline\"><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>q<\/mi><msub><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt ist. Da letzteres gilt, erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen damit die so definierten rationalen Zahlen die <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen<\/span> <span class=\"ecti-1095\">Erweiterungs- und K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzungsregeln.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Des Weiteren l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">als Teilmenge von <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">auffassen. In der Tat ist die Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>m<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <\/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\">injektiv, denn die Gleichung <\/span><math display=\"inline\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">per Definition<\/span> <span class=\"ecti-1095\">genau dann erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt, wenn <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">ist. Wir identifizieren <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit dem Bild obiger Abbildung und schreiben insbesondere<\/span> <math display=\"inline\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen kurz zu allgemeinen Quotienten zur\u00fcckkehren, also sei <math display=\"inline\"><mi>X<\/mi><\/math> eine Menge und <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> eine \u00c4quivalenzrelation auf <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> H\u00e4ufig will man eine Funktion auf <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> unter Verwendung der Elemente von <math display=\"inline\"><mi>X<\/mi><\/math> (also der Repr\u00e4sentanten der \u00c4quivalenzklassen) definieren. Zum Beispiel m\u00f6chten wir in obigem Beispiel in der Lage sein, zus\u00e4tzliche Abbildungen auf <math display=\"inline\"><mi>\u211a<\/mi><\/math> zu definieren (unter anderem die Addition und die Multiplikation). <\/p><p class=\"indent\">Konkreter, wenn <math display=\"inline\"><mi>Y<\/mi> <\/math> eine weitere Menge ist und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion ist, wollen wir m\u00f6glicherweise durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mi>X<\/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\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine Funktion definieren. Dies ist aber nur dann m\u00f6glich, wenn <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> (also <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math>) auch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> impliziert. In diesem Fall ist <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> unabh\u00e4ngig von der Wahl des Repr\u00e4sentanten <math display=\"inline\"><mi>x<\/mi><\/math> der \u00c4quivalenzklasse <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Also definiert dies in der Tat eine Funktion <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">,<\/span> die jedem Element <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math> des Definitionsbereichs <math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> ein eindeutig bestimmtes Element <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zuordnet. Wie bereits erw\u00e4hnt, sagen wir zur Betonung dieser (f\u00fcr Funktionen notwendiger) Eigenschaft, dass <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <\/math> <span class=\"ecbx-1095\">wohldefiniert <\/span>ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z11e432c05f84\"> <a id=\"x1-20044r72\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 1.72 <\/span>(Addition und Multiplikation auf den rationalen Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir definieren nun zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzliche Strukturen auf<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Abbildungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/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 sind. Verifizieren Sie des Weiteren die Rechenregeln<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen in dieser Aufgabe zwar alle <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechenregeln und Eigenschaften von<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">verwenden, aber<\/span> <span class=\"ecti-1095\">nicht jene von <\/span><math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">(da wir letztere ja definieren wollen).<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir zeigen am Beispiel der Addition, dass die Aufl<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung bekannte Rechengesetze verwendet.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">,<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Dann gilt nach<\/span> <span class=\"ecti-1095\">Definition <\/span><math display=\"inline\"><mi>m<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>n<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>p<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir verwenden nun Erweitern (siehe Beispiel <\/span><a href=\"..\/..\/chapter\/aequivalenzrelationen#x1-20043r71\"><span class=\"ecti-1095\">1.71<\/span><\/a><span class=\"ecti-1095\">), Umformungen des Z<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlers (welcher in<\/span> <math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">liegt und wo die <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechengesetze angenommen werden), K<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rzen, und erhalten<\/span> <span class=\"ecti-1095\">dadurch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mi>q<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>m<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>p<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>q<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><mi>q<\/mi><\/mrow> <mrow><mi>n<\/mi><mi>q<\/mi><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><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\">Dies zeigt, dass die definierte Abbildung <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">in der Tat wohldefiniert ist. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb3f2a2646496\"> <a id=\"x1-20045r73\"><\/a> <span class=\"ecbx-1095\">Beispiel 1.73.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Angenommen wir wollen die reellen Zahlen<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mittels<\/span> <span class=\"ecti-1095\">Dezimalbruchentwicklungen definieren<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Es gibt auch mehrere andere M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten, siehe auch<\/span> <span class=\"ecti-1095\">Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"#x1-2960002\"><span class=\"ecti-1095\">A.2<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/span><\/span><span class=\"ecti-1095\">. Ebenso wollen wir auch noch, dass alle <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blichen Rechenregeln erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sein sollten.<\/span> <span class=\"ecti-1095\">Dann muss man eine reelle Zahl als eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse von Dezimalbr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chen definieren. Der<\/span> <span class=\"ecti-1095\">Grund daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ist, dass auf Grund <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">blicher Rechenmethoden<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und ebenso<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mn>3<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2217<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">da kein <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bertrag notwendig ist. Aber eigentlich sollte<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2217<\/mo> <mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">sein. Damit also unsere Rechenmethoden Sinn machen, muss also<\/span> <math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <\/math> <span class=\"ecti-1095\">identifiziert werden. Anders formuliert, muss also eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">auf der Menge der Dezimalbruchentwicklungen definiert werden, so dass<\/span> <math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">\u223c<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math><span class=\"ecti-1095\">. Die reelle Zahl<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist dann eigentlich<\/span> <span class=\"ecti-1095\">die <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzklasse <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mn>0<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mn>9<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <a id=\"x1-20046r20\"><\/a> \n","protected":false},"author":1089,"menu_order":6,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-29","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/29","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\/29\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/29\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=29"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=29"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=29"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}