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(de)"> <link rel="EditURI" type="application/rsd+xml" href="//de.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://de.wikipedia.org/wiki/Kommutativgesetz"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de"> <link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Kommutativgesetz rootpage-Kommutativgesetz skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kommutativgesetz</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Commutativity_of_binary_operations_(without_question_mark).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Commutativity_of_binary_operations_%28without_question_mark%29.svg/220px-Commutativity_of_binary_operations_%28without_question_mark%29.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Commutativity_of_binary_operations_%28without_question_mark%29.svg/330px-Commutativity_of_binary_operations_%28without_question_mark%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Commutativity_of_binary_operations_%28without_question_mark%29.svg/440px-Commutativity_of_binary_operations_%28without_question_mark%29.svg.png 2x" data-file-width="248" data-file-height="142" /></a><figcaption>Eine Verknüpfung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> ist kommutativ, wenn <i>stets</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\circ y=y\circ x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∘</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y=y\circ x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2578d2ab8b7ab36781793b5484c664d7f283e93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.458ex; height:2.009ex;" alt="{\displaystyle x\circ y=y\circ x}"></span> gilt. In dieser Abbildung wird die Vorstellung einer Operation als Maschine genutzt, die aus zwei Eingaben ein Ergebnis macht. Wenn die Verknüpfung kommutativ ist, dann ist es egal, in welcher Reihenfolge die Eingaben <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> auftreten – das Ergebnis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\circ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ba86902ee98c41deb1275ddb8693977f27e1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.68ex; height:2.009ex;" alt="{\displaystyle x\circ y}"></span> ist dasselbe wie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\circ x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∘</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\circ x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd2f4a92fab96a47269e979f0659f85ee538a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.68ex; height:2.009ex;" alt="{\displaystyle y\circ x}"></span>.</figcaption></figure> <p>Das <b>Kommutativgesetz</b> (<span style="font-style:normal;font-weight:normal"><a href="/wiki/Latein" title="Latein">lateinisch</a></span> <span lang="la-Latn" style="font-style:italic">commutare</span> <span lang="de" style="font-style:normal;font-weight:normal">‚vertauschen‘</span>), auf Deutsch <i>Vertauschungsgesetz,</i> ist eine Regel aus der <a href="/wiki/Mathematik" title="Mathematik">Mathematik</a>. Wenn sie gilt, können die <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Argumente</a> einer <a href="/wiki/Operator_(Mathematik)" title="Operator (Mathematik)">Operation</a> vertauscht werden, ohne dass sich das Ergebnis verändert. Mathematische Operationen, die dem Kommutativgesetz unterliegen, nennt man <i>kommutativ.</i> </p><p>Das Kommutativgesetz bildet mit dem <a href="/wiki/Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetz</a> und dem <a href="/wiki/Distributivgesetz" title="Distributivgesetz">Distributivgesetz</a> grundlegende Regeln der <a href="/wiki/Algebra" title="Algebra">Algebra</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Formale_Definition"><span class="tocnumber">1</span> <span class="toctext">Formale Definition</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Beispiele_und_Gegenbeispiele"><span class="tocnumber">2</span> <span class="toctext">Beispiele und Gegenbeispiele</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Reelle_Zahlen"><span class="tocnumber">2.1</span> <span class="toctext">Reelle Zahlen</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Skalarprodukte"><span class="tocnumber">2.2</span> <span class="toctext">Skalarprodukte</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Mengenoperation"><span class="tocnumber">2.3</span> <span class="toctext">Mengenoperation</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Matrizenrechnung"><span class="tocnumber">2.4</span> <span class="toctext">Matrizenrechnung</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Gruppentheorie"><span class="tocnumber">2.5</span> <span class="toctext">Gruppentheorie</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Aussagenlogik"><span class="tocnumber">2.6</span> <span class="toctext">Aussagenlogik</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Weitere_Beispiele"><span class="tocnumber">2.7</span> <span class="toctext">Weitere Beispiele</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Antikommutativität"><span class="tocnumber">3</span> <span class="toctext">Antikommutativität</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Anmerkungen"><span class="tocnumber">4</span> <span class="toctext">Anmerkungen</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Siehe_auch"><span class="tocnumber">5</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Literatur"><span class="tocnumber">6</span> <span class="toctext">Literatur</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Formale_Definition">Formale Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=1" title="Abschnitt bearbeiten: Formale Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Formale Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es seien <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> Mengen. Eine <a href="/wiki/Bin%C3%A4re_Verkn%C3%BCpfung" class="mw-redirect" title="Binäre Verknüpfung">binäre Verknüpfung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\colon A\times A\to X,\;(a,b)\mapsto a*b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mo>:</mo> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\colon A\times A\to X,\;(a,b)\mapsto a*b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0709efd7b4a68d891bbbe2b8b3db0f78f959dc5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.903ex; height:2.843ex;" alt="{\displaystyle *\colon A\times A\to X,\;(a,b)\mapsto a*b}"></span> heißt kommutativ, wenn für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923af80638e57592b33fa715fb6c23651729fb73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.845ex; height:2.509ex;" alt="{\displaystyle a,b\in A}"></span> die Gleichheit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a*b=b*a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>∗</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b=b*a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7a1235743322cc522c4ecddbc3bb6ca354eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.943ex; height:2.176ex;" alt="{\displaystyle a*b=b*a}"></span> gilt. </p> <div class="mw-heading mw-heading2"><h2 id="Beispiele_und_Gegenbeispiele">Beispiele und Gegenbeispiele</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=2" title="Abschnitt bearbeiten: Beispiele und Gegenbeispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Beispiele und Gegenbeispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Vector_Addition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/220px-Vector_Addition.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/330px-Vector_Addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/440px-Vector_Addition.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Die Vektoraddition ist kommutativ, weil <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2b7482da2847013fa5de8900757c561b98c815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.427ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}"></span> ist.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Reelle_Zahlen">Reelle Zahlen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=3" title="Abschnitt bearbeiten: Reelle Zahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Reelle Zahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Commutative_Addition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Commutative_Addition.svg/220px-Commutative_Addition.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Commutative_Addition.svg/330px-Commutative_Addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Commutative_Addition.svg/440px-Commutative_Addition.svg.png 2x" data-file-width="600" data-file-height="300" /></a><figcaption>Die Addition natürlicher Zahlen ist kommutativ.</figcaption></figure> <p>Für <a href="/wiki/Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reelle Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span> gilt stets </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7dede7493e0231b3ad6ff9b54f4eae954108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.911ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a}"></span>,</dd></dl> <p>die Operationen <a href="/wiki/Addition" title="Addition">Addition</a> und <a href="/wiki/Multiplikation" title="Multiplikation">Multiplikation</a> sind also kommutativ. Die erste Formel wird auch <i>Kommutativgesetz der Addition,</i> die zweite <i>Kommutativgesetz der Multiplikation</i> genannt. Die <a href="/wiki/Subtraktion" title="Subtraktion">Subtraktion</a> und die <a href="/wiki/Division_(Mathematik)" title="Division (Mathematik)">Division</a> reeller Zahlen sind dagegen keine kommutativen Operationen. Auch die <a href="/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenzierung</a> ist nicht kommutativ (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}\neq 3^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≠</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}\neq 3^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170cfbfd71bd6828dbe7fb72a9eb6f135814693e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.532ex; height:3.176ex;" alt="{\displaystyle 2^{3}\neq 3^{2}}"></span> ist ein Gegenbeispiel). </p><p>Die älteste überlieferte Form des Kommutativgesetzes der Addition ist die sumerische <i><a href="/wiki/Fabel_vom_klugen_Wolf_und_den_neun_dummen_W%C3%B6lfen" title="Fabel vom klugen Wolf und den neun dummen Wölfen">Fabel vom klugen Wolf und den neun dummen Wölfen</a></i>. </p> <div class="mw-heading mw-heading3"><h3 id="Skalarprodukte">Skalarprodukte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=4" title="Abschnitt bearbeiten: Skalarprodukte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Skalarprodukte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Das <a href="/wiki/Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> in einem reellen <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a> ist kommutativ, es gilt also stets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle a,b\rangle =\langle b,a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle a,b\rangle =\langle b,a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1055277b7d0952c68369d4e0c0189b0a571419ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.24ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle =\langle b,a\rangle }"></span>.</li> <li>Das Skalarprodukt in einem komplexen Vektorraum ist dagegen nicht kommutativ, es gilt vielmehr <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle a,b\rangle ={\overline {\langle b,a\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle a,b\rangle ={\overline {\langle b,a\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dcf05f5e3ce517f3ec40f1c984cb35e3a4e02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.355ex; height:3.676ex;" alt="{\displaystyle \langle a,b\rangle ={\overline {\langle b,a\rangle }}}"></span>, wobei der Überstrich die <a href="/wiki/Komplexe_Konjugation" title="Komplexe Konjugation">komplexe Konjugation</a> bezeichnet.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Mengenoperation">Mengenoperation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=5" title="Abschnitt bearbeiten: Mengenoperation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Mengenoperation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Mengenlehre" title="Mengenlehre">Mengenlehre</a> sind die <a href="/wiki/Vereinigung_(Mengenlehre)" class="mw-redirect" title="Vereinigung (Mengenlehre)">Vereinigung</a> und der <a href="/wiki/Schnittmenge" class="mw-redirect" title="Schnittmenge">Schnitt</a> kommutative Operationen; für Mengen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span> gilt also stets: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B=B\cup A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mo>∪</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B=B\cup A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fdb5dfc00d1e5850310af370e406c817267287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.278ex; height:2.176ex;" alt="{\displaystyle A\cup B=B\cup A}"></span> (Vereinigung)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=B\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B=B\cap A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9192707b022c63da4833c865e2ce5c7ced05860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.278ex; height:2.176ex;" alt="{\displaystyle A\cap B=B\cap A}"></span> (Schnitt)</dd></dl> <p>Dagegen ist die <a href="/wiki/Differenzmenge" class="mw-redirect" title="Differenzmenge">Differenz</a> nicht kommutativ. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef797ed5deb971321592e34281d9fac27c3249d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle A\setminus B}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\setminus A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\setminus A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9fc9ff1d2b0248677aed6da1a89025a859476c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle B\setminus A}"></span> sind also manchmal verschiedene Mengen, z. B. für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908d8e94bd40754d0635a5292896b33d9523b83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.525ex; height:2.843ex;" alt="{\displaystyle A=\{1,2\}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c6ea8fb0140ce94e7f8183a7b2618d0718a0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.35ex; height:2.843ex;" alt="{\displaystyle B=\{2\}}"></span>, denn dann wäre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=\{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B=\{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8126157d504b9390213765f494ef63228b5322d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.288ex; height:2.843ex;" alt="{\displaystyle A\setminus B=\{1\}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\setminus A=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\setminus A=\emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3c6e80b6bc3532182196143dba53aa45bb923c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle B\setminus A=\emptyset }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Matrizenrechnung">Matrizenrechnung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=6" title="Abschnitt bearbeiten: Matrizenrechnung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Matrizenrechnung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Addition von <a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a> über einem <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ring</a> oder <a href="/wiki/K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> ist kommutativ. Die <a href="/wiki/Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a> ist dagegen nicht kommutativ: Die Faktoren sind zwar <i>manchmal,</i> aber nicht <i>immer</i> vertauschbar. </p><p>Ebenfalls kommutativ sind die <a href="/wiki/Skalarmultiplikation" title="Skalarmultiplikation">Multiplikation von Matrizen mit Skalaren</a> und die Matrizenmultiplikation im Unterring der <a href="/wiki/Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrizen</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Gruppentheorie">Gruppentheorie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=7" title="Abschnitt bearbeiten: Gruppentheorie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Gruppentheorie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Allgemein nennt man eine <a href="/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a>, bei der die <a href="/wiki/Verkn%C3%BCpfung_(Mathematik)" title="Verknüpfung (Mathematik)">Verknüpfung</a> von Gruppenelementen kommutativ ist, <a href="/wiki/Abelsche_Gruppe" title="Abelsche Gruppe">abelsch</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Aussagenlogik">Aussagenlogik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=8" title="Abschnitt bearbeiten: Aussagenlogik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Aussagenlogik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Aussagenlogik" title="Aussagenlogik">Aussagenlogik</a></i></div> <p>In der Aussagenlogik gilt für die <a href="/wiki/Junktor" title="Junktor">Junktoren</a>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> („oder“) ist kommutativ.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span> („und“) ist kommutativ.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span> („<a href="/wiki/Logische_%C3%84quivalenz" title="Logische Äquivalenz">logische Äquivalenz</a>“) ist kommutativ.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span> („wenn …, dann …“; siehe <a href="/wiki/Implikation" title="Implikation">Implikation</a>) ist nicht kommutativ.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Weitere_Beispiele">Weitere Beispiele</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=9" title="Abschnitt bearbeiten: Weitere Beispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Weitere Beispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weitere Beispiele für nichtkommutative Operationen sind das <a href="/wiki/Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a> in Vektorräumen oder die Multiplikation von <a href="/wiki/Quaternionen" class="mw-redirect" title="Quaternionen">Quaternionen</a>. </p><p>Kommutativität ist außerdem eine wichtige Grundeigenschaft in der <a href="/wiki/Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a>, das Kommutieren zweier <a href="/wiki/Observable" title="Observable">Observablen</a> bedeutet physikalisch deren gleichzeitige genaue Messbarkeit. Nicht alle Observablen kommutieren. </p> <div class="mw-heading mw-heading2"><h2 id="Antikommutativität"><span id="Antikommutativit.C3.A4t"></span>Antikommutativität</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=10" title="Abschnitt bearbeiten: Antikommutativität" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Antikommutativität"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Cross_product_vector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Cross_product_vector.svg/220px-Cross_product_vector.svg.png" decoding="async" width="220" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Cross_product_vector.svg/330px-Cross_product_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Cross_product_vector.svg/440px-Cross_product_vector.svg.png 2x" data-file-width="484" data-file-height="673" /></a><figcaption>Das Kreuzprodukt ist antikommutativ (hier ein <a href="/wiki/Rechtssystem_(Mathematik)" title="Rechtssystem (Mathematik)">Rechtssystem</a>)</figcaption></figure> <p>In einigen Strukturen mit zwei Operationen, beispielsweise beim <a href="/wiki/Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> in Vektorräumen, gilt nicht das Kommutativgesetz, sondern stattdessen eine Art <i>Gegensatz</i> davon: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times b=-(b\times a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>×</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times b=-(b\times a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecee58b46128370161812bd6a85c897bcfcec495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.851ex; height:2.843ex;" alt="{\displaystyle a\times b=-(b\times a)}"></span>.</dd></dl> <p>Allgemeiner erfüllt das Produkt auf einer <a href="/wiki/Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a>, das als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> geschrieben wird, die Antikommutativität. </p> <div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=11" title="Abschnitt bearbeiten: Anmerkungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Anmerkungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Symmetrische_Relation" title="Symmetrische Relation">Symmetrische Relation</a></dt></dl> <p>Die Kommutativität, die das Vertauschen von Argumenten bei einer <i>Operation</i> erlaubt, weist Ähnlichkeit mit der Symmetrie-Eigenschaft von <a href="/wiki/Relation_(Mathematik)" title="Relation (Mathematik)">Relationen</a> auf, die das Vertauschen der verglichenen Elemente bzgl. der <i>Relation</i> erlaubt: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>R</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xRy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324aab4e2674bb19cc073ea887888b98f0fc63d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.249ex; height:2.509ex;" alt="{\displaystyle xRy}"></span> genau dann, wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yRx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mi>R</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle yRx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b423dd05da0ad3628b519a2289cee17e1c10df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.249ex; height:2.509ex;" alt="{\displaystyle yRx}"></span>. </p> <dl><dt><a href="/wiki/Flexibilit%C3%A4tsgesetz" class="mw-redirect" title="Flexibilitätsgesetz">Flexibilitätsgesetz</a></dt></dl> <p>Eine alternative Möglichkeit des „Um-Klammerns“ bietet das Flexibilitätsgesetz für eine <a href="/wiki/Verkn%C3%BCpfung_(Mathematik)" title="Verknüpfung (Mathematik)">Verknüpfung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a*\left(b*a\right)=\left(a*b\right)*a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>∗</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>∗</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>∗</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*\left(b*a\right)=\left(a*b\right)*a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf79b0c1d3de382a138a4fd5e335ea0a7e6a8d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.41ex; height:2.843ex;" alt="{\displaystyle a*\left(b*a\right)=\left(a*b\right)*a}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=12" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Symmetrische_Funktion" title="Symmetrische Funktion">Symmetrische Funktion</a></li> <li><a href="/wiki/Kommutatives_Diagramm" title="Kommutatives Diagramm">Kommutatives Diagramm</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kommutativgesetz&veaction=edit&section=13" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kommutativgesetz&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Otto_Forster" title="Otto Forster">Otto Forster</a>: <cite style="font-style:italic">Differential- und Integralrechnung einer Veränderlichen</cite> (= <cite style="font-style:italic">Analysis</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>1</span>). 10. Auflage. 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interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D8%AA%D8%A8%D8%AF%D9%8A%D9%84%D9%8A%D8%A9" title="عملية تبديلية – Arabisch" lang="ar" hreflang="ar" data-title="عملية تبديلية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Conmutativid%C3%A1" title="Conmutatividá – Asturisch" lang="ast" hreflang="ast" data-title="Conmutatividá" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kommutativlik_xass%C9%99si" title="Kommutativlik xassəsi – Aserbaidschanisch" lang="az" hreflang="az" data-title="Kommutativlik xassəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1" title="Коммутативлыҡ – Baschkirisch" lang="ba" hreflang="ba" data-title="Коммутативлыҡ" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BC%D1%83%D1%82%D0%B0%D1%82%D1%8B%D1%9E%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%8B%D1%8F" title="Камутатыўная аперацыя – Belarussisch" lang="be" hreflang="be" data-title="Камутатыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Комутативност – Bulgarisch" lang="bg" hreflang="bg" data-title="Комутативност" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%A8%E0%A6%BF%E0%A6%AE%E0%A6%AF%E0%A6%BC_%E0%A6%AC%E0%A7%88%E0%A6%B6%E0%A6%BF%E0%A6%B7%E0%A7%8D%E0%A6%9F%E0%A7%8D%E0%A6%AF" title="বিনিময় বৈশিষ্ট্য – Bengalisch" lang="bn" hreflang="bn" data-title="বিনিময় বৈশিষ্ট্য" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Bosnisch" lang="bs" hreflang="bs" data-title="Komutativnost" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://ca.wikipedia.org/wiki/Propietat_commutativa" title="Propietat commutativa – Katalanisch" lang="ca" hreflang="ca" data-title="Propietat commutativa" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B3%DB%8C%DB%95%D8%AA%DB%8C_%D8%A6%D8%A7%DA%B5%D9%88%DA%AF%DB%86%DA%95" title="خاسیەتی ئاڵوگۆڕ – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="خاسیەتی ئاڵوگۆڕ" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Komutativita" title="Komutativita – Tschechisch" lang="cs" hreflang="cs" data-title="Komutativita" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%C4%83_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8" title="Коммутативлă операци – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Коммутативлă операци" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Dänisch" lang="da" hreflang="da" data-title="Kommutativitet" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CF%84%CE%B9%CE%BC%CE%B5%CF%84%CE%B1%CE%B8%CE%B5%CF%84%CE%B9%CE%BA%CE%AE_%CE%B9%CE%B4%CE%B9%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Αντιμεταθετική ιδιότητα – Griechisch" lang="el" hreflang="el" data-title="Αντιμεταθετική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://en.wikipedia.org/wiki/Commutative_property" title="Commutative property – Englisch" lang="en" hreflang="en" data-title="Commutative property" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Komuteco" title="Komuteco – Esperanto" lang="eo" hreflang="eo" data-title="Komuteco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conmutatividad" title="Conmutatividad – Spanisch" lang="es" hreflang="es" data-title="Conmutatividad" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kommutatiivsus" title="Kommutatiivsus – Estnisch" lang="et" hreflang="et" data-title="Kommutatiivsus" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Trukakortasun" title="Trukakortasun – Baskisch" lang="eu" hreflang="eu" data-title="Trukakortasun" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5%DB%8C%D8%AA_%D8%AC%D8%A7%D8%A8%D9%87%E2%80%8C%D8%AC%D8%A7%DB%8C%DB%8C" title="خاصیت جابه‌جایی – Persisch" lang="fa" hreflang="fa" data-title="خاصیت جابه‌جایی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vaihdannaisuus" title="Vaihdannaisuus – Finnisch" lang="fi" hreflang="fi" data-title="Vaihdannaisuus" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_commutative" title="Loi commutative – Französisch" lang="fr" hreflang="fr" data-title="Loi commutative" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Komutatiifgesets" title="Komutatiifgesets – Nordfriesisch" lang="frr" hreflang="frr" data-title="Komutatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="Nordfriesisch" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Oibr%C3%ADocht_ch%C3%B3mhalartach" title="Oibríocht chómhalartach – Irisch" lang="ga" hreflang="ga" data-title="Oibríocht chómhalartach" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Co-iomlaideachd" title="Co-iomlaideachd – Gälisch (Schottland)" lang="gd" hreflang="gd" data-title="Co-iomlaideachd" data-language-autonym="Gàidhlig" data-language-local-name="Gälisch (Schottland)" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Conmutatividade" title="Conmutatividade – Galicisch" lang="gl" hreflang="gl" data-title="Conmutatividade" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%A7%D7%95%D7%9E%D7%95%D7%98%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה קומוטטיבית – Hebräisch" lang="he" hreflang="he" data-title="פעולה קומוטטיבית" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%B5%E0%A4%BF%E0%A4%A8%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF%E0%A4%A4%E0%A4%BE" title="क्रमविनिमेयता – Hindi" lang="hi" hreflang="hi" data-title="क्रमविनिमेयता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Kroatisch" lang="hr" hreflang="hr" data-title="Komutativnost" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kommutativit%C3%A1s" title="Kommutativitás – Ungarisch" lang="hu" hreflang="hu" data-title="Kommutativitás" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A5%D5%B2%D5%A1%D6%83%D5%B8%D5%AD%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Տեղափոխականություն – Armenisch" lang="hy" hreflang="hy" data-title="Տեղափոխականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Commutativitate" title="Commutativitate – Interlingua" lang="ia" hreflang="ia" data-title="Commutativitate" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_komutatif" title="Sifat komutatif – Indonesisch" lang="id" hreflang="id" data-title="Sifat komutatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/V%C3%ADxlregla" title="Víxlregla – Isländisch" lang="is" hreflang="is" data-title="Víxlregla" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Commutativit%C3%A0" title="Commutatività – Italienisch" lang="it" hreflang="it" data-title="Commutatività" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%A4%E6%8F%9B%E6%B3%95%E5%89%87" title="交換法則 – Japanisch" lang="ja" hreflang="ja" data-title="交換法則" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%83%D1%8B%D1%81%D1%82%D1%8B%D1%80%D1%8B%D0%BC%D0%B4%D1%8B%D0%BB%D1%8B%D2%9B" title="Ауыстырымдылық – Kasachisch" lang="kk" hreflang="kk" data-title="Ауыстырымдылық" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%90%ED%99%98%EB%B2%95%EC%B9%99" title="교환법칙 – Koreanisch" lang="ko" hreflang="ko" data-title="교환법칙" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Commutativitas_(mathematica)" title="Commutativitas (mathematica) – Latein" lang="la" hreflang="la" data-title="Commutativitas (mathematica)" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Komutatyvumas" title="Komutatyvumas – Litauisch" lang="lt" hreflang="lt" data-title="Komutatyvumas" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komutativit%C4%81te" title="Komutativitāte – Lettisch" lang="lv" hreflang="lv" data-title="Komutativitāte" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Комутативност – Mazedonisch" lang="mk" hreflang="mk" data-title="Комутативност" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="ക്രമനിയമം – Malayalam" lang="ml" hreflang="ml" data-title="ക്രമനിയമം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kalis_tukar_tertib" title="Kalis tukar tertib – Malaiisch" lang="ms" hreflang="ms" data-title="Kalis tukar tertib" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Commutativiteit" title="Commutativiteit – Niederländisch" lang="nl" hreflang="nl" data-title="Commutativiteit" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Kommutativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kommutativ_lov" title="Kommutativ lov – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Kommutativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przemienno%C5%9B%C4%87" title="Przemienność – Polnisch" lang="pl" hreflang="pl" data-title="Przemienność" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Comutatividade" title="Comutatividade – Portugiesisch" lang="pt" hreflang="pt" data-title="Comutatividade" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Comutativitate" title="Comutativitate – Rumänisch" lang="ro" hreflang="ro" data-title="Comutativitate" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Коммутативность – Russisch" lang="ru" hreflang="ru" data-title="Коммутативность" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Serbokroatisch" lang="sh" hreflang="sh" data-title="Komutativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%AF%E0%B7%9A%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BA_%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B6%BA" title="න්‍යාදේශ්‍ය න්‍යාය – Singhalesisch" lang="si" hreflang="si" data-title="න්‍යාදේශ්‍ය න්‍යාය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Commutative_property" title="Commutative property – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Commutative property" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Komutat%C3%ADvnos%C5%A5" title="Komutatívnosť – Slowakisch" lang="sk" hreflang="sk" data-title="Komutatívnosť" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Slowenisch" lang="sl" hreflang="sl" data-title="Komutativnost" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vetia_e_nd%C3%ABrrimit" title="Vetia e ndërrimit – Albanisch" lang="sq" hreflang="sq" data-title="Vetia e ndërrimit" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Комутативност – Serbisch" lang="sr" hreflang="sr" data-title="Комутативност" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Schwedisch" lang="sv" hreflang="sv" data-title="Kommutativitet" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%BF%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88" title="பரிமாற்றுத்தன்மை – Tamil" lang="ta" hreflang="ta" data-title="பரிமாற்றுத்தன்மை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%9A%E0%B8%B1%E0%B8%95%E0%B8%B4%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AA%E0%B8%A5%E0%B8%B1%E0%B8%9A%E0%B8%97%E0%B8%B5%E0%B9%88" title="สมบัติการสลับที่ – Thailändisch" lang="th" hreflang="th" data-title="สมบัติการสลับที่" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/De%C4%9Fi%C5%9Fme_%C3%B6zelli%C4%9Fi" title="Değişme özelliği – Türkisch" lang="tr" hreflang="tr" data-title="Değişme özelliği" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D0%BA" title="Коммутативлык – Tatarisch" lang="tt" hreflang="tt" data-title="Коммутативлык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarisch" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Комутативність – Ukrainisch" lang="uk" hreflang="uk" data-title="Комутативність" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Propiet%C3%A0_comutativa" title="Propietà comutativa – Venetisch" lang="vec" hreflang="vec" data-title="Propietà comutativa" data-language-autonym="Vèneto" data-language-local-name="Venetisch" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADnh_giao_ho%C3%A1n" title="Tính giao hoán – Vietnamesisch" lang="vi" hreflang="vi" data-title="Tính giao hoán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BA%A4%E6%8D%A2%E5%BE%8B" title="交换律 – Wu" lang="wuu" hreflang="wuu" data-title="交换律" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" title="交換律 – Chinesisch" lang="zh" hreflang="zh" data-title="交換律" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" title="交換律 – Kantonesisch" lang="yue" hreflang="yue" data-title="交換律" data-language-autonym="粵語" data-language-local-name="Kantonesisch" 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