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class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Commutative property</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Commutative_property" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Commutative_property" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button 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edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Commutative&redirect=no" class="mw-redirect" title="Commutative">Commutative</a>)</span></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Commutative" redirects here. For other uses, see <a href="/wiki/Commutative_(disambiguation)" class="mw-redirect mw-disambig" title="Commutative (disambiguation)">Commutative (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> is <b>commutative</b> if changing the order of the <a href="/wiki/Operand" title="Operand">operands</a> does not change the result. It is a fundamental property of many binary operations, and many <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proofs</a> depend on it. Perhaps most familiar as a property of arithmetic, e.g. <span class="nowrap">"3 + 4 = 4 + 3"</span> or <span class="nowrap">"2 × 5 = 5 × 2"</span>, the property can also be used in more advanced settings. The name is needed because there are operations, such as <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> and <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, that do not have it (for example, <span class="nowrap">"3 − 5 ≠ 5 − 3"</span>); such operations are <i>not</i> commutative, and so are referred to as <i>noncommutative operations</i>. The idea that simple operations, such as the <a href="/wiki/Multiplication_(mathematics)" class="mw-redirect" title="Multiplication (mathematics)">multiplication</a> and <a href="/wiki/Addition" title="Addition">addition</a> of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.<sup id="cite_ref-Cabillón_1-0" class="reference"><a href="#cite_note-Cabill%C3%B3n-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Flood11_2-0" class="reference"><a href="#cite_note-Flood11-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A similar property exists for <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>; a binary relation is said to be <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a> if the relation applies regardless of the order of its operands; for example, <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a> is symmetric as two equal mathematical objects are equal regardless of their order.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;">Commutative property</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File: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></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Property_(mathematics)" title="Property (mathematics)">Property</a></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><a href="/wiki/Algebra" title="Algebra">Algebra</a></td></tr><tr><th scope="row" class="infobox-label">Statement</th><td class="infobox-data">A <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> is <i>commutative</i> if changing the order of the <a href="/wiki/Operand" title="Operand">operands</a> does not change the result.</td></tr><tr><th scope="row" class="infobox-label">Symbolic statement</th><td class="infobox-data"><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*y=y*x\quad \forall x,y\in S.}"> <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> <mspace width="1em"></mspace> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=y*x\quad \forall x,y\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b4ac067a9814abdfa618c913586a9ee9f56c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.579ex; height:2.509ex;" alt="{\displaystyle x*y=y*x\quad \forall x,y\in S.}"></span></td></tr></tbody></table> <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="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Mathematical_definitions"><span class="tocnumber">1</span> <span class="toctext">Mathematical definitions</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Examples"><span class="tocnumber">2</span> <span class="toctext">Examples</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Commutative_operations"><span class="tocnumber">2.1</span> <span class="toctext">Commutative operations</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Noncommutative_operations"><span class="tocnumber">2.2</span> <span class="toctext">Noncommutative operations</span></a> <ul> <li class="toclevel-3 tocsection-5"><a href="#Division,_subtraction,_and_exponentiation"><span class="tocnumber">2.2.1</span> <span class="toctext">Division, subtraction, and exponentiation</span></a></li> <li class="toclevel-3 tocsection-6"><a href="#Truth_functions"><span class="tocnumber">2.2.2</span> <span class="toctext">Truth functions</span></a></li> <li class="toclevel-3 tocsection-7"><a href="#Function_composition_of_linear_functions"><span class="tocnumber">2.2.3</span> <span class="toctext">Function composition of linear functions</span></a></li> <li class="toclevel-3 tocsection-8"><a href="#Matrix_multiplication"><span class="tocnumber">2.2.4</span> <span class="toctext">Matrix multiplication</span></a></li> <li class="toclevel-3 tocsection-9"><a href="#Vector_product"><span class="tocnumber">2.2.5</span> <span class="toctext">Vector product</span></a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#History_and_etymology"><span class="tocnumber">3</span> <span class="toctext">History and etymology</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Propositional_logic"><span class="tocnumber">4</span> <span class="toctext">Propositional logic</span></a> <ul> <li class="toclevel-2 tocsection-12"><a href="#Rule_of_replacement"><span class="tocnumber">4.1</span> <span class="toctext">Rule of replacement</span></a></li> <li class="toclevel-2 tocsection-13"><a href="#Truth_functional_connectives"><span class="tocnumber">4.2</span> <span class="toctext">Truth functional connectives</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#Set_theory"><span class="tocnumber">5</span> <span class="toctext">Set theory</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Mathematical_structures_and_commutativity"><span class="tocnumber">6</span> <span class="toctext">Mathematical structures and commutativity</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Related_properties"><span class="tocnumber">7</span> <span class="toctext">Related properties</span></a> <ul> <li class="toclevel-2 tocsection-17"><a href="#Associativity"><span class="tocnumber">7.1</span> <span class="toctext">Associativity</span></a></li> <li class="toclevel-2 tocsection-18"><a href="#Distributivity"><span class="tocnumber">7.2</span> <span class="toctext">Distributivity</span></a></li> <li class="toclevel-2 tocsection-19"><a href="#Symmetry"><span class="tocnumber">7.3</span> <span class="toctext">Symmetry</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-20"><a href="#Non-commuting_operators_in_quantum_mechanics"><span class="tocnumber">8</span> <span class="toctext">Non-commuting operators in quantum mechanics</span></a></li> <li class="toclevel-1 tocsection-21"><a href="#See_also"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-22"><a href="#Notes"><span class="tocnumber">10</span> <span class="toctext">Notes</span></a></li> <li class="toclevel-1 tocsection-23"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a> <ul> <li class="toclevel-2 tocsection-24"><a href="#Books"><span class="tocnumber">11.1</span> <span class="toctext">Books</span></a></li> <li class="toclevel-2 tocsection-25"><a href="#Articles"><span class="tocnumber">11.2</span> <span class="toctext">Articles</span></a></li> <li class="toclevel-2 tocsection-26"><a href="#Online_resources"><span class="tocnumber">11.3</span> <span class="toctext">Online resources</span></a></li> </ul> </li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Mathematical_definitions">Mathematical definitions</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=1" title="Edit section: Mathematical definitions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>A <a href="/wiki/Binary_operation" title="Binary operation">binary operation</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><noscript><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 *}"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 1.509ex;vertical-align: 0.079ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" data-alt="{\displaystyle *}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i> is called <i>commutative</i> if<sup id="cite_ref-Krowne,_p.1_4-0" class="reference"><a href="#cite_note-Krowne,_p.1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}"> <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> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b366462929cecad76945a77bb8ffaec7274c1e21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.013ex; height:2.509ex;" alt="{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}"></noscript><span class="lazy-image-placeholder" style="width: 32.013ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b366462929cecad76945a77bb8ffaec7274c1e21" data-alt="{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called <i>noncommutative</i>. </p><p>One says that <span class="texhtml mvar" style="font-style:italic;">x</span> <i>commutes</i> with <span class="texhtml"><i>y</i></span> or that <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> <i>commute</i> under <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><noscript><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 *}"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 1.509ex;vertical-align: 0.079ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" data-alt="{\displaystyle *}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x*y=y*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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=y*x.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/848e86c4b6f3f24d6cce3a5fb50665c08466cce6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.105ex; height:2.009ex;" alt="{\displaystyle x*y=y*x.}"></noscript><span class="lazy-image-placeholder" style="width: 13.105ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/848e86c4b6f3f24d6cce3a5fb50665c08466cce6" data-alt="{\displaystyle x*y=y*x.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Examples">Examples</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=2" title="Edit section: Examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Commutative_Addition.svg" class="mw-file-description"><noscript><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" data-file-width="600" data-file-height="300"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 110px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Commutative_Addition.svg/220px-Commutative_Addition.svg.png" data-width="220" data-height="110" data-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-class="mw-file-element"> </span></a><figcaption>The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Commutative_operations">Commutative operations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=3" title="Edit section: Commutative operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_Addition.svg" class="mw-file-description"><noscript><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" data-file-width="400" data-file-height="400"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 220px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/220px-Vector_Addition.svg.png" data-width="220" data-height="220" data-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-class="mw-file-element"> </span></a><figcaption>The addition of vectors is commutative, because <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcaa01e9959d82da83b5b2ddea6cd9f33e64878d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.074ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.}"></noscript><span class="lazy-image-placeholder" style="width: 14.074ex;height: 3.009ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcaa01e9959d82da83b5b2ddea6cd9f33e64878d" data-alt="{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></figcaption></figure> <ul><li><a href="/wiki/Addition" title="Addition">Addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> are commutative in most <a href="/wiki/Number_system" class="mw-redirect" title="Number system">number systems</a>, and, in particular, between <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, <a href="/wiki/Real_number" title="Real number">real numbers</a> and <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. This is also true in every <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>.</li> <li>Addition is commutative in every <a href="/wiki/Vector_space" title="Vector space">vector space</a> and in every <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a>.</li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a> and <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> are commutative operations on <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>.</li> <li>"<a href="/wiki/And_(logic)" class="mw-redirect" title="And (logic)">And</a>" and "<a href="/wiki/Or_(logic)" class="mw-redirect" title="Or (logic)">or</a>" are commutative <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">logical operations</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Noncommutative_operations">Noncommutative operations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=4" title="Edit section: Noncommutative operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <div class="mw-heading mw-heading4"><h4 id="Division,_subtraction,_and_exponentiation"><span id="Division.2C_subtraction.2C_and_exponentiation"></span>Division, subtraction, and exponentiation</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=5" title="Edit section: Division, subtraction, and exponentiation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equation_xy_%3D_yx" title="Equation xy = yx">Equation xy = yx</a></div> <p><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a> is noncommutative, since <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 1\div 2\neq 2\div 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>÷</mo> <mn>2</mn> <mo>≠</mo> <mn>2</mn> <mo>÷</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\div 2\neq 2\div 1}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7784e4bb12fc8a2f7730b431aac47ed174c6b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.429ex; height:2.676ex;" alt="{\displaystyle 1\div 2\neq 2\div 1}"></noscript><span class="lazy-image-placeholder" style="width: 13.429ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7784e4bb12fc8a2f7730b431aac47ed174c6b2d" data-alt="{\displaystyle 1\div 2\neq 2\div 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p><p><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a> is noncommutative, since <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 0-1\neq 1-0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>−</mo> <mn>1</mn> <mo>≠</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0-1\neq 1-0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb89fedf760b003313ae21bb70bab6f6a12169c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.429ex; height:2.676ex;" alt="{\displaystyle 0-1\neq 1-0}"></noscript><span class="lazy-image-placeholder" style="width: 13.429ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb89fedf760b003313ae21bb70bab6f6a12169c" data-alt="{\displaystyle 0-1\neq 1-0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. However it is classified more precisely as <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">anti-commutative</a>, since <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 0-1=-(1-0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0-1=-(1-0)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6022731a4f1dab82e9131c617525ef10e9c4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.047ex; height:2.843ex;" alt="{\displaystyle 0-1=-(1-0)}"></noscript><span class="lazy-image-placeholder" style="width: 17.047ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6022731a4f1dab82e9131c617525ef10e9c4dc" data-alt="{\displaystyle 0-1=-(1-0)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p><p><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> is noncommutative, since <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><noscript><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}}"></noscript><span class="lazy-image-placeholder" style="width: 7.532ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170cfbfd71bd6828dbe7fb72a9eb6f135814693e" data-alt="{\displaystyle 2^{3}\neq 3^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. This property leads to two different "inverse" operations of exponentiation (namely, the <a href="/wiki/Nth_root" title="Nth root"><i>n</i>th-root</a> operation and the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> operation), whereas multiplication only has one inverse operation.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Truth_functions">Truth functions</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=6" title="Edit section: Truth functions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Some <a href="/wiki/Truth_function" title="Truth function">truth functions</a> are noncommutative, since the <a href="/wiki/Truth_table" title="Truth table">truth tables</a> for the functions are different when one changes the order of the operands. For example, the truth tables for <span class="texhtml">(A ⇒ B) = (¬A ∨ B)</span> and <span class="texhtml">(B ⇒ A) = (A ∨ ¬B)</span> are </p> <dl><dd><table class="wikitable" style="text-align:center; width:20%;"><tbody><tr style="vertical-align:top"><th scope="col"> <span class="texhtml">A</span></th><th scope="col"> <span class="texhtml">B</span></th><th scope="col" style="width:30%;"> <span class="texhtml">A ⇒ B</span></th><th scope="col" style="width:30%;"> <span class="texhtml">B ⇒ A</span></th></tr><tr style="vertical-align:top"><td> F</td><td> F</td><td style="width:30%;"> T</td><td style="width:30%;"> T</td></tr><tr style="vertical-align:top"><td> F</td><td> T</td><td style="width:30%;"> T</td><td style="width:30%;"> F</td></tr><tr style="vertical-align:top"><td> T</td><td> F</td><td style="width:30%;"> F</td><td style="width:30%;"> T</td></tr><tr style="vertical-align:top"><td> T</td><td> T</td><td style="width:30%;"> T</td><td style="width:30%;"> T</td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Function_composition_of_linear_functions">Function composition of linear functions</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=7" title="Edit section: Function composition of linear functions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p><a href="/wiki/Function_composition" title="Function composition">Function composition</a> of <a href="/wiki/Linear_function" title="Linear function">linear functions</a> from the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> to the real numbers is almost always noncommutative. For example, let <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 f(x)=2x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2x+1}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ca6b62bf1326a2e8672de9d2a8bfa95240fd76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.011ex; height:2.843ex;" alt="{\displaystyle f(x)=2x+1}"></noscript><span class="lazy-image-placeholder" style="width: 14.011ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ca6b62bf1326a2e8672de9d2a8bfa95240fd76" data-alt="{\displaystyle f(x)=2x+1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <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 g(x)=3x+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=3x+7}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23e30e18d8cb03b82552b98e9174bf5d8bc4f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.848ex; height:2.843ex;" alt="{\displaystyle g(x)=3x+7}"></noscript><span class="lazy-image-placeholder" style="width: 13.848ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23e30e18d8cb03b82552b98e9174bf5d8bc4f61" data-alt="{\displaystyle g(x)=3x+7}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d79e32c51f1826f0e734c67bc2f94fb9f2f17e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.303ex; height:2.843ex;" alt="{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}"></noscript><span class="lazy-image-placeholder" style="width: 47.303ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d79e32c51f1826f0e734c67bc2f94fb9f2f17e" data-alt="{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>7</mn> <mo>=</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c495d76e3a656080cd77df098f2c4105339ccf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.303ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}"></noscript><span class="lazy-image-placeholder" style="width: 47.303ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c495d76e3a656080cd77df098f2c4105339ccf" data-alt="{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> This also applies more generally for <a href="/wiki/Linear_map" title="Linear map">linear</a> and <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a> from a <a href="/wiki/Vector_space" title="Vector space">vector space</a> to itself (see below for the Matrix representation). </p> <div class="mw-heading mw-heading4"><h4 id="Matrix_multiplication">Matrix multiplication</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=8" title="Edit section: Matrix multiplication" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Matrix multiplication</a> of <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a> is almost always noncommutative, for example: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668291a5a8ba23c6bf436b331db5ccbcf3395719" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.419ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 56.419ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668291a5a8ba23c6bf436b331db5ccbcf3395719" data-alt="{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> </p> <div class="mw-heading mw-heading4"><h4 id="Vector_product">Vector product</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=9" title="Edit section: Vector product" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The vector product (or <a href="/wiki/Cross_product" title="Cross product">cross product</a>) of two vectors in three dimensions is <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">anti-commutative</a>; i.e., <i>b</i> × <i>a</i> = −(<i>a</i> × <i>b</i>). </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="History_and_etymology">History and etymology</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=10" title="Edit section: History and etymology" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Commutative_Word_Origin.PNG" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Commutative_Word_Origin.PNG/250px-Commutative_Word_Origin.PNG" decoding="async" width="250" height="130" class="mw-file-element" data-file-width="521" data-file-height="271"></noscript><span class="lazy-image-placeholder" style="width: 250px;height: 130px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Commutative_Word_Origin.PNG/250px-Commutative_Word_Origin.PNG" data-width="250" data-height="130" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Commutative_Word_Origin.PNG/375px-Commutative_Word_Origin.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Commutative_Word_Origin.PNG/500px-Commutative_Word_Origin.PNG 2x" data-class="mw-file-element"> </span></a><figcaption>The first known use of the term was in a French Journal published in 1814</figcaption></figure> <p>Records of the implicit use of the commutative property go back to ancient times. The <a href="/wiki/Egypt" title="Egypt">Egyptians</a> used the commutative property of <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> to simplify computing <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">products</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Euclid" title="Euclid">Euclid</a> is known to have assumed the commutative property of multiplication in his book <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics. </p><p>The first recorded use of the term <i>commutative</i> was in a memoir by <a href="/wiki/Fran%C3%A7ois-Joseph_Servois" title="François-Joseph Servois">François Servois</a> in 1814,<sup id="cite_ref-Cabillón_1-1" class="reference"><a href="#cite_note-Cabill%C3%B3n-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> which used the word <i>commutatives</i> when describing functions that have what is now called the commutative property. <i>Commutative</i> is the feminine form of the French adjective <i>commutatif</i>, which is derived from the French noun <i>commutation</i> and the French verb <i>commuter</i>, meaning "to exchange" or "to switch", a cognate of <i>to commute</i>. The term then appeared in English in 1838.<sup id="cite_ref-Flood11_2-1" class="reference"><a href="#cite_note-Flood11-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Duncan_Gregory" title="Duncan Gregory">Duncan Gregory</a>'s article entitled "On the real nature of symbolical algebra" published in 1840 in the <a href="/wiki/Royal_Society_of_Edinburgh" title="Royal Society of Edinburgh">Transactions of the Royal Society of Edinburgh</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Propositional_logic">Propositional logic</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=11" title="Edit section: Propositional logic" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output 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screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style> <div class="mw-heading mw-heading3"><h3 id="Rule_of_replacement">Rule of replacement</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=12" title="Edit section: Rule of replacement" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>In truth-functional propositional logic, <i>commutation</i>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> or <i>commutativity</i><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> refer to two <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">rules of replacement</a>. The rules allow one to transpose <a href="/wiki/Propositional_variable" title="Propositional variable">propositional variables</a> within <a href="/wiki/Well-formed_formula" title="Well-formed formula">logical expressions</a> in <a href="/wiki/Formal_proof" title="Formal proof">logical proofs</a>. The rules are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b532501a03771c233967e3cba1f6157908a38a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.565ex; height:2.843ex;" alt="{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}"></noscript><span class="lazy-image-placeholder" style="width: 19.565ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b532501a03771c233967e3cba1f6157908a38a" data-alt="{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e0fe7ca4b48406c2a7cf15d8c8a052755e2460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.565ex; height:2.843ex;" alt="{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}"></noscript><span class="lazy-image-placeholder" style="width: 19.565ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e0fe7ca4b48406c2a7cf15d8c8a052755e2460" data-alt="{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> where "<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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" 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 }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>" is a <a href="/wiki/Metalogic" title="Metalogic">metalogical</a> <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbol</a> representing "can be replaced in a <a href="/wiki/Formal_proof" title="Formal proof">proof</a> with". </p> <div class="mw-heading mw-heading3"><h3 id="Truth_functional_connectives">Truth functional connectives</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=13" title="Edit section: Truth functional connectives" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p><i>Commutativity</i> is a property of some <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a> of truth functional <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. The following <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalences</a> demonstrate that commutativity is a property of particular connectives. The following are truth-functional <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a>. </p> <dl><dt>Commutativity of conjunction</dt> <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 (P\land Q)\leftrightarrow (Q\land P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68a37da164fe75209f4998cbe69839a241d8c45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.565ex; height:2.843ex;" alt="{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}"></noscript><span class="lazy-image-placeholder" style="width: 19.565ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68a37da164fe75209f4998cbe69839a241d8c45" data-alt="{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd> <dt>Commutativity of disjunction</dt> <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 (P\lor Q)\leftrightarrow (Q\lor P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f4ed76e18831ba1a9bab333b467883e32a40f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.565ex; height:2.843ex;" alt="{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}"></noscript><span class="lazy-image-placeholder" style="width: 19.565ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f4ed76e18831ba1a9bab333b467883e32a40f4" data-alt="{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd> <dt>Commutativity of implication (also called the law of permutation)</dt> <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 {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>P</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>Q</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a92e89b62e33fae44f3ffc5708c6ece9b7e9b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.644ex; height:3.176ex;" alt="{\displaystyle {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}"></noscript><span class="lazy-image-placeholder" style="width: 36.644ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a92e89b62e33fae44f3ffc5708c6ece9b7e9b5" data-alt="{\displaystyle {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd> <dt>Commutativity of equivalence (also called the complete commutative law of equivalence)</dt> <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 (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">↔</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e78176e9a7a335f3a577a0355614d17a62f591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.628ex; height:2.843ex;" alt="{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}"></noscript><span class="lazy-image-placeholder" style="width: 21.628ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e78176e9a7a335f3a577a0355614d17a62f591" data-alt="{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Set_theory">Set theory</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=14" title="Edit section: Set theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>In <a href="/wiki/Group_theory" title="Group theory">group</a> and <a href="/wiki/Set_theory" title="Set theory">set theory</a>, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> the commutativity of well-known operations (such as <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> on real and complex numbers) is often used (or implicitly assumed) in proofs.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gallian,_p._34_16-0" class="reference"><a href="#cite_note-Gallian,_p._34-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Mathematical_structures_and_commutativity">Mathematical structures and commutativity</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=15" title="Edit section: Mathematical structures and commutativity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li>A <a href="/wiki/Commutative_semigroup" class="mw-redirect" title="Commutative semigroup">commutative semigroup</a> is a set endowed with a total, <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative</a> and commutative operation.</li> <li>If the operation additionally has an <a href="/wiki/Identity_element" title="Identity element">identity element</a>, we have a <a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoid</a></li> <li>An <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, or <i>commutative group</i> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> whose group operation is commutative.<sup id="cite_ref-Gallian,_p._34_16-1" class="reference"><a href="#cite_note-Gallian,_p._34-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li> <li>A <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> is a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> whose <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> is commutative. (Addition in a ring is always commutative.)<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li>In a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> both addition and multiplication are commutative.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Related_properties">Related properties</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=16" title="Edit section: Related properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <div class="mw-heading mw-heading3"><h3 id="Associativity">Associativity</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=17" title="Edit section: Associativity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Associative_property" title="Associative property">Associative property</a></div> <p>The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result. </p><p>Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)={\frac {x+y}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)={\frac {x+y}{2}},}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98de0952342197fe32bd0736e79b1ec0073da81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.514ex; height:5.176ex;" alt="{\displaystyle f(x,y)={\frac {x+y}{2}},}"></noscript><span class="lazy-image-placeholder" style="width: 16.514ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98de0952342197fe32bd0736e79b1ec0073da81" data-alt="{\displaystyle f(x,y)={\frac {x+y}{2}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> which is clearly commutative (interchanging <i>x</i> and <i>y</i> does not affect the result), but it is not associative (since, for example, <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 f(-4,f(0,+4))=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-4,f(0,+4))=-1}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77da612e7edca77a81e11eac2f0db9b4a703c2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.416ex; height:2.843ex;" alt="{\displaystyle f(-4,f(0,+4))=-1}"></noscript><span class="lazy-image-placeholder" style="width: 21.416ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77da612e7edca77a81e11eac2f0db9b4a703c2ce" data-alt="{\displaystyle f(-4,f(0,+4))=-1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> but <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 f(f(-4,0),+4)=+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(f(-4,0),+4)=+1}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8003a91094abcea66bc81ab7d4bb13019abce867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.416ex; height:2.843ex;" alt="{\displaystyle f(f(-4,0),+4)=+1}"></noscript><span class="lazy-image-placeholder" style="width: 21.416ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8003a91094abcea66bc81ab7d4bb13019abce867" data-alt="{\displaystyle f(f(-4,0),+4)=+1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). More such examples may be found in <a href="/wiki/Commutative_non-associative_magmas" class="mw-redirect" title="Commutative non-associative magmas">commutative non-associative magmas</a>. Furthermore, associativity does not imply commutativity either – for example multiplication of <a href="/wiki/Quaternion#Algebraic_properties" title="Quaternion">quaternions</a> or of <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrices</a> is always associative but not always commutative. </p> <div class="mw-heading mw-heading3"><h3 id="Distributivity">Distributivity</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=18" title="Edit section: Distributivity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distributive_property" title="Distributive property">Distributive property</a></div> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=19" title="Edit section: Symmetry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Symmetry_Of_Addition.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Symmetry_Of_Addition.svg/200px-Symmetry_Of_Addition.svg.png" decoding="async" width="200" height="155" class="mw-file-element" data-file-width="129" data-file-height="100"></noscript><span class="lazy-image-placeholder" style="width: 200px;height: 155px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Symmetry_Of_Addition.svg/200px-Symmetry_Of_Addition.svg.png" data-width="200" data-height="155" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Symmetry_Of_Addition.svg/300px-Symmetry_Of_Addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Symmetry_Of_Addition.svg/400px-Symmetry_Of_Addition.svg.png 2x" data-class="mw-file-element"> </span></a><figcaption>Graph showing the symmetry of the addition function</figcaption></figure> <p>Some forms of <a href="/wiki/Symmetry_in_mathematics" title="Symmetry in mathematics">symmetry</a> can be directly linked to commutativity. When a commutative operation is written as a <a href="/wiki/Binary_function" title="Binary function">binary function</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 z=f(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y),}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1269bfd7a8dbf5a109363ce2a7992efdf8e406a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.44ex; height:2.843ex;" alt="{\displaystyle z=f(x,y),}"></noscript><span class="lazy-image-placeholder" style="width: 11.44ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1269bfd7a8dbf5a109363ce2a7992efdf8e406a9" data-alt="{\displaystyle z=f(x,y),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> then this function is called a <a href="/wiki/Symmetric_function" title="Symmetric function">symmetric function</a>, and its <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> is symmetric across the plane <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=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=x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></noscript><span class="lazy-image-placeholder" style="width: 5.584ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" data-alt="{\displaystyle y=x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. For example, if the function <span class="texhtml"><i>f</i></span> is defined as <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 f(x,y)=x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=x+y}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56416bef45cdf8f146d331dbea5872fa1ed4acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.031ex; height:2.843ex;" alt="{\displaystyle f(x,y)=x+y}"></noscript><span class="lazy-image-placeholder" style="width: 15.031ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56416bef45cdf8f146d331dbea5872fa1ed4acb" data-alt="{\displaystyle f(x,y)=x+y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> then <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a symmetric function. </p><p>For relations, a <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric relation</a> is analogous to a commutative operation, in that if a relation <i>R</i> is symmetric, then <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 aRb\Leftrightarrow bRa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mo stretchy="false">⇔</mo> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRb\Leftrightarrow bRa}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5734e3e9c97749481edd4a7327ae973c4c6c36b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.597ex; height:2.176ex;" alt="{\displaystyle aRb\Leftrightarrow bRa}"></noscript><span class="lazy-image-placeholder" style="width: 11.597ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5734e3e9c97749481edd4a7327ae973c4c6c36b0" data-alt="{\displaystyle aRb\Leftrightarrow bRa}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Non-commuting_operators_in_quantum_mechanics">Non-commuting operators in quantum mechanics</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=20" title="Edit section: Non-commuting operators in quantum mechanics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">Canonical commutation relation</a></div> <p>In <a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">quantum mechanics</a> as formulated by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a>, physical variables are represented by <a href="/wiki/Linear_operators" class="mw-redirect" title="Linear operators">linear operators</a> such as <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (meaning multiply by <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), and <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="{\textstyle {\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d}{dx}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ce97a0649139023076f33481001a29d8d4ea4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.636ex; height:3.843ex;" alt="{\textstyle {\frac {d}{dx}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.636ex;height: 3.843ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ce97a0649139023076f33481001a29d8d4ea4f" data-alt="{\textstyle {\frac {d}{dx}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. These two operators do not commute as may be seen by considering the effect of their <a href="/wiki/Function_composition" title="Function composition">compositions</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="{\textstyle x{\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x{\frac {d}{dx}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9e94277a782f4c0ff189a8262dc93c7f0d0347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.966ex; height:3.843ex;" alt="{\textstyle x{\frac {d}{dx}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.966ex;height: 3.843ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9e94277a782f4c0ff189a8262dc93c7f0d0347" data-alt="{\textstyle x{\frac {d}{dx}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <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="{\textstyle {\frac {d}{dx}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {d}{dx}}x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a993151be6c3159033cce9e5ba8033789a63050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.966ex; height:3.843ex;" alt="{\textstyle {\frac {d}{dx}}x}"></noscript><span class="lazy-image-placeholder" style="width: 3.966ex;height: 3.843ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a993151be6c3159033cce9e5ba8033789a63050" data-alt="{\textstyle {\frac {d}{dx}}x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (also called products of operators) on a one-dimensional <a href="/wiki/Wave_function" title="Wave function">wave function</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 \psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"></noscript><span class="lazy-image-placeholder" style="width: 4.652ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" data-alt="{\displaystyle \psi (x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mtext> </mtext> <mo>≠</mo> <mtext> </mtext> <mi>ψ</mi> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>⋅</mo> <mi>ψ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f528887aba007e3af62da9c8c76b13b79c320f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.381ex; height:5.509ex;" alt="{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}"></noscript><span class="lazy-image-placeholder" style="width: 43.381ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f528887aba007e3af62da9c8c76b13b79c320f5" data-alt="{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> </p><p>According to the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> of <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually <a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">complementary</a>, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear <a href="/wiki/Momentum" title="Momentum">momentum</a> in the <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-direction of a particle are represented by the operators <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <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 -i\hbar {\frac {\partial }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.401ex; height:5.509ex;" alt="{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}"></noscript><span class="lazy-image-placeholder" style="width: 7.401ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf" data-alt="{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, respectively (where <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 \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></noscript><span class="lazy-image-placeholder" style="width: 1.306ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" data-alt="{\displaystyle \hbar }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the <a href="/wiki/Planck_constant" title="Planck constant">reduced Planck constant</a>). This is the same example except for the constant <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 -i\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c20ffda7dcb2d7857a3ae8f1da581ad799d517c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.917ex; height:2.343ex;" alt="{\displaystyle -i\hbar }"></noscript><span class="lazy-image-placeholder" style="width: 3.917ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c20ffda7dcb2d7857a3ae8f1da581ad799d517c" data-alt="{\displaystyle -i\hbar }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=21" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="512" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 40px;height: 40px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" data-alt="" data-width="40" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-class="mw-file-element"> </span></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/commutative_property" class="extiw" title="wiktionary:Special:Search/commutative property">commutative property</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Anticommutative_property" title="Anticommutative property">Anticommutative property</a></li> <li><a href="/wiki/Centralizer" class="mw-redirect" title="Centralizer">Centralizer and normalizer</a> (also called a commutant)</li> <li><a href="/wiki/Commutative_diagram" title="Commutative diagram">Commutative diagram</a></li> <li><a href="/wiki/Commutative_(neurophysiology)" class="mw-redirect" title="Commutative (neurophysiology)">Commutative (neurophysiology)</a></li> <li><a href="/wiki/Commutator" title="Commutator">Commutator</a></li> <li><a href="/wiki/Parallelogram_law" title="Parallelogram law">Parallelogram law</a></li> <li><a href="/wiki/Particle_statistics" title="Particle statistics">Particle statistics</a> (for commutativity in <a href="/wiki/Physics" title="Physics">physics</a>)</li> <li><a href="/wiki/Proofs_involving_the_addition_of_natural_numbers#Proof_of_commutativity" title="Proofs involving the addition of natural numbers">Proof that Peano's axioms imply the commutativity of the addition of natural numbers</a></li> <li><a href="/wiki/Quasi-commutative_property" title="Quasi-commutative property">Quasi-commutative property</a></li> <li><a href="/wiki/Trace_monoid" title="Trace monoid">Trace monoid</a></li> <li><a href="/wiki/Commuting_probability" title="Commuting probability">Commuting probability</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Notes">Notes</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=22" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Cabillón-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cabill%C3%B3n_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cabill%C3%B3n_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCabill%C3%B3nMiller">Cabillón & Miller</a>, <i>Commutative and Distributive</i></span> </li> <li id="cite_note-Flood11-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Flood11_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Flood11_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFloodRiceWilson2011" class="citation book cs1">Flood, Raymond; Rice, Adrian; <a href="/wiki/Robin_Wilson_(mathematician)" title="Robin Wilson (mathematician)">Wilson, Robin</a>, eds. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YruifIx88AQC&pg=PA4"><i>Mathematics in Victorian Britain</i></a>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780191627941" title="Special:BookSources/9780191627941"><bdi>9780191627941</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+Victorian+Britain&rft.pages=4&rft.pub=Oxford+University+Press&rft.date=2011&rft.isbn=9780191627941&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYruifIx88AQC%26pg%3DPA4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Symmetric_Relation"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SymmetricRelation.html">"Symmetric Relation"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Symmetric+Relation&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSymmetricRelation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></span></span> </li> <li id="cite_note-Krowne,_p.1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Krowne,_p.1_4-0">^</a></b></span> <span class="reference-text">Krowne, p. 1</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Weisstein, <i>Commute</i>, p. 1</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/users/29949/mathematicalorchid">"User MathematicalOrchid"</a>. <i>Mathematics Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">20 January</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematics+Stack+Exchange&rft.atitle=User+MathematicalOrchid&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fusers%2F29949%2Fmathematicalorchid&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFLumpkin1997">Lumpkin 1997</a>, p. 11</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFGayShute1987">Gay & Shute 1987</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">O'Conner & Robertson <i>Real Numbers</i></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">O'Conner & Robertson, <i>Servois</i></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGregory1840" class="citation journal cs1">Gregory, D. F. (1840). <a rel="nofollow" class="external text" href="https://archive.org/details/transactionsofro14royal">"On the real nature of symbolical algebra"</a>. <i>Transactions of the Royal Society of Edinburgh</i>. <b>14</b>: 208–216.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Royal+Society+of+Edinburgh&rft.atitle=On+the+real+nature+of+symbolical+algebra&rft.volume=14&rft.pages=208-216&rft.date=1840&rft.aulast=Gregory&rft.aufirst=D.+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftransactionsofro14royal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Moore and Parker</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFCopiCohen2005">Copi & Cohen 2005</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFHurleyWatson2016">Hurley & Watson 2016</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler1997">Axler 1997</a>, p. 2</span> </li> <li id="cite_note-Gallian,_p._34-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gallian,_p._34_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gallian,_p._34_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGallian2006">Gallian 2006</a>, p. 34</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFGallian2006">Gallian 2006</a>, pp. 26, 87</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFGallian2006">Gallian 2006</a>, p. 236</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFGallian2006">Gallian 2006</a>, p. 250</span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=23" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=24" title="Edit section: Books" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler1997" class="citation book cs1">Axler, Sheldon (1997). <i>Linear Algebra Done Right, 2e</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98258-2" title="Special:BookSources/0-387-98258-2"><bdi>0-387-98258-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right%2C+2e&rft.pub=Springer&rft.date=1997&rft.isbn=0-387-98258-2&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCopiCohen2005" class="citation book cs1">Copi, Irving M.; Cohen, Carl (2005). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780536179364"><i>Introduction to Logic</i></a></span> (12th ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780131898349" title="Special:BookSources/9780131898349"><bdi>9780131898349</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Logic&rft.edition=12th&rft.pub=Prentice+Hall&rft.date=2005&rft.isbn=9780131898349&rft.aulast=Copi&rft.aufirst=Irving+M.&rft.au=Cohen%2C+Carl&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780536179364&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2006" class="citation book cs1">Gallian, Joseph (2006). <i>Contemporary Abstract Algebra</i> (6e ed.). Houghton Mifflin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-618-51471-6" title="Special:BookSources/0-618-51471-6"><bdi>0-618-51471-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Abstract+Algebra&rft.edition=6e&rft.pub=Houghton+Mifflin&rft.date=2006&rft.isbn=0-618-51471-6&rft.aulast=Gallian&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodman2003" class="citation book cs1">Goodman, Frederick (2003). <i>Algebra: Abstract and Concrete, Stressing Symmetry</i> (2e ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-067342-0" title="Special:BookSources/0-13-067342-0"><bdi>0-13-067342-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra%3A+Abstract+and+Concrete%2C+Stressing+Symmetry&rft.edition=2e&rft.pub=Prentice+Hall&rft.date=2003&rft.isbn=0-13-067342-0&rft.aulast=Goodman&rft.aufirst=Frederick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Abstract algebra theory. Uses commutativity property throughout book.</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHurleyWatson2016" class="citation book cs1">Hurley, Patrick J.; Watson, Lori (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l-W5DQAAQBAJ&pg=PA675"><i>A Concise Introduction to Logic</i></a> (12th ed.). Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-337-51478-1" title="Special:BookSources/978-1-337-51478-1"><bdi>978-1-337-51478-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Logic&rft.edition=12th&rft.pub=Cengage+Learning&rft.date=2016&rft.isbn=978-1-337-51478-1&rft.aulast=Hurley&rft.aufirst=Patrick+J.&rft.au=Watson%2C+Lori&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dl-W5DQAAQBAJ%26pg%3DPA675&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles">Articles</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=25" title="Edit section: Articles" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLumpkin1997" class="citation web cs1">Lumpkin, B. (1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070713072942/http://www.ethnomath.org/resources/lumpkin1997.pdf">"The Mathematical Legacy of Ancient Egypt – A Response To Robert Palter"</a> <span class="cs1-format">(PDF)</span> (Unpublished manuscript). Archived from <a rel="nofollow" class="external text" href="http://www.ethnomath.org/resources/lumpkin1997.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 13 July 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Mathematical+Legacy+of+Ancient+Egypt+%E2%80%93+A+Response+To+Robert+Palter&rft.date=1997&rft.aulast=Lumpkin&rft.aufirst=B.&rft_id=http%3A%2F%2Fwww.ethnomath.org%2Fresources%2Flumpkin1997.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Article describing the mathematical ability of ancient civilizations.</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGayShute1987" class="citation book cs1"><a href="/wiki/Gay_Robins" title="Gay Robins">Gay, Robins R.</a>; <a href="/wiki/Charles_Shute_(academic)" title="Charles Shute (academic)">Shute, Charles C. D.</a> (1987). <i>The Rhind Mathematical Papyrus: An Ancient Egyptian Text</i>. British Museum. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7141-0944-4" title="Special:BookSources/0-7141-0944-4"><bdi>0-7141-0944-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus%3A+An+Ancient+Egyptian+Text&rft.pub=British+Museum&rft.date=1987&rft.isbn=0-7141-0944-4&rft.aulast=Gay&rft.aufirst=Robins+R.&rft.au=Shute%2C+Charles+C.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Translation and interpretation of the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>.</i></dd></dl></li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_resources">Online resources</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Commutative_property&action=edit&section=26" title="Edit section: Online resources" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Commutativity">"Commutativity"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Commutativity&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DCommutativity&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></li> <li>Krowne, Aaron, <a rel="nofollow" class="external text" href="https://planetmath.org/Commutative">Commutative</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>., Accessed 8 August 2007. <dl><dd><i>Definition of commutativity and examples of commutative operations</i></dd></dl></li> <li><span class="citation mathworld" id="Reference-Mathworld-Commute"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Commute.html">"Commute"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Commute&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCommute.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span></span>, Accessed 8 August 2007. <dl><dd><i>Explanation of the term commute</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYark" class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/?op=getuser&id=2760">"Yark"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Yark&rft_id=http%3A%2F%2Fplanetmath.org%2F%3Fop%3Dgetuser%26id%3D2760&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://planetmath.org/ExampleOfCommutative">Examples of non-commutative operations</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>., Accessed 8 August 2007 <dl><dd><i>Examples proving some noncommutative operations</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnerRobertson" class="citation web cs1">O'Conner, J.J.; Robertson, E.F. <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html">"History of real numbers"</a>. <i>MacTutor</i><span class="reference-accessdate">. Retrieved <span class="nowrap">8 August</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor&rft.atitle=History+of+real+numbers&rft.aulast=O%27Conner&rft.aufirst=J.J.&rft.au=Robertson%2C+E.F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FHistTopics%2FReal_numbers_1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Article giving the history of the real numbers</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCabillónMiller" class="citation web cs1">Cabillón, Julio; Miller, Jeff. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/c.html">"Earliest Known Uses of Mathematical Terms"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">22 November</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earliest+Known+Uses+of+Mathematical+Terms&rft.aulast=Cabill%C3%B3n&rft.aufirst=Julio&rft.au=Miller%2C+Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Page covering the earliest uses of mathematical terms</i></dd></dl></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnerRobertson" class="citation web cs1">O'Conner, J.J.; Robertson, E.F. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090902063524/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html">"biography of François Servois"</a>. <i>MacTutor</i>. Archived from <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html">the original</a> on 2 September 2009<span class="reference-accessdate">. Retrieved <span class="nowrap">8 August</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor&rft.atitle=biography+of+Fran%C3%A7ois+Servois&rft.aulast=O%27Conner&rft.aufirst=J.J.&rft.au=Robertson%2C+E.F.&rft_id=http%3A%2F%2Fwww-groups.dcs.st-and.ac.uk%2F~history%2FBiographies%2FServois.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACommutative+property" class="Z3988"></span> <dl><dd><i>Biography of Francois Servois, who first used the term</i></dd></dl></li></ul> <p><br> </p>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Commutative_property&oldid=1254713647">https://en.wikipedia.org/w/index.php?title=Commutative_property&oldid=1254713647</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Commutative_property&action=history"> <div class="post-content last-modified-bar__content"> <span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="80.215.93.19" data-user-gender="unknown" data-timestamp="1730452330"> <span>Last edited on 1 November 2024, at 09:12</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Kommutatiewe_bewerking" title="Kommutatiewe bewerking – Afrikaans" lang="af" hreflang="af" data-title="Kommutatiewe bewerking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link 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 – Arabic" lang="ar" hreflang="ar" data-title="عملية تبديلية" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Asturian" lang="ast" hreflang="ast" data-title="Conmutatividá" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Kommutativlik xassəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</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 – Bangla" lang="bn" hreflang="bn" data-title="বিনিময় বৈশিষ্ট্য" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Bashkir" lang="ba" hreflang="ba" data-title="Коммутативлыҡ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" 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 – Belarusian" lang="be" hreflang="be" data-title="Камутатыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Комутативност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Bosnian" lang="bs" hreflang="bs" data-title="Komutativnost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ca.wikipedia.org/wiki/Propietat_commutativa" title="Propietat commutativa – Catalan" lang="ca" hreflang="ca" data-title="Propietat commutativa" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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 – Chuvash" lang="cv" hreflang="cv" data-title="Коммутативлă операци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Komutativita" title="Komutativita – Czech" lang="cs" hreflang="cs" data-title="Komutativita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Danish" lang="da" hreflang="da" data-title="Kommutativitet" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kommutativgesetz" title="Kommutativgesetz – German" lang="de" hreflang="de" data-title="Kommutativgesetz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kommutatiivsus" title="Kommutatiivsus – Estonian" lang="et" hreflang="et" data-title="Kommutatiivsus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Greek" lang="el" hreflang="el" data-title="Αντιμεταθετική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conmutatividad" title="Conmutatividad – Spanish" lang="es" hreflang="es" data-title="Conmutatividad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Trukakortasun" title="Trukakortasun – Basque" lang="eu" hreflang="eu" data-title="Trukakortasun" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="خاصیت جابه‌جایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_commutative" title="Loi commutative – French" lang="fr" hreflang="fr" data-title="Loi commutative" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Irish" lang="ga" hreflang="ga" data-title="Oibríocht chómhalartach" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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 – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Co-iomlaideachd" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" 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 – Galician" lang="gl" hreflang="gl" data-title="Conmutatividade" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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 – Korean" lang="ko" hreflang="ko" data-title="교환법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Armenian" lang="hy" hreflang="hy" data-title="Տեղափոխականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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 – Croatian" lang="hr" hreflang="hr" data-title="Komutativnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_komutatif" title="Sifat komutatif – Indonesian" lang="id" hreflang="id" data-title="Sifat komutatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/V%C3%ADxlregla" title="Víxlregla – Icelandic" lang="is" hreflang="is" data-title="Víxlregla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Commutatività" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="פעולה קומוטטיבית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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 – Kazakh" lang="kk" hreflang="kk" data-title="Ауыстырымдылық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Commutativitas_(mathematica)" title="Commutativitas (mathematica) – Latin" lang="la" hreflang="la" data-title="Commutativitas (mathematica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Komutativit%C4%81te" title="Komutativitāte – Latvian" lang="lv" hreflang="lv" data-title="Komutativitāte" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Komutatyvumas" title="Komutatyvumas – Lithuanian" lang="lt" hreflang="lt" data-title="Komutatyvumas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kommutativit%C3%A1s" title="Kommutativitás – Hungarian" lang="hu" hreflang="hu" data-title="Kommutativitás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Комутативност" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Kalis tukar tertib" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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 – Dutch" lang="nl" hreflang="nl" data-title="Commutativiteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="交換法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Komutatiifgesets" title="Komutatiifgesets – Northern Frisian" lang="frr" hreflang="frr" data-title="Komutatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kommutativ_lov" title="Kommutativ lov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kommutativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kommutativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</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 – Polish" lang="pl" hreflang="pl" data-title="Przemienność" data-language-autonym="Polski" data-language-local-name="Polish" 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 – Portuguese" lang="pt" hreflang="pt" data-title="Comutatividade" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Comutativitate" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Коммутативность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</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 – Albanian" lang="sq" hreflang="sq" data-title="Vetia e ndërrimit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</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 – Sinhala" lang="si" hreflang="si" data-title="න්‍යාදේශ්‍ය න්‍යාය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" 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 – Simple English" lang="en-simple" hreflang="en-simple" data-title="Commutative property" data-language-autonym="Simple English" data-language-local-name="Simple English" 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 – Slovak" lang="sk" hreflang="sk" data-title="Komutatívnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Komutativnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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 – Central Kurdish" lang="ckb" hreflang="ckb" data-title="خاسیەتی ئاڵوگۆڕ" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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 – Serbian" lang="sr" hreflang="sr" data-title="Комутативност" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Komutativnost" title="Komutativnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Komutativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vaihdannaisuus" title="Vaihdannaisuus – Finnish" lang="fi" hreflang="fi" data-title="Vaihdannaisuus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kommutativitet" title="Kommutativitet – Swedish" lang="sv" hreflang="sv" data-title="Kommutativitet" data-language-autonym="Svenska" data-language-local-name="Swedish" 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-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 – Tatar" lang="tt" hreflang="tt" data-title="Коммутативлык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</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 – Thai" lang="th" hreflang="th" data-title="สมบัติการสลับที่" data-language-autonym="ไทย" data-language-local-name="Thai" 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 – Turkish" lang="tr" hreflang="tr" data-title="Değişme özelliği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Комутативність" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Venetian" lang="vec" hreflang="vec" data-title="Propietà comutativa" data-language-autonym="Vèneto" data-language-local-name="Venetian" 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 – Vietnamese" lang="vi" hreflang="vi" data-title="Tính giao hoán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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-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 – Cantonese" lang="yue" hreflang="yue" data-title="交換律" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</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 – Chinese" lang="zh" hreflang="zh" data-title="交換律" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 1 November 2024, at 09:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a 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