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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalences,_elimination,_and_introduction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalences,_elimination,_and_introduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Equivalences, elimination, and introduction</span> </div> </a> <ul id="toc-Equivalences,_elimination,_and_introduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negation_of_the_operator" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Negation_of_the_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Negation of the operator</span> </div> </a> <ul id="toc-Negation_of_the_operator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_modern_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_modern_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relation to modern algebra</span> </div> </a> <ul id="toc-Relation_to_modern_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exclusive_or_in_natural_language" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exclusive_or_in_natural_language"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Exclusive or in natural language</span> </div> </a> <ul id="toc-Exclusive_or_in_natural_language-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_symbols" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_symbols"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alternative symbols</span> </div> </a> <ul id="toc-Alternative_symbols-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Computer science</span> </div> </a> <button aria-controls="toc-Computer_science-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computer science subsection</span> </button> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> <li id="toc-Bitwise_operation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bitwise_operation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Bitwise operation</span> </div> </a> <ul id="toc-Bitwise_operation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Encodings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Encodings"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Encodings</span> </div> </a> <ul id="toc-Encodings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Exclusive or</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 32 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Disjunci%C3%B3_exclusiva" title="Disjunció exclusiva – Catalan" lang="ca" hreflang="ca" data-title="Disjunció exclusiva" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Exkluzivn%C3%AD_disjunkce" title="Exkluzivní disjunkce – Czech" lang="cs" hreflang="cs" data-title="Exkluzivní disjunkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kontravalenz" title="Kontravalenz – German" lang="de" hreflang="de" data-title="Kontravalenz" 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/V%C3%A4listav_disjunktsioon" title="Välistav disjunktsioon – Estonian" lang="et" hreflang="et" data-title="Välistav disjunktsioon" 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%CF%80%CE%BF%CE%BA%CE%BB%CE%B5%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CE%AC%CE%B6%CE%B5%CF%85%CE%BE%CE%B7" title="Αποκλειστική διάζευξη – Greek" lang="el" hreflang="el" data-title="Αποκλειστική διάζευξη" 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/Disyunci%C3%B3n_exclusiva" title="Disyunción exclusiva – Spanish" lang="es" hreflang="es" data-title="Disyunción exclusiva" 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/Disa%C5%ADo" title="Disaŭo – Esperanto" lang="eo" hreflang="eo" data-title="Disaŭo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%8C%D8%A7%DB%8C_%D8%A7%D9%86%D8%AD%D8%B5%D8%A7%D8%B1%DB%8C" title="یای انحصاری – Persian" lang="fa" hreflang="fa" data-title="یای انحصاری" 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/Fonction_OU_exclusif" title="Fonction OU exclusif – French" lang="fr" hreflang="fr" data-title="Fonction OU exclusif" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/OR_exclusivo" title="OR exclusivo – Galician" lang="gl" hreflang="gl" data-title="OR exclusivo" 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/%EB%B0%B0%ED%83%80%EC%A0%81_%EB%85%BC%EB%A6%AC%ED%95%A9" title="배타적 논리합 – Korean" lang="ko" hreflang="ko" data-title="배타적 논리합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Kontrevalenco" title="Kontrevalenco – Ido" lang="io" hreflang="io" data-title="Kontrevalenco" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Disgiunzione_esclusiva" title="Disgiunzione esclusiva – Italian" lang="it" hreflang="it" data-title="Disgiunzione esclusiva" 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/XOR" title="XOR – Hebrew" lang="he" hreflang="he" data-title="XOR" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D1%81%D0%BA%D0%BB%D1%83%D1%87%D0%B8%D1%82%D0%B5%D0%BB%D0%BD%D0%B0_%D0%B4%D0%B8%D1%81%D1%98%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Исклучителна дисјункција – Macedonian" lang="mk" hreflang="mk" data-title="Исклучителна дисјункција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Exclusieve_disjunctie" title="Exclusieve disjunctie – Dutch" lang="nl" hreflang="nl" data-title="Exclusieve disjunctie" 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/%E6%8E%92%E4%BB%96%E7%9A%84%E8%AB%96%E7%90%86%E5%92%8C" title="排他的論理和 – Japanese" lang="ja" hreflang="ja" data-title="排他的論理和" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eksklusiv_disjunksjon" title="Eksklusiv disjunksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Eksklusiv disjunksjon" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Disgionsion_esclusiva" title="Disgionsion esclusiva – Piedmontese" lang="pms" hreflang="pms" data-title="Disgionsion esclusiva" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Alternatywa_roz%C5%82%C4%85czna" title="Alternatywa rozłączna – Polish" lang="pl" hreflang="pl" data-title="Alternatywa rozłączna" 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/Ou_exclusivo" title="Ou exclusivo – Portuguese" lang="pt" hreflang="pt" data-title="Ou exclusivo" 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/Disjunc%C8%9Bie_exclusiv%C4%83" title="Disjuncție exclusivă – Romanian" lang="ro" hreflang="ro" data-title="Disjuncție exclusivă" 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%98%D1%81%D0%BA%D0%BB%D1%8E%D1%87%D0%B0%D1%8E%D1%89%D0%B5%D0%B5_%C2%AB%D0%B8%D0%BB%D0%B8%C2%BB" title="Исключающее «или» – Russian" lang="ru" hreflang="ru" data-title="Исключающее «или»" 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/XOR" title="XOR – Albanian" lang="sq" hreflang="sq" data-title="XOR" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Exclusive_disjunction" title="Exclusive disjunction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Exclusive disjunction" 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/Vylu%C4%8Duj%C3%BAce_alebo" title="Vylučujúce alebo – Slovak" lang="sk" hreflang="sk" data-title="Vylučujúce alebo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D1%81%D0%BA%D1%99%D1%83%D1%87%D0%B8%D0%B2%D0%B0_%D0%B4%D0%B8%D1%81%D1%98%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Искључива дисјункција – Serbian" lang="sr" hreflang="sr" data-title="Искључива дисјункција" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Eksklusiivinen_disjunktio" title="Eksklusiivinen disjunktio – Finnish" lang="fi" hreflang="fi" data-title="Eksklusiivinen disjunktio" 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/Exklusiv_disjunktion" title="Exklusiv disjunktion – Swedish" lang="sv" hreflang="sv" data-title="Exklusiv disjunktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%BB%D1%8E%D1%87%D0%BD%D0%B0_%D0%B4%D0%B8%D0%B7%27%D1%8E%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Виключна диз'юнкція – Ukrainian" lang="uk" hreflang="uk" data-title="Виключна диз'юнкція" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E7%95%B0%E6%88%96" title="邏輯異或 – Cantonese" lang="yue" hreflang="yue" data-title="邏輯異或" 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/%E9%80%BB%E8%BE%91%E5%BC%82%E6%88%96" title="逻辑异或 – Chinese" lang="zh" hreflang="zh" data-title="逻辑异或" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q498186#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Exclusive_or" title="View the content 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">True when either but not both inputs are true</div> <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">"XOR" redirects here. For the logic gate, see <a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a>. For other uses, see <a href="/wiki/XOR_(disambiguation)" class="mw-disambig" title="XOR (disambiguation)">XOR (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Exclusive_or" title="Special:EditPage/Exclusive or">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Exclusive+or%22">"Exclusive or"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Exclusive+or%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Exclusive+or%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Exclusive+or%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Exclusive+or%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Exclusive+or%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <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><table class="infobox"><caption class="infobox-title" style="background:navy; color:white;">Exclusive or</caption><tbody><tr><th colspan="2" class="infobox-above">XOR</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description" title="Venn diagram of Exclusive or"><img alt="Venn diagram of Exclusive or" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/150px-Venn0110.svg.png" decoding="async" width="150" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/225px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/300px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Truth_table" title="Truth table">Truth table</a></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 (0110)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0110</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0110)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b63ef3858fcb11ec9e1cf9a3f0e8f300da58c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.459ex; height:2.843ex;" alt="{\displaystyle (0110)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></th><td class="infobox-data"><span typeof="mw:File"><a href="/wiki/File:XOR_ANSI.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/XOR_ANSI.svg/70px-XOR_ANSI.svg.png" decoding="async" width="70" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/XOR_ANSI.svg/105px-XOR_ANSI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/XOR_ANSI.svg/140px-XOR_ANSI.svg.png 2x" data-file-width="100" data-file-height="50" /></a></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;">Normal forms</th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Disjunctive_normal_form" title="Disjunctive normal form">Disjunctive</a></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 {\overline {x}}\cdot y+x\cdot {\overline {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}\cdot y+x\cdot {\overline {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbb13548109264bd80e8539ec12239e94acc062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.409ex; height:2.676ex;" alt="{\displaystyle {\overline {x}}\cdot y+x\cdot {\overline {y}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conjunctive_normal_form" title="Conjunctive normal form">Conjunctive</a></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 ({\overline {x}}+{\overline {y}})\cdot (x+y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\overline {x}}+{\overline {y}})\cdot (x+y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8569d88cefecb443c9ff0a61b9ed5e70da0a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.189ex; height:2.843ex;" alt="{\displaystyle ({\overline {x}}+{\overline {y}})\cdot (x+y)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Zhegalkin_polynomial" title="Zhegalkin polynomial">Zhegalkin polynomial</a></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\oplus y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊕</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\oplus y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10fc94462e7622639c0c464161a1f0c8fc057999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x\oplus y}"></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;"><a href="/wiki/Post%27s_lattice" title="Post's lattice"><span style="color:white;">Post's lattices</span></a></th></tr><tr><th scope="row" class="infobox-label">0-preserving</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label">1-preserving</th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Monotonic_function" title="Monotonic function">Monotone</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Affine_transformation" title="Affine transformation">Affine</a></th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label">Self-dual</th><td class="infobox-data">no</td></tr><tr><td colspan="2" class="infobox-navbar"><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 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></th></tr><tr><td class="sidebar-content"> <table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.3em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Negation" title="Negation">NOT</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 \neg A,-A,{\overline {A}},\sim A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>,</mo> <mo>−</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mo>∼</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A,-A,{\overline {A}},\sim A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eab858e54d8de87e36fc80a991b32e74201a600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.001ex; height:3.343ex;" alt="{\displaystyle \neg A,-A,{\overline {A}},\sim A}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mtext> </mtext> <mi mathvariant="normal">&</mi> <mtext> </mtext> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mtext> </mtext> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c041e99940ccd418648ea18d200af37e2b3548d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.68ex; height:2.509ex;" alt="{\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">↑</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05374b45c2316947f052c6a46ca0f1d9381ed0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.98ex; height:3.509ex;" alt="{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B,A+B,A\mid B,A\parallel B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>∥</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B,A+B,A\mid B,A\parallel B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a262d8ab1dd1738c2b888661fe847101b624992d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle A\lor B,A+B,A\mid B,A\parallel B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">↓</mo> <mi>B</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/331ccd940d0039678505e971d3e13a63fca14354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.663ex; height:3.343ex;" alt="{\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊙</mo> <mi>B</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5a7f5c2cebe8c2903dea347e6ce9223cc47e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.669ex; height:3.843ex;" alt="{\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">⇔</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">⇋</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73fd8a2bddea3e7553e1905a4b2b8944269d5430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a class="mw-selflink selflink">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\underline {\lor }}B,A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B,A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48ea5022d9d865ea81c6f954cf73429be684009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:12.441ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B,A\oplus B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≢</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⇎</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>↮</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e31480781c46a0001e81f596615bc56e20d8aaa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⊃</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2d4ee4d40286755cb17f11743dcece3224fa90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_nonimplication" title="Material nonimplication">nonimplication (NIMPLY)</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⇏</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⊅</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>↛</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d66f3ed3dc468f35292dfe91a75d59b3b5d4915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇐</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">←</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128eb93aed65dd2e3aa1a4aaef4171a44f9a6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_nonimplication" title="Converse nonimplication">converse nonimplication</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⇍</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>⊄</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo>↚</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651dce7a12fa2331a8c610ee47b32982552a01f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}"></span></td></tr></tbody></table></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Related concepts</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="line-height:1.3em;"><ul><li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li><li><a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li><li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li><li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="/wiki/Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Applications</th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="/wiki/Programming_language" title="Programming language">Programming languages</a></li><li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li><li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td> </tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;"> <span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Logical_connectives" title="Category:Logical connectives">Category</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Logical_connectives_sidebar" title="Template:Logical connectives sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Logical_connectives_sidebar&action=edit&redlink=1" class="new" title="Template talk:Logical connectives sidebar (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives_sidebar" title="Special:EditPage/Template:Logical connectives sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_0110_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/220px-Venn_0110_1001.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/330px-Venn_0110_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/440px-Venn_0110_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B\oplus C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B\oplus C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecbdac83c3c26c4129785c69c8a77277683dba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.954ex; height:2.343ex;" alt="{\displaystyle A\oplus B\oplus C}"></span></figcaption></figure> <p><b>Exclusive or</b>, <b>exclusive disjunction</b>, <b>exclusive alternation</b>, <b>logical non-equivalence</b>, or <a href="/wiki/Logical_equality#Inequality" title="Logical equality">logical inequality</a> is a <a href="/wiki/Logical_connective" title="Logical connective">logical operator</a> whose negation is the <a href="/wiki/Logical_biconditional" title="Logical biconditional">logical biconditional</a>. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a>.<sup id="cite_ref-wolfram_1-0" class="reference"><a href="#cite_note-wolfram-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>It gains the name "exclusive or" because the meaning of "or" is ambiguous when both <a href="/wiki/Operand" title="Operand">operands</a> are true. XOR <i>excludes</i> that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B". </p><p>It is <a href="/wiki/Table_of_logic_symbols" class="mw-redirect" title="Table of logic symbols">symbolized</a> by the prefix operator <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span><sup id="cite_ref-Bochenski1949_2-0" class="reference"><a href="#cite_note-Bochenski1949-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 16">: 16 </span></sup> and by the <a href="/wiki/Infix_operator" class="mw-redirect" title="Infix operator">infix operators</a> <b>XOR</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/ɛ/: 'e' in 'dress'">ɛ</span><span title="'k' in 'kind'">k</span><span title="'s' in 'sigh'">s</span></span><span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɔː/: 'au' in 'fraud'">ɔː</span><span title="'r' in 'rye'">r</span></span>/</a></span></span>, <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/ɛ/: 'e' in 'dress'">ɛ</span><span title="'k' in 'kind'">k</span><span title="'s' in 'sigh'">s</span></span><span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɔː/: 'au' in 'fraud'">ɔː</span></span>/</a></span></span>, <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'k' in 'kind'">k</span><span title="'s' in 'sigh'">s</span><span title="/ɔː/: 'au' in 'fraud'">ɔː</span><span title="'r' in 'rye'">r</span></span>/</a></span></span> or <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'k' in 'kind'">k</span><span title="'s' in 'sigh'">s</span><span title="/ɔː/: 'au' in 'fraud'">ɔː</span></span>/</a></span></span>), <b>EOR</b>, <b>EXOR</b>, <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 {\dot {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e489691ea7d6ac0c6598efef8525ed22d1b54c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.509ex;" alt="{\displaystyle {\dot {\vee }}}"></span>, <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 {\overline {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9bdf4cb18d1b79d370c396dc425e80f8340f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}"></span>, <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 {\underline {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a328a1eeedacd21188476352ae3ffb47fd8165d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:1.553ex; height:3.009ex;" alt="{\displaystyle {\underline {\vee }}}"></span>, <b><span style="font-size:120%;">⩛</span></b>, <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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span>, <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 \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></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 \not \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d130bfc3eff6deb5c732a636f866cd9e373c197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.676ex;" alt="{\displaystyle \not \equiv }"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Variadic_logical_XOR.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Variadic_logical_XOR.svg/220px-Variadic_logical_XOR.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Variadic_logical_XOR.svg/330px-Variadic_logical_XOR.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Variadic_logical_XOR.svg/440px-Variadic_logical_XOR.svg.png 2x" data-file-width="1807" data-file-height="1452" /></a><figcaption>Each row of this binary <a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh matrix</a> is the truth table of the <a href="/wiki/Variadic_function" title="Variadic function">variadic</a> XOR of the arguments shown on the left. <small>E.g. row AB corresponds to the 2-circle, and row ABC to the 3-circle Venn diagram shown above. (As in the Venn diagrams, white is false, and red is true.)</small></figcaption></figure> <p>The <a href="/wiki/Truth_table" title="Truth table">truth table</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span> shows that it outputs true whenever the inputs differ: </p> <style data-mw-deduplicate="TemplateStyles:r1259796377">.mw-parser-output .two-ary-truth-table th{font-weight:normal}.mw-parser-output .two-ary-truth-table abbr{text-decoration:none}.mw-parser-output .two-ary-truth-table tr td:nth-child(1),.mw-parser-output .two-ary-truth-table tr td:nth-child(2){font-weight:bold}.mw-parser-output .two-ary-truth-table td{text-align:center;padding-left:14px;padding-right:14px}.mw-parser-output .two-ary-truth-table-false{background-color:var(--background-color-base,#fff)}.mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,90%)}.mw-parser-output .two-ary-truth-table-border{border-left:2px solid var(--border-color-interactive,#72777d)}@media screen{html.skin-theme-clientpref-night .mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,10%)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,10%)}}</style><table class="wikitable sortable two-ary-truth-table"><tbody><tr><th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></th><th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th><th class="unsortable two-ary-truth-table-border"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></th></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-false"><abbr title="false">F</abbr></td></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-false"><abbr title="false">F</abbr></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Equivalences,_elimination,_and_introduction"><span id="Equivalences.2C_elimination.2C_and_introduction"></span>Equivalences, elimination, and introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=2" title="Edit section: Equivalences, elimination, and introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction <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\nleftrightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42d587b5c154e4ff8b71ce713c005b9aad49eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}"></span>, also denoted 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 p\operatorname {?} q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>?</mo> </mrow> <mo>⁡</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\operatorname {?} q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53eb7c7ecb204557416ab10f09659860b926ba69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.2ex; height:2.509ex;" alt="{\displaystyle p\operatorname {?} q}"></span> or <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 Jpq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>p</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Jpq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba158b8b07114c247402a9dddd17fc2ea7fc64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle Jpq}"></span>, can be expressed in terms of the <a href="/wiki/Logical_conjunction" title="Logical conjunction">logical conjunction</a> ("logical 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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span>), the <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> ("logical or", <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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span>), and the <a href="/wiki/Negation" title="Negation">negation</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 \lnot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099107443792f5fec9bebe39b919a690db7198c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \lnot }"></span>) as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2975eb6d72eb2ad04f71d835c2c022eb45aac1d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.452ex; height:2.843ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}"></span></dd></dl> <p>The exclusive disjunction <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\nleftrightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42d587b5c154e4ff8b71ce713c005b9aad49eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}"></span> can also be expressed in the following way: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d33e818b1ed45a9319e81cb768e7919b4c84a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.003ex; height:2.843ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}"></span></dd></dl> <p>This representation of XOR may be found useful when constructing a circuit or network, because it has only one <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 \lnot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099107443792f5fec9bebe39b919a690db7198c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \lnot }"></span> operation and small number of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span> 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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span> operations. A proof of this identity is given below: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.7em 0.7em 0.7em 0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∨</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0b32f858658f10e29d829f1f86a61b727e7e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:60.762ex; height:19.176ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}"></span></dd></dl> <p>It is sometimes useful to write <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\nleftrightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42d587b5c154e4ff8b71ce713c005b9aad49eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}"></span> in the following way: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1375d40a528b9fd69e0dfaa0f947a45900c2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.362ex; height:2.843ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}"></span></dd></dl> <p>or: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89ba504dd9fa6a3d415a21919a7e62ab438dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.003ex; height:2.843ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}"></span></dd></dl> <p>This equivalence can be established by applying <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> twice to the fourth line of the above proof. </p><p>The exclusive or is also equivalent to the negation of a <a href="/wiki/Logical_biconditional" title="Logical biconditional">logical biconditional</a>, by the rules of material implication (a <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a> is equivalent to the disjunction of the negation of its <a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a> and its consequence) and <a href="/wiki/If_and_only_if" title="If and only if">material equivalence</a>. </p><p>In summary, we have, in mathematical and in engineering notation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.7em 0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>↮</mo> <mi>q</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∨</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>q</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>∧</mo> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mi>q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3871ca4f609dc7d0ee3b21540dd249cf55c749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:57.648ex; height:11.176ex;" alt="{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Negation_of_the_operator">Negation of the operator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=3" title="Edit section: Negation of the operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By applying the spirit of <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a>, we get: <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 \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>↮</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>↮</mo> <mi>q</mi> <mo stretchy="false">⇔</mo> <mi>p</mi> <mo>↮</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225a9a55694a530c1b7c64fcbd170b6b8b9d24b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.894ex; height:2.843ex;" alt="{\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_modern_algebra">Relation to modern algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=4" title="Edit section: Relation to modern algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operators</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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> (<a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a>) 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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span> (<a href="/wiki/Logical_disjunction" title="Logical disjunction">disjunction</a>) are very useful in logic systems, they fail a more generalizable structure in the following way: </p><p>The systems <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 (\{T,F\},\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>,</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\{T,F\},\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b934dddbe05f344fdc7ad09e6c48040a2ed98743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.129ex; height:2.843ex;" alt="{\displaystyle (\{T,F\},\wedge )}"></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 (\{T,F\},\lor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>,</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∨</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\{T,F\},\lor )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b2cd44bc42ef5617916d4ea70e42daa662f494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.129ex; height:2.843ex;" alt="{\displaystyle (\{T,F\},\lor )}"></span> are <a href="/wiki/Monoid" title="Monoid">monoids</a>, but neither is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. This unfortunately prevents the combination of these two systems into larger structures, such as a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">mathematical ring</a>. </p><p>However, the system using exclusive or <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 (\{T,F\},\oplus )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>,</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⊕</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\{T,F\},\oplus )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e0ed6eb05cef736fc9ae20ab8bba028fd5b425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.387ex; height:2.843ex;" alt="{\displaystyle (\{T,F\},\oplus )}"></span> <i>is</i> an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>. The combination of 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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> over elements <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 \{T,F\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>,</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{T,F\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e794e3d3c71ade40aa94383ff0da56ac72b9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.736ex; height:2.843ex;" alt="{\displaystyle \{T,F\}}"></span> produce the well-known <a href="/wiki/GF(2)" title="GF(2)">two-element field <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 \mathbb {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}"></span></a>. This field can represent any logic obtainable with the system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\land ,\lor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\land ,\lor )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce12e9cbe1e10df747c4f19962e18833a8cda39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.944ex; height:2.843ex;" alt="{\displaystyle (\land ,\lor )}"></span> and has the added benefit of the arsenal of algebraic analysis tools for fields. </p><p>More specifically, if one associates <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> with 0 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> with 1, one can interpret the logical "AND" operation as multiplication on <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 \mathbb {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}"></span> and the "XOR" operation as addition on <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 \mathbb {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.7em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> <mo>=</mo> <mi>p</mi> <mo>∧</mo> <mi>q</mi> </mtd> <mtd> <mo stretchy="false">⇔</mo> </mtd> <mtd> <mi>r</mi> <mo>=</mo> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>=</mo> <mi>p</mi> <mo>⊕</mo> <mi>q</mi> </mtd> <mtd> <mo stretchy="false">⇔</mo> </mtd> <mtd> <mi>r</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7811b12106fb2efca43f94e8ea67a96e0f9068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.73ex; height:7.176ex;" alt="{\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}"></span></dd></dl> <p>The description of a <a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a> as a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in <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 \mathbb {F} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}"></span>, using this basis, is called the function's <a href="/wiki/Algebraic_normal_form" title="Algebraic normal form">algebraic normal form</a>.<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> <div class="mw-heading mw-heading2"><h2 id="Exclusive_or_in_natural_language">Exclusive or in natural language</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=5" title="Edit section: Exclusive or in natural language"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Disjunction is often understood exclusively in <a href="/wiki/Natural_language" title="Natural language">natural languages</a>. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.<sup id="cite_ref-alonisep_4-0" class="reference"><a href="#cite_note-alonisep-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> </p> <dl><dd>1. Mary is a singer or a poet.</dd></dl> <p>However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be <a href="/wiki/Felicity_(pragmatics)" title="Felicity (pragmatics)">felicitously</a> used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under <a href="/wiki/Downward_entailing" title="Downward entailing">downward entailing</a> contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.<sup id="cite_ref-alonisep_4-1" class="reference"><a href="#cite_note-alonisep-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>2. Mary is either a singer or a poet or both.</dd> <dd>3. Nobody ate either rice or beans.</dd></dl> <p>Examples such as the above have motivated analyses of the exclusivity inference as <a href="/wiki/Pragmatics" title="Pragmatics">pragmatic</a> <a href="/wiki/Conversational_implicature" class="mw-redirect" title="Conversational implicature">conversational implicatures</a> calculated on the basis of an inclusive <a href="/wiki/Formal_semantics_(linguistics)" class="mw-redirect" title="Formal semantics (linguistics)">semantics</a>. Implicatures are typically <a href="/wiki/Cancellable_(linguistics)" class="mw-redirect" title="Cancellable (linguistics)">cancellable</a> and do not arise in downward entailing contexts if their calculation depends on the <a href="/wiki/Cooperative_principle#Maxim_of_quantity_(content_length_and_depth)" title="Cooperative principle">Maxim of Quantity</a>. However, some researchers have treated exclusivity as a bona fide semantic <a href="/wiki/Entailment" class="mw-redirect" title="Entailment">entailment</a> and proposed nonclassical logics which would validate it.<sup id="cite_ref-alonisep_4-2" class="reference"><a href="#cite_note-alonisep-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French <i>soit... soit</i>.<sup id="cite_ref-alonisep_4-3" class="reference"><a href="#cite_note-alonisep-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_symbols">Alternative symbols</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=6" title="Edit section: Alternative symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> was used by <a href="/wiki/George_Boole" title="George Boole">George Boole</a> in 1847.<sup id="cite_ref-boole1847_6-0" class="reference"><a href="#cite_note-boole1847-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Although Boole used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> mainly on classes, he also considered the case that <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> are propositions in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb4441dccedb5ede51a213408b17cf83eec9a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x+y}"></span>, and at the time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> is almost fixedly used nowadays) and exclusive disjunction, and may also bring about confusions with its other uses, some classical and modern textbooks still keep such use.<sup id="cite_ref-enderton2001_7-0" class="reference"><a href="#cite_note-enderton2001-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-rautenberg2010_8-0" class="reference"><a href="#cite_note-rautenberg2010-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9bdf4cb18d1b79d370c396dc425e80f8340f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}"></span> was used by <a href="/wiki/Christine_Ladd-Franklin" title="Christine Ladd-Franklin">Christine Ladd-Franklin</a> in 1883.<sup id="cite_ref-ladd1883_9-0" class="reference"><a href="#cite_note-ladd1883-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Strictly speaking, Ladd used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\operatorname {\overline {\vee }} B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mo>⁡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\operatorname {\overline {\vee }} B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a76cc75925d10fc19e6408e56f0870e95395e476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.947ex; height:2.843ex;" alt="{\displaystyle A\operatorname {\overline {\vee }} B}"></span> to express "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is-not <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>" or "No <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>", i.e., used <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 {\overline {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9bdf4cb18d1b79d370c396dc425e80f8340f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}"></span> as exclusions, while implicitly <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 {\overline {\vee }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\vee }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9bdf4cb18d1b79d370c396dc425e80f8340f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}"></span> has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \not =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf8afec7dacd23ce8632c3a4b4abb3873afa97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \not =}"></span>, denoting the negation of <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalence</a>, was used by <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a> in 1890,<sup id="cite_ref-schroder1890_10-0" class="reference"><a href="#cite_note-schroder1890-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 307">: 307 </span></sup> Although the usage of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> as equivalence could be dated back to <a href="/wiki/George_Boole" title="George Boole">George Boole</a> in 1847,<sup id="cite_ref-boole1847_6-1" class="reference"><a href="#cite_note-boole1847-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> during the 40 years after Boole, his followers, such as <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, <a href="/wiki/Hugh_MacColl" title="Hugh MacColl">Hugh MacColl</a>, <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> and so on, did not use <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 \not =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf8afec7dacd23ce8632c3a4b4abb3873afa97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \not =}"></span> as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> was used by <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> in 1894: "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\circ b=a-b\,\cup \,b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∘</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mo>∪</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\circ b=a-b\,\cup \,b-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71787593f2e19c150d99893373e800d36975cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.013ex; height:2.343ex;" alt="{\displaystyle a\circ b=a-b\,\cup \,b-a}"></span>. The sign <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> corresponds to Latin <i>aut</i>; the sign <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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"></span> to <i>vel</i>."<sup id="cite_ref-peano1894_11-0" class="reference"><a href="#cite_note-peano1894-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 10">: 10 </span></sup> Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use <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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"></span> as inclusive disjunction.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f744895a8e6d6d25e5d6fb5262a3bc76f9272e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.009ex;" alt="{\displaystyle \vee \vee }"></span> was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.<sup id="cite_ref-gradshtein1959_12-0" class="reference"><a href="#cite_note-gradshtein1959-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 76">: 76 </span></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> was used by <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> in 1938.<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> Shannon borrowed the symbol as exclusive disjunction from <a href="/wiki/Edward_Vermilye_Huntington" title="Edward Vermilye Huntington">Edward Vermilye Huntington</a> in 1904.<sup id="cite_ref-huntington1904_14-0" class="reference"><a href="#cite_note-huntington1904-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Huntington borrowed the symbol from <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in 1890 (the original date is not definitely known, but almost certainly it is written after 1685; and 1890 is the publishing time).<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> While both Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction (logical sum) too, and in 1933 used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> as inclusive disjunction.<sup id="cite_ref-huntington1933_16-0" class="reference"><a href="#cite_note-huntington1933-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \not \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d130bfc3eff6deb5c732a636f866cd9e373c197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.676ex;" alt="{\displaystyle \not \equiv }"></span>, also denoting the negation of <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalence</a>, was used by <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> in 1944.<sup id="cite_ref-church1944_17-0" class="reference"><a href="#cite_note-church1944-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> (as a <a href="/wiki/Polish_notation" title="Polish notation">prefix operator</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 J\phi \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>ϕ</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\phi \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf1bca8d96a9d832bc24751a96ef7528ad324fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.37ex; height:2.509ex;" alt="{\displaystyle J\phi \psi }"></span>) was used by <a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Józef Maria Bocheński</a> in 1949.<sup id="cite_ref-Bochenski1949_2-1" class="reference"><a href="#cite_note-Bochenski1949-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 16">: 16 </span></sup> Somebody<sup id="cite_ref-Craig_1998_18-0" class="reference"><a href="#cite_note-Craig_1998-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> may mistake that it is <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a> who is the first to use <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929<sup id="cite_ref-lukasiewicz1929_19-0" class="reference"><a href="#cite_note-lukasiewicz1929-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of <a href="/wiki/Polish_notation" title="Polish notation">Polish notation</a> that names all 16 binary <a href="/wiki/Logical_connective" title="Logical connective">connectives</a> of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> for exclusive disjunction appeared at the first time. Bocheński's usage of <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.</li> <li><samp>^</samp>, the <a href="/wiki/Caret" title="Caret">caret</a>, has been used in several <a href="/wiki/Programming_language" title="Programming language">programming languages</a> to denote the <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise</a> exclusive or operator, beginning with <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and also including <a href="/wiki/C%2B%2B" title="C++">C++</a>, <a href="/wiki/C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a>, <a href="/wiki/D_(programming_language)" title="D (programming language)">D</a>, <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="/wiki/Perl" title="Perl">Perl</a>, <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>, <a href="/wiki/PHP" title="PHP">PHP</a> and <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>.</li> <li>The <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> of two sets <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, which may be interpreted as their elementwise exclusive or, has variously been denoted 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 S\ominus T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊖</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\ominus T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7d22af18094c49a6d3acad282491b2d45dc22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.976ex; height:2.343ex;" alt="{\displaystyle S\ominus T}"></span>, <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 S\mathop {\triangledown } T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow class="MJX-TeXAtom-OP"> <mi class="MJX-variant">▽</mi> </mrow> <mo>⁡</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\mathop {\triangledown } T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402744ade3ed5a6c858fb5afcad8183aecbf055b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.588ex; height:2.176ex;" alt="{\displaystyle S\mathop {\triangledown } T}"></span>, or <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 S\mathop {\vartriangle } T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow class="MJX-TeXAtom-OP"> <mo class="MJX-variant">△</mo> </mrow> <mo>⁡</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\mathop {\vartriangle } T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580e1d12d1687f5410d803d861bbd367306a7c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.588ex; height:2.176ex;" alt="{\displaystyle S\mathop {\vartriangle } T}"></span>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1228772891">.mw-parser-output .glossary dt{margin-top:0.4em}.mw-parser-output .glossary dt+dt{margin-top:-0.2em}.mw-parser-output .glossary .templatequote{margin-top:0;margin-bottom:-0.5em}</style> <dl class="glossary"> <dt id="commutativity:_yes"><dfn><a href="/wiki/Commutative_property" title="Commutative property">Commutativity</a>: yes</dfn></dt><dd> <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\oplus A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊕</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\oplus A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae8cf8e0fc9b626cbc1fa3e8518ad3fce08b985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle B\oplus A}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/50px-Venn0110.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/75px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/100px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/50px-Venn0110.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/75px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/100px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td></tr></tbody></table> </dd> <dt id="associativity:_yes"><dfn><a href="/wiki/Associative_property" title="Associative property">Associativity</a>: yes</dfn></dt><dd> <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></span></td><td> <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 ~~~\oplus ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\oplus ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f07ca1a23ec64b131075da4ebb6f162f1460094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.324ex; height:2.176ex;" alt="{\displaystyle ~~~\oplus ~~~}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\oplus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\oplus C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d4800638f9d7524cd268e8e9443f12bf67afac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.18ex; height:2.843ex;" alt="{\displaystyle (B\oplus C)}"></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td></td><td></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\oplus B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\oplus B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966575de8d90ba0483c7eb54bd43ec27f7404e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.157ex; height:2.843ex;" alt="{\displaystyle (A\oplus B)}"></span></td><td> <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 ~~~\oplus ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\oplus ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f07ca1a23ec64b131075da4ebb6f162f1460094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.324ex; height:2.176ex;" alt="{\displaystyle ~~~\oplus ~~~}"></span></td><td> <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 ~C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f52ed2496dc4077efa433abb4685684a158d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.176ex;" alt="{\displaystyle ~C}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td> <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 ~~~\oplus ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\oplus ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f07ca1a23ec64b131075da4ebb6f162f1460094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.324ex; height:2.176ex;" alt="{\displaystyle ~~~\oplus ~~~}"></span></td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0011_1100.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/50px-Venn_0011_1100.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/75px-Venn_0011_1100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/100px-Venn_0011_1100.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0110_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/50px-Venn_0110_1001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/75px-Venn_0110_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/100px-Venn_0110_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0110_0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/50px-Venn_0110_0110.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/75px-Venn_0110_0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/100px-Venn_0110_0110.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td> <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 ~~~\oplus ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\oplus ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f07ca1a23ec64b131075da4ebb6f162f1460094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.324ex; height:2.176ex;" alt="{\displaystyle ~~~\oplus ~~~}"></span></td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0000_1111.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/50px-Venn_0000_1111.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/75px-Venn_0000_1111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/100px-Venn_0000_1111.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td></tr></tbody></table> </dd> <dt id="distributivity:"><dfn><a href="/wiki/Distributive_property" title="Distributive property">Distributivity</a>:</dfn></dt><dd>The exclusive or does not distribute over any binary function (not even itself), but <a href="/wiki/Logical_conjunction#Properties" title="Logical conjunction">logical conjunction distributes over exclusive or</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 C\land (A\oplus B)=(C\land A)\oplus (C\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7515efafcfd282a75a9e10f44b8723d5ea0955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.268ex; height:2.843ex;" alt="{\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)}"></span> (Conjunction and exclusive or form the multiplication and addition operations of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <a href="/wiki/GF(2)" title="GF(2)">GF(2)</a>, and as in any field they obey the distributive law.)</dd> <dt id="idempotency:_no"><dfn><a href="/wiki/Idempotence" title="Idempotence">Idempotency</a>: no</dfn></dt><dd> <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></span></td><td> <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 ~\oplus ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\oplus ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767eb37cbb7ba0cb1dd95c05efe9daf890806a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.002ex; height:2.176ex;" alt="{\displaystyle ~\oplus ~}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <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~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~0~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d7a0026db5c6cb729f05a18071b31816c11cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~0~}"></span></td><td>     <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 \nLeftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇎</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nLeftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeebe3d2aa323d1bd18a7c3ba7ff3179f7931471" 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 \nLeftrightarrow }"></span>    </td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span></td><td> <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 ~\oplus ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\oplus ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767eb37cbb7ba0cb1dd95c05efe9daf890806a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.002ex; height:2.176ex;" alt="{\displaystyle ~\oplus ~}"></span></td><td> <span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/36px-Venn00.svg.png" decoding="async" width="36" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/54px-Venn00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Venn00.svg/72px-Venn00.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span></td><td>     <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 \nLeftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇎</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nLeftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeebe3d2aa323d1bd18a7c3ba7ff3179f7931471" 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 \nLeftrightarrow }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span></td></tr></tbody></table> </dd> <dt id="monotonicity:_no"><dfn><a href="/wiki/Monotone_Boolean_function" class="mw-redirect" title="Monotone Boolean function">Monotonicity</a>: no</dfn></dt><dd> <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\rightarrow B}"></span></td><td>     <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 \nRightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇏</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nRightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>    </td><td></td><td></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\oplus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\oplus C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dafae2dad55444ab6ea1a036b65ca4dd88d3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.159ex; height:2.843ex;" alt="{\displaystyle (A\oplus C)}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\oplus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\oplus C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d4800638f9d7524cd268e8e9443f12bf67afac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.18ex; height:2.843ex;" alt="{\displaystyle (B\oplus C)}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn_1011_1011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/50px-Venn_1011_1011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/75px-Venn_1011_1011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/100px-Venn_1011_1011.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td>     <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 \nRightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇏</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nRightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_1011_1101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Venn_1011_1101.svg/50px-Venn_1011_1101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Venn_1011_1101.svg/75px-Venn_1011_1101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Venn_1011_1101.svg/100px-Venn_1011_1101.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0101_1010.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Venn_0101_1010.svg/50px-Venn_0101_1010.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Venn_0101_1010.svg/75px-Venn_0101_1010.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Venn_0101_1010.svg/100px-Venn_0101_1010.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></td><td> <span typeof="mw:File"><a href="/wiki/File:Venn_0011_1100.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/50px-Venn_0011_1100.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/75px-Venn_0011_1100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/100px-Venn_0011_1100.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></td></tr></tbody></table> </dd> <dt id="truth-preserving:_no"><dfn>Truth-preserving: no</dfn></dt><dd>When all inputs are true, the output is not true. <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span></td><td>     <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 \nRightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇏</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nRightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>    </td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/75px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/100px-Venn0001.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td><td>     <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 \nRightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇏</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nRightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/60px-Venn0110.svg.png" decoding="async" width="60" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/90px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/120px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td></tr></tbody></table> </dd> <dt id="falsehood-preserving:_yes"><dfn>Falsehood-preserving: yes</dfn></dt><dd>When all inputs are false, the output is false. <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span>    </td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/60px-Venn0110.svg.png" decoding="async" width="60" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/90px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/120px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn0111.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/50px-Venn0111.svg.png" decoding="async" width="50" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/75px-Venn0111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/100px-Venn0111.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></td></tr></tbody></table> </dd> <dt id="walsh_spectrum:_(2,0,0,−2)"><dfn><a href="/wiki/Hadamard_transform" title="Hadamard transform">Walsh spectrum</a>: (2,0,0,−2)</dfn></dt> <dt id="non-linearity:_0"><dfn>Non-<a href="/wiki/Linear#Boolean_functions" class="mw-redirect" title="Linear">linearity</a>: 0</dfn></dt><dd>The function is linear.</dd> <dt id="involution:"><dfn>Involution:</dfn></dt><dd>Exclusive or with one specified input, as a function of the other input, is an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> or self-inverse function; applying it twice leaves the variable input unchanged. <table style="text-align:center; border:1px solid darkgrey;"><tbody><tr style=";"><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A\oplus B~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\oplus B~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdeddbf102bbc267ca7017a2ba59e0d991d503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.509ex; height:2.343ex;" alt="{\displaystyle ~A\oplus B~}"></span></td><td> <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 ~\oplus ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\oplus ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767eb37cbb7ba0cb1dd95c05efe9daf890806a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.002ex; height:2.176ex;" alt="{\displaystyle ~\oplus ~}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~B~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>B</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~B~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956e638fba76c96f7fe763b97efbf2bac229169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.925ex; height:2.176ex;" alt="{\displaystyle ~B~}"></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></span></td></tr><tr style=";"><td> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/50px-Venn0110.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/75px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/100px-Venn0110.svg.png 2x" data-file-width="384" data-file-height="280" /></a></span></td><td> <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 ~\oplus ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>⊕</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\oplus ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767eb37cbb7ba0cb1dd95c05efe9daf890806a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.002ex; height:2.176ex;" alt="{\displaystyle ~\oplus ~}"></span></td><td> <span typeof="mw:File"><a href="/wiki/File:Venn0011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/50px-Venn0011.svg.png" decoding="async" width="50" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/75px-Venn0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn0011.svg/100px-Venn0011.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></td><td>     <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 }"></span>    </td><td> <span typeof="mw:File"><a href="/wiki/File:Venn0101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/50px-Venn0101.svg.png" decoding="async" width="50" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/75px-Venn0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Venn0101.svg/100px-Venn0101.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></td></tr></tbody></table> </dd> </dl> <p>If using <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> values for true (1) and false (0), then <i>exclusive or</i> works exactly like <a href="/wiki/Addition" title="Addition">addition</a> <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 2. </p> <div class="mw-heading mw-heading2"><h2 id="Computer_science">Computer science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=8" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:XOR_ANSI_Labelled.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/XOR_ANSI_Labelled.svg/114px-XOR_ANSI_Labelled.svg.png" decoding="async" width="114" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/XOR_ANSI_Labelled.svg/171px-XOR_ANSI_Labelled.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/XOR_ANSI_Labelled.svg/228px-XOR_ANSI_Labelled.svg.png 2x" data-file-width="120" data-file-height="50" /></a><figcaption>Traditional symbolic representation of an XOR <a href="/wiki/Logic_gate" title="Logic gate">logic gate</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Bitwise_operation">Bitwise operation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=9" title="Edit section: Bitwise operation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></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/Bitwise_operation" title="Bitwise operation">Bitwise operation</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Z2%5E4;_Cayley_table;_binary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Z2%5E4%3B_Cayley_table%3B_binary.svg/220px-Z2%5E4%3B_Cayley_table%3B_binary.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Z2%5E4%3B_Cayley_table%3B_binary.svg/330px-Z2%5E4%3B_Cayley_table%3B_binary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Z2%5E4%3B_Cayley_table%3B_binary.svg/440px-Z2%5E4%3B_Cayley_table%3B_binary.svg.png 2x" data-file-width="1127" data-file-height="1407" /></a><figcaption><a href="/wiki/Nimber" title="Nimber">Nimber</a> addition is the <i>exclusive or</i> of <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a> in <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> representation. This is also the vector addition in <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 (\mathbb {Z} /2\mathbb {Z} )^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e6377ca086aedf49b5e9f4a375b512a4235224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.289ex; height:3.176ex;" alt="{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}"></span>.</figcaption></figure> <p>Exclusive disjunction is often used for bitwise operations. Examples: </p> <ul><li>1 XOR 1 = 0</li> <li>1 XOR 0 = 1</li> <li>0 XOR 1 = 1</li> <li>0 XOR 0 = 0</li> <li><span class="nowrap"><span data-sort-value="7001140000000000000♠" style="display:none"></span>1110<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> XOR <span class="nowrap"><span data-sort-value="7000900000000000000♠" style="display:none"></span>1001<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> = <span class="nowrap"><span data-sort-value="7000700000000000000♠" style="display:none"></span>0111<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> (this is equivalent to addition without <a href="/wiki/Carry_(arithmetic)" title="Carry (arithmetic)">carry</a>)</li></ul> <p>As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two <i>n</i>-bit strings is identical to the standard vector of addition in the <a href="/wiki/Vector_space" title="Vector space">vector space</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 (\mathbb {Z} /2\mathbb {Z} )^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5686cb9b950687c6d056088cb2314829751e9ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.453ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}}"></span>. </p><p>In computer science, exclusive disjunction has several uses: </p> <ul><li>It tells whether two bits are unequal.</li> <li>It is a controllable bit-flipper (the control input chooses whether or not to invert the data input).</li> <li>It tells whether there is an <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> number of 1 bits (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B\oplus C\oplus D\oplus E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> <mo>⊕</mo> <mi>D</mi> <mo>⊕</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B\oplus C\oplus D\oplus E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535f2ba0a85d234e262cf865cb6332fca094d431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.335ex; height:2.343ex;" alt="{\displaystyle A\oplus B\oplus C\oplus D\oplus E}"></span> is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> an odd number of the variables are true), which is equal to the <a href="/wiki/Parity_bit" title="Parity bit">parity bit</a> returned by a <a href="/wiki/Parity_function" title="Parity function">parity function</a>.</li></ul> <p>In logical circuits, a simple <a href="/wiki/Adder_(electronics)" title="Adder (electronics)">adder</a> can be made with an <a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a> to add the numbers, and a series of AND, OR and NOT gates to create the carry output. </p><p>On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) than to load and store the value zero. </p><p>In <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>, XOR is sometimes used as a simple, self-inverse mixing function, such as in <a href="/wiki/One-time_pad" title="One-time pad">one-time pad</a> or <a href="/wiki/Feistel_cipher" title="Feistel cipher">Feistel network</a> systems.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2015)">citation needed</span></a></i>]</sup> XOR is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR). </p><p>In simple threshold-activated <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural networks</a>, modeling the XOR function requires a second layer because XOR is not a <a href="/wiki/Linear_separability" title="Linear separability">linearly separable</a> function. </p><p>Similarly, XOR can be used in generating <a href="/wiki/Entropy_pool" class="mw-redirect" title="Entropy pool">entropy pools</a> for <a href="/wiki/Hardware_random_number_generator" title="Hardware random number generator">hardware random number generators</a>. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>XOR is used in <a href="/wiki/RAID" title="RAID">RAID</a> 3–6 for creating parity information. For example, RAID can "back up" bytes <span class="nowrap"><span data-sort-value="7002156000000000000♠" style="display:none"></span>10011100<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> and <span class="nowrap"><span data-sort-value="7002108000000000000♠" style="display:none"></span>01101100<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> from two (or more) hard drives by XORing the just mentioned bytes, resulting in (<span class="nowrap"><span data-sort-value="7002240000000000000♠" style="display:none"></span>11110000<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span>) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing <span class="nowrap"><span data-sort-value="7002108000000000000♠" style="display:none"></span>01101100<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> is lost, <span class="nowrap"><span data-sort-value="7002156000000000000♠" style="display:none"></span>10011100<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> and <span class="nowrap"><span data-sort-value="7002240000000000000♠" style="display:none"></span>11110000<span style="vertical-align:sub; font-size:smaller; line-height:normal;">2</span></span> can be XORed to recover the lost byte.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow. </p><p>XOR can be used to swap two numeric variables in computers, using the <a href="/wiki/XOR_swap_algorithm" title="XOR swap algorithm">XOR swap algorithm</a>; however this is regarded as more of a curiosity and not encouraged in practice. </p><p><a href="/wiki/XOR_linked_list" title="XOR linked list">XOR linked lists</a> leverage XOR properties in order to save space to represent <a href="/wiki/Doubly_linked_list" title="Doubly linked list">doubly linked list</a> data structures. </p><p>In <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, XOR-based drawing methods are often used to manage such items as <a href="/wiki/Bounding_volume" title="Bounding volume">bounding boxes</a> and <a href="/wiki/Cursor_(computers)" class="mw-redirect" title="Cursor (computers)">cursors</a> on systems without <a href="/wiki/Alpha_compositing" title="Alpha compositing">alpha channels</a> or overlay planes. </p> <div class="mw-heading mw-heading2"><h2 id="Encodings">Encodings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=10" title="Edit section: Encodings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is also called "not left-right arrow" (<code>\nleftrightarrow</code>) in <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>-based markdown (<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 \nleftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce85ed756bc5a6cdf0f62892f57a6a1f96803ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftrightarrow }"></span>). Apart from the ASCII codes, the operator is encoded at <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+22BB</span> </span><span style="font-size:125%;line-height:1em">⊻</span> <span style="font-variant: small-caps; text-transform: lowercase;">XOR</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&veebar;</span>) and <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2295</span> </span><span style="font-size:125%;line-height:1em">⊕</span> <span style="font-variant: small-caps; text-transform: lowercase;">CIRCLED PLUS</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&CirclePlus;, &oplus;</span>), both in block <a href="/wiki/Mathematical_operators_and_symbols_in_Unicode#Mathematical_Operators" title="Mathematical operators and symbols in Unicode">mathematical operators</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Material_conditional" title="Material conditional">Material conditional</a> • <a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">(Paradox)</a></li> <li><a href="/wiki/Affirming_a_disjunct" title="Affirming a disjunct">Affirming a disjunct</a></li> <li><a href="/wiki/Ampheck" class="mw-redirect" title="Ampheck">Ampheck</a></li> <li><a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">Controlled NOT gate</a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism">Disjunctive syllogism</a></li> <li><a href="/wiki/Inclusive_or" class="mw-redirect" title="Inclusive or">Inclusive or</a></li> <li><a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">Involution</a></li> <li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">List of Boolean algebra topics</a></li> <li><a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li> <li><a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">Logical value</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Rule_90" title="Rule 90">Rule 90</a></li> <li><a href="/wiki/XOR_cipher" title="XOR cipher">XOR cipher</a></li> <li><a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a></li> <li><a href="/wiki/XOR_linked_list" title="XOR linked list">XOR linked list</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-wolfram-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-wolfram_1-0">^</a></b></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="CITEREFGermundssonWeisstein" class="citation web cs1">Germundsson, Roger; Weisstein, Eric. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/XOR.html">"XOR"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 June</span> 2015</span>.</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=XOR&rft.aulast=Germundsson&rft.aufirst=Roger&rft.au=Weisstein%2C+Eric&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FXOR.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-Bochenski1949-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bochenski1949_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bochenski1949_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBocheński1949" class="citation book cs1 cs1-prop-foreign-lang-source">Bocheński, J. M. (1949). <a rel="nofollow" class="external text" href="https://burjcdigital.urjc.es/bitstream/handle/10115/1425/PRECIS_DE_LOGIQUE_MATHEMATIQUE.pdf?sequence=1&isAllowed=y"><i>Précis de logique mathématique</i></a> <span class="cs1-format">(PDF)</span> (in French). The Netherlands: F. G. Kroonder, Bussum, Pays-Bas.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pr%C3%A9cis+de+logique+math%C3%A9matique&rft.place=The+Netherlands&rft.pub=F.+G.+Kroonder%2C+Bussum%2C+Pays-Bas&rft.date=1949&rft.aulast=Boche%C5%84ski&rft.aufirst=J.+M.&rft_id=https%3A%2F%2Fburjcdigital.urjc.es%2Fbitstream%2Fhandle%2F10115%2F1425%2FPRECIS_DE_LOGIQUE_MATHEMATIQUE.pdf%3Fsequence%3D1%26isAllowed%3Dy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span> Translated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBocheński1959" class="citation book cs1">Bocheński, J. M. (1959). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/precisofmathemat0000boch/"><i>A Precis of Mathematical Logic</i></a></span>. Translated by Bird, O. Dordrecht, Holland: D. Reidel Publishing Company. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-0592-9">10.1007/978-94-017-0592-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-8329-6" title="Special:BookSources/978-90-481-8329-6"><bdi>978-90-481-8329-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Precis+of+Mathematical+Logic&rft.place=Dordrecht%2C+Holland&rft.pub=D.+Reidel+Publishing+Company&rft.date=1959&rft_id=info%3Adoi%2F10.1007%2F978-94-017-0592-9&rft.isbn=978-90-481-8329-6&rft.aulast=Boche%C5%84ski&rft.aufirst=J.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprecisofmathemat0000boch%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoux2009" class="citation book cs1">Joux, Antoine (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=buQajqt-_iUC&pg=PA285">"9.2: Algebraic normal forms of Boolean functions"</a>. <i>Algorithmic Cryptanalysis</i>. CRC Press. pp. <span class="nowrap">285–</span>286. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420070033" title="Special:BookSources/9781420070033"><bdi>9781420070033</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=9.2%3A+Algebraic+normal+forms+of+Boolean+functions&rft.btitle=Algorithmic+Cryptanalysis&rft.pages=%3Cspan+class%3D%22nowrap%22%3E285-%3C%2Fspan%3E286&rft.pub=CRC+Press&rft.date=2009&rft.isbn=9781420070033&rft.aulast=Joux&rft.aufirst=Antoine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbuQajqt-_iUC%26pg%3DPA285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-alonisep-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-alonisep_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-alonisep_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-alonisep_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-alonisep_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAloni2016" class="citation encyclopaedia cs1"><a href="/wiki/Maria_Aloni" title="Maria Aloni">Aloni, Maria</a> (2016). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/win2016/entries/disjunction/">"Disjunction"</a>. In Zalta, Edward N. (ed.). <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2016 ed.). Metaphysics Research Lab, Stanford University<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Disjunction&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Winter+2016&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2016&rft.aulast=Aloni&rft.aufirst=Maria&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fwin2016%2Fentries%2Fdisjunction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></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">Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See Chapter 3, "The First Myth of 'Or'":<br /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJennings1994" class="citation book cs1">Jennings, R. E. (1994). <i>The Genealogy of Disjunction</i>. New York: Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Genealogy+of+Disjunction&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1994&rft.aulast=Jennings&rft.aufirst=R.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-boole1847-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-boole1847_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-boole1847_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoole1847" class="citation book cs1">Boole, G. (1847). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalanal00booluoft"><i>The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning</i></a>. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Analysis+of+Logic%2C+Being+an+Essay+Towards+a+Calculus+of+Deductive+Reasoning&rft.place=Cambridge%2FLondon&rft.pages=17&rft.pub=Macmillan%2C+Barclay%2C+%26+Macmillan%2FGeorge+Bell&rft.date=1847&rft.aulast=Boole&rft.aufirst=G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalanal00booluoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-enderton2001-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-enderton2001_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton2001" class="citation book cs1">Enderton, H. (2001) [1972]. <i>A Mathematical Introduction to Logic</i> (2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.place=San+Diego%2C+New+York%2C+Boston%2C+London%2C+Toronto%2C+Sydney+and+Tokyo&rft.pages=51&rft.edition=2&rft.pub=A+Harcourt+Science+and+Technology+Company&rft.date=2001&rft.aulast=Enderton&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-rautenberg2010-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-rautenberg2010_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRautenberg2010" class="citation book cs1">Rautenberg, W. (2010) [2006]. <i>A Concise Introduction to Mathematical Logic</i> (3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3.</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+Mathematical+Logic&rft.place=New+York%2C+Dordrecht%2C+Heidelberg+and+London&rft.pages=3&rft.edition=3&rft.pub=Springer&rft.date=2010&rft.aulast=Rautenberg&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-ladd1883-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-ladd1883_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLadd1883" class="citation encyclopaedia cs1">Ladd, Christine (1883). <a rel="nofollow" class="external text" href="https://archive.org/details/studiesinlogic00peiruoft/page/16">"On the Algebra of Logic"</a>. In Peirce, C. S. (ed.). <i>Studies in Logic by Members of the Johns Hopkins University</i>. Boston: Little, Brown & Company. pp. <span class="nowrap">17–</span>71.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+Algebra+of+Logic&rft.btitle=Studies+in+Logic+by+Members+of+the+Johns+Hopkins+University&rft.place=Boston&rft.pages=%3Cspan+class%3D%22nowrap%22%3E17-%3C%2Fspan%3E71&rft.pub=Little%2C+Brown+%26+Company&rft.date=1883&rft.aulast=Ladd&rft.aufirst=Christine&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstudiesinlogic00peiruoft%2Fpage%2F16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-schroder1890-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-schroder1890_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchröder1890" class="citation book cs1 cs1-prop-foreign-lang-source">Schröder, E. (1890). <i>Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band</i> (in German). Leipzig: Druck und Verlag B. G. Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+die+Algebra+der+Logik+%28Exakte+Logik%29%2C+Erster+Band&rft.place=Leipzig&rft.pub=Druck+und+Verlag+B.+G.+Teubner&rft.date=1890&rft.aulast=Schr%C3%B6der&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span> Reprinted by Thoemmes Press in 2000.</span> </li> <li id="cite_note-peano1894-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-peano1894_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1894" class="citation book cs1">Peano, G. (1894). <i>Notations de logique mathématique. Introduction au formulaire de mathématique</i>. Turin: Fratelli Boccna.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notations+de+logique+math%C3%A9matique.+Introduction+au+formulaire+de+math%C3%A9matique&rft.place=Turin&rft.pub=Fratelli+Boccna.&rft.date=1894&rft.aulast=Peano&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span> Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1958" class="citation book cs1">Peano, G. (1958). <a rel="nofollow" class="external text" href="https://archive.org/details/operescelte0002gius/page/n5/mode/2up"><i>Opere Scelte, Volume II</i></a>. Roma: Edizioni Cremonese. pp. <span class="nowrap">123–</span>176.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Opere+Scelte%2C+Volume+II&rft.place=Roma&rft.pages=%3Cspan+class%3D%22nowrap%22%3E123-%3C%2Fspan%3E176&rft.pub=Edizioni+Cremonese&rft.date=1958&rft.aulast=Peano&rft.aufirst=G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foperescelte0002gius%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-gradshtein1959-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-gradshtein1959_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFГРАДШТЕЙН1959" class="citation book cs1 cs1-prop-foreign-lang-source">ГРАДШТЕЙН, И. С. (1959) [1936]. <a rel="nofollow" class="external text" href="https://www.mathedu.ru/text/gradshteyn_pryamaya_i_obratnaya_teoremy_1959/p0/"><i>ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ</i></a> (in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%D0%9F%D0%A0%D0%AF%D0%9C%D0%90%D0%AF+%D0%98+%D0%9E%D0%91%D0%A0%D0%90%D0%A2%D0%9D%D0%90%D0%AF+%D0%A2%D0%95%D0%9E%D0%A0%D0%95%D0%9C%D0%AB%3A+%D0%AD%D0%9B%D0%95%D0%9C%D0%95%D0%9D%D0%A2%D0%AB+%D0%90%D0%9B%D0%93%D0%95%D0%91%D0%A0%D0%AB+%D0%9B%D0%9E%D0%93%D0%98%D0%9A%D0%98&rft.place=%D0%9C%D0%9E%D0%A1%D0%9A%D0%92%D0%90&rft.edition=3&rft.pub=%D0%93%D0%9E%D0%A1%D0%A3%D0%94%D0%90%D0%A0%D0%A1%D0%A2%D0%92%D0%95%D0%9D%D0%9D%D0%9E%D0%95+%D0%98%D0%97%D0%94%D0%90%D0%A2%D0%95%D0%9B%D0%AC%D0%A1%D0%A2%D0%92%D0%9E+%D0%A4%D0%98%D0%97%D0%98%D0%9A%D0%B0-%D0%9C%D0%90%D0%A2%D0%95%D0%9C%D0%90%D0%A2%D0%98%D0%A7%D0%95%D0%A1%D0%9A%D0%9E%D0%99+%D0%9B%D0%98%D0%A2%D0%95%D0%A0%D0%90%D0%A2%D0%A3%D0%A0%D0%AB&rft.date=1959&rft.aulast=%D0%93%D0%A0%D0%90%D0%94%D0%A8%D0%A2%D0%95%D0%99%D0%9D&rft.aufirst=%D0%98.+%D0%A1.&rft_id=https%3A%2F%2Fwww.mathedu.ru%2Ftext%2Fgradshteyn_pryamaya_i_obratnaya_teoremy_1959%2Fp0%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span> Translated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGradshtein1963" class="citation book cs1">Gradshtein, I. S. (1963). <i>Direct and Converse Theorems: The Elements of Symbolic Logic</i>. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Direct+and+Converse+Theorems%3A+The+Elements+of+Symbolic+Logic&rft.place=Oxford%2C+London%2C+New+York+and+Paris&rft.pub=Pergamon+Press&rft.date=1963&rft.aulast=Gradshtein&rft.aufirst=I.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShannon1938" class="citation journal cs1"><a href="/wiki/Claude_Elwood_Shannon" class="mw-redirect" title="Claude Elwood Shannon">Shannon, C. E.</a> (1938). <a rel="nofollow" class="external text" href="https://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf">"A Symbolic Analysis of Relay and Switching Circuits"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Institute of Electrical Engineers</i>. <b>57</b> (12): <span class="nowrap">713–</span>723. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FT-AIEE.1938.5057767">10.1109/T-AIEE.1938.5057767</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F11173">1721.1/11173</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51638483">51638483</a>.</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+American+Institute+of+Electrical+Engineers&rft.atitle=A+Symbolic+Analysis+of+Relay+and+Switching+Circuits&rft.volume=57&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E713-%3C%2Fspan%3E723&rft.date=1938&rft_id=info%3Ahdl%2F1721.1%2F11173&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51638483%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FT-AIEE.1938.5057767&rft.aulast=Shannon&rft.aufirst=C.+E.&rft_id=https%3A%2F%2Fwww.cs.virginia.edu%2F~evans%2Fgreatworks%2Fshannon38.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-huntington1904-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-huntington1904_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuntington1904" class="citation journal cs1">Huntington, E. V. (1904). 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W. (1890) [16??/17??]. Gerhardt, C. I. (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/diephilosophisc01leibgoog/page/n11/mode/2up"><i>Die philosophischen Schriften, Siebter Band</i></a> (in German). Berlin: Weidmann. p. 237<span class="reference-accessdate">. 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"New Sets of Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica". <i>Transactions of the American Mathematical Society</i>. <b>35</b> (1): <span class="nowrap">274–</span>304.</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+American+Mathematical+Society&rft.atitle=New+Sets+of+Independent+Postulates+for+the+Algebra+of+Logic%2C+With+Special+Reference+to+Whitehead+and+Russell%27s+Principia+Mathematica&rft.volume=35&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E274-%3C%2Fspan%3E304&rft.date=1933&rft.aulast=Huntington&rft.aufirst=E.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-church1944-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-church1944_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1996" class="citation book cs1">Church, A. (1996) [1944]. <i>Introduction to Mathematical Logic</i>. New Jersey: Princeton University Press. p. 37.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.place=New+Jersey&rft.pages=37&rft.pub=Princeton+University+Press&rft.date=1996&rft.aulast=Church&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-Craig_1998-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Craig_1998_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCraig1998" class="citation book cs1">Craig, Edward (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mxpFwcAplaAC&pg=PA496"><i>Routledge Encyclopedia of Philosophy, Volume 8</i></a>. <a href="/wiki/Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. p. 496. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-41507310-3" title="Special:BookSources/978-0-41507310-3"><bdi>978-0-41507310-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Routledge+Encyclopedia+of+Philosophy%2C+Volume+8&rft.pages=496&rft.pub=Taylor+%26+Francis&rft.date=1998&rft.isbn=978-0-41507310-3&rft.aulast=Craig&rft.aufirst=Edward&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmxpFwcAplaAC%26pg%3DPA496&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-lukasiewicz1929-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-lukasiewicz1929_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŁukasiewicz1929" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz, Jan</a> (1929). <i>Elementy logiki matematycznej</i> [<i>Elements of Mathematical Logic</i>] (in Polish) (1 ed.). 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Prentice-Hall. pp. <span class="nowrap">44–</span>46.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.9%3A+Bitwise+logical+operators&rft.btitle=The+C+Programming+Language&rft.pages=%3Cspan+class%3D%22nowrap%22%3E44-%3C%2Fspan%3E46&rft.pub=Prentice-Hall&rft.date=1978&rft.aulast=Kernighan&rft.aufirst=Brian+W.&rft.au=Ritchie%2C+Dennis+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FTheCProgrammingLanguageFirstEdition%2Fpage%2Fn51&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Symmetric_Difference"><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/SymmetricDifference.html">"Symmetric Difference"</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+Difference&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSymmetricDifference.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavies2002" class="citation web cs1">Davies, Robert B (28 February 2002). <a rel="nofollow" class="external text" href="http://www.robertnz.net/pdf/xor2.pdf">"Exclusive OR (XOR) and hardware random number generators"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">28 August</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Exclusive+OR+%28XOR%29+and+hardware+random+number+generators&rft.date=2002-02-28&rft.aulast=Davies&rft.aufirst=Robert+B&rft_id=http%3A%2F%2Fwww.robertnz.net%2Fpdf%2Fxor2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNobel2011" class="citation web cs1">Nobel, Rickard (26 July 2011). <a rel="nofollow" class="external text" href="http://rickardnobel.se/how-raid5-works">"How RAID 5 actually works"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">23 March</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+RAID+5+actually+works&rft.date=2011-07-26&rft.aulast=Nobel&rft.aufirst=Rickard&rft_id=http%3A%2F%2Frickardnobel.se%2Fhow-raid5-works&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExclusive+or" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exclusive_or&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Logical_connectives" title="Template:Logical connectives"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Logical_connectives" title="Template talk:Logical connectives"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives" title="Special:EditPage/Template:Logical connectives"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Common_logical_connectives157" style="font-size:114%;margin:0 4em">Common <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a>/<a href="/wiki/Logical_truth" title="Logical truth">True</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 \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span></li></ul> </div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/80px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="80" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/120px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/160px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> (<a href="/wiki/NAND_gate" title="NAND gate">NAND gate</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 \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></li> <li><a href="/wiki/Converse_(logic)" title="Converse (logic)">Converse implication</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 \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" 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 \leftarrow }"></span></li> <li><a href="/wiki/Material_conditional" title="Material conditional">Implication</a> (<a href="/wiki/IMPLY_gate" title="IMPLY gate">IMPLY gate</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 \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction</a> (<a href="/wiki/OR_gate" title="OR gate">OR gate</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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation" title="Negation">Negation</a> (<a href="/wiki/Inverter_(logic_gate)" title="Inverter (logic gate)">NOT gate</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 \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></li> <li><a class="mw-selflink selflink">Exclusive or</a> (<a href="/wiki/XOR_gate" title="XOR gate">XOR gate</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 \not \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363ed81fd02da85c658dde9f17737c13b7263e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \not \leftrightarrow }"></span></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">Biconditional</a> (<a href="/wiki/XNOR_gate" title="XNOR gate">XNOR gate</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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a> (<a href="/wiki/Digital_buffer" title="Digital buffer">Digital buffer</a>)</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Logical_NOR" title="Logical NOR">Joint denial</a> (<a href="/wiki/NOR_gate" title="NOR gate">NOR gate</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 \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></li> <li><a href="/wiki/Material_nonimplication" title="Material nonimplication">Nonimplication</a> (<a href="/wiki/NIMPLY_gate" title="NIMPLY gate">NIMPLY gate</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 \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></span></li> <li><a href="/wiki/Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</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 \nleftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7694c9fc8eebe8a57c8156dd3c2caf022a619439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftarrow }"></span></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction</a> (<a href="/wiki/AND_gate" title="AND gate">AND gate</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 \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a>/<a href="/wiki/False_(logic)" title="False (logic)">False</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 \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; 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