De Morgan's laws - Wikipedia

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id="toc-Proof_for_Boolean_algebra-sublist" class="vector-toc-list"> <li id="toc-Negation_of_a_disjunction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negation_of_a_disjunction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Negation of a disjunction</span> </div> </a> <ul id="toc-Negation_of_a_disjunction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negation_of_a_conjunction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negation_of_a_conjunction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Negation of a conjunction</span> </div> </a> <ul id="toc-Negation_of_a_conjunction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_for_set_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_for_set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Proof for set theory</span> </div> </a> <button aria-controls="toc-Proof_for_set_theory-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 Proof for set theory subsection</span> </button> <ul id="toc-Proof_for_set_theory-sublist" class="vector-toc-list"> <li id="toc-Part_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Part_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Part 1</span> </div> </a> <ul id="toc-Part_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Part_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Part_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Part 2</span> </div> </a> <ul id="toc-Part_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conclusion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conclusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Conclusion</span> </div> </a> <ul id="toc-Conclusion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalising_De_Morgan_duality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalising_De_Morgan_duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalising De Morgan duality</span> </div> </a> <ul id="toc-Generalising_De_Morgan_duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_to_predicate_and_modal_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_to_predicate_and_modal_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Extension to predicate and modal logic</span> </div> </a> <ul id="toc-Extension_to_predicate_and_modal_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_intuitionistic_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_intuitionistic_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In intuitionistic logic</span> </div> </a> <ul id="toc-In_intuitionistic_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_computer_engineering" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_computer_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>In computer engineering</span> </div> </a> <ul id="toc-In_computer_engineering-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">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-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">11</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" > <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">De Morgan's laws</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 47 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-47" 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">47 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%88%D8%A7%D9%86%D9%8A%D9%86_%D8%AF%D9%8A_%D9%85%D9%88%D8%B1%D8%BA%D8%A7%D9%86" title="قوانين دي مورغان – Arabic" lang="ar" hreflang="ar" data-title="قوانين دي مورغان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/De_Morqan_qanunlar%C4%B1" title="De Morqan qanunları – Azerbaijani" lang="az" hreflang="az" data-title="De Morqan qanunları" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/De_Morgan_%C3%AA_hoat-chek" title="De Morgan ê hoat-chek – Minnan" lang="nan" hreflang="nan" data-title="De Morgan ê hoat-chek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%94%D1%8D_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD%D0%B0" title="Законы Дэ Моргана – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Законы Дэ Моргана" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8_%D0%BD%D0%B0_%D0%94%D0%B5_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD" title="Закони на Де Морган – Bulgarian" lang="bg" hreflang="bg" data-title="Закони на Де Морган" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Lleis_de_De_Morgan" title="Lleis de De Morgan – Catalan" lang="ca" hreflang="ca" data-title="Lleis de De Morgan" 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/De_Morganovy_z%C3%A1kony" title="De Morganovy zákony – Czech" lang="cs" hreflang="cs" data-title="De Morganovy zákony" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deddfau_De_Morgan" title="Deddfau De Morgan – Welsh" lang="cy" hreflang="cy" data-title="Deddfau De Morgan" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/De_Morgans_love" title="De Morgans love – Danish" lang="da" hreflang="da" data-title="De Morgans love" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/De-morgansche_Gesetze" title="De-morgansche Gesetze – German" lang="de" hreflang="de" data-title="De-morgansche Gesetze" 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/De_Morgani_seadused" title="De Morgani seadused – Estonian" lang="et" hreflang="et" data-title="De Morgani seadused" 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%A4%CF%8D%CF%80%CE%BF%CE%B9_%CE%9D%CF%84%CE%B5_%CE%9C%CF%8C%CF%81%CE%B3%CE%BA%CE%B1%CE%BD" title="Τύποι Ντε Μόργκαν – Greek" lang="el" hreflang="el" data-title="Τύποι Ντε Μόργκαν" 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/Leyes_de_De_Morgan" title="Leyes de De Morgan – Spanish" lang="es" hreflang="es" data-title="Leyes de De Morgan" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/De_Morganen_legeak" title="De Morganen legeak – Basque" lang="eu" hreflang="eu" data-title="De Morganen legeak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D9%88%D8%A7%D9%86%DB%8C%D9%86_%D8%AF%D9%85%D9%88%D8%B1%DA%AF%D8%A7%D9%86" title="قوانین دمورگان – Persian" lang="fa" hreflang="fa" data-title="قوانین دمورگان" 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/Lois_de_De_Morgan" title="Lois de De Morgan – French" lang="fr" hreflang="fr" data-title="Lois de De Morgan" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%93%9C_%EB%AA%A8%EB%A5%B4%EA%B0%84%EC%9D%98_%EB%B2%95%EC%B9%99" title="드 모르간의 법칙 – Korean" lang="ko" hreflang="ko" data-title="드 모르간의 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5_%D5%84%D5%B8%D6%80%D5%A3%D5%A1%D5%B6%D5%AB_%D5%A2%D5%A1%D5%B6%D5%A1%D5%B1%D6%87%D5%A5%D6%80" title="Դե Մորգանի բանաձևեր – Armenian" lang="hy" hreflang="hy" data-title="Դե Մորգանի բանաձևեր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%AE%E0%A5%89%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%A8_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="डिमॉर्गन नियम – Hindi" lang="hi" hreflang="hi" data-title="डिमॉर्गन नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_De_Morgan" title="Hukum De Morgan – Indonesian" lang="id" hreflang="id" data-title="Hukum De Morgan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/De_Morgan-reglan" title="De Morgan-reglan – Icelandic" lang="is" hreflang="is" data-title="De Morgan-reglan" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Leggi_di_De_Morgan" title="Leggi di De Morgan – Italian" lang="it" hreflang="it" data-title="Leggi di De Morgan" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%9C%D7%9C%D7%99_%D7%93%D7%94_%D7%9E%D7%95%D7%A8%D7%92%D7%9F" title="כללי דה מורגן – Hebrew" lang="he" hreflang="he" data-title="כללי דה מורגן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/De_Morgana_likumi" title="De Morgana likumi – Latvian" lang="lv" hreflang="lv" data-title="De Morgana likumi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dualioji_funkcija" title="Dualioji funkcija – Lithuanian" lang="lt" hreflang="lt" data-title="Dualioji funkcija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_di_De_Morgan" title="Teorema di De Morgan – Lombard" lang="lmo" hreflang="lmo" data-title="Teorema di De Morgan" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/De_Morgan-azonoss%C3%A1gok" title="De Morgan-azonosságok – Hungarian" lang="hu" hreflang="hu" data-title="De Morgan-azonosságok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8" title="Де Морганови закони – Macedonian" lang="mk" hreflang="mk" data-title="Де Морганови закони" 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/Wetten_van_De_Morgan" title="Wetten van De Morgan – Dutch" lang="nl" hreflang="nl" data-title="Wetten van De Morgan" 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/%E3%83%89%E3%83%BB%E3%83%A2%E3%83%AB%E3%82%AC%E3%83%B3%E3%81%AE%E6%B3%95%E5%89%87" title="ド・モルガンの法則 – Japanese" lang="ja" hreflang="ja" data-title="ド・モルガンの法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/De_Morgans_lover" title="De Morgans lover – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="De Morgans lover" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawa_De_Morgana" title="Prawa De Morgana – Polish" lang="pl" hreflang="pl" data-title="Prawa De Morgana" 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/Teoremas_de_De_Morgan" title="Teoremas de De Morgan – Portuguese" lang="pt" hreflang="pt" data-title="Teoremas de De Morgan" 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/Formulele_lui_De_Morgan" title="Formulele lui De Morgan – Romanian" lang="ro" hreflang="ro" data-title="Formulele lui De Morgan" 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%97%D0%B0%D0%BA%D0%BE%D0%BD%D1%8B_%D0%B4%D0%B5_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD%D0%B0" title="Законы де Моргана – Russian" lang="ru" hreflang="ru" data-title="Законы де Моргана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/De_Morgan%27s_laws" title="De Morgan's laws – Simple English" lang="en-simple" hreflang="en-simple" data-title="De Morgan's laws" 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/De_Morganove_z%C3%A1kony" title="De Morganove zákony – Slovak" lang="sk" hreflang="sk" data-title="De Morganove zákony" 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%94%D0%B5_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8" title="Де Морганови закони – Serbian" lang="sr" hreflang="sr" data-title="Де Морганови закони" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/De_Morganovi_zakoni" title="De Morganovi zakoni – Serbo-Croatian" lang="sh" hreflang="sh" data-title="De Morganovi zakoni" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/De_Morganin_lait" title="De Morganin lait – Finnish" lang="fi" hreflang="fi" data-title="De Morganin lait" 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/De_Morgans_lagar" title="De Morgans lagar – Swedish" lang="sv" hreflang="sv" data-title="De Morgans lagar" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4_%E0%AE%AE%E0%AF%8B%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AE%A9%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="த மோர்கனின் விதி – Tamil" lang="ta" hreflang="ta" data-title="த மோர்கனின் விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B9%80%E0%B8%94%E0%B8%AD%E0%B8%A1%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B9%81%E0%B8%81%E0%B8%99" title="กฎเดอมอร์แกน – Thai" lang="th" hreflang="th" data-title="กฎเดอมอร์แกน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/De_Morgan_yasas%C4%B1" title="De Morgan yasası – Turkish" lang="tr" hreflang="tr" data-title="De Morgan yasası" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%B0_%D0%B4%D0%B5_%D0%9C%D0%BE%D1%80%D0%B3%D0%B0%D0%BD%D0%B0" title="Правила де Моргана – Ukrainian" lang="uk" hreflang="uk" data-title="Правила де Моргана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Lu%E1%BA%ADt_De_Morgan" title="Luật De Morgan – Vietnamese" lang="vi" hreflang="vi" data-title="Luật De Morgan" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Rule_of_inference" title="Rule of inference">Transformation rules</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;"> <a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Conditional_proof" title="Conditional proof"><span>Implication introduction</span></a> / <a href="/wiki/Modus_ponens" title="Modus ponens"><span title="A→B,   A   ⊢   B">elimination (<i>modus ponens</i>)</span></a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction"><span title="A→B,   B→A   ⊢   A↔B">Biconditional introduction</span></a> / <a href="/wiki/Biconditional_elimination" title="Biconditional elimination"><span title="A↔B   ⊢   A→B">elimination</span></a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction"><span title="A,   B   ⊢   A∧B">Conjunction introduction</span></a> / <a href="/wiki/Conjunction_elimination" title="Conjunction elimination"><span title="A∧B   ⊢   A">elimination</span></a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction"><span title="A   ⊢   A∨B">Disjunction introduction</span></a> / <a href="/wiki/Disjunction_elimination" title="Disjunction elimination"><span title="A∨B,   A→C,   B→C   ⊢   C">elimination</span></a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism"><span title="A∨B,   ¬A   ⊢   B">Disjunctive</span></a> / <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism"><span title="A→B,   B→C   ⊢   A→C">hypothetical syllogism</span></a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma"><span title="A→P,   B→Q,   A∨B   ⊢   P∨Q">Constructive</span></a> / <a href="/wiki/Destructive_dilemma" title="Destructive dilemma"><span title="A→P,   B→Q,   ¬P∨¬Q   ⊢   ¬A∨¬B">destructive dilemma</span></a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)"><span title="A→B   ⊢   A→A∧B">Absorption</span></a> / <a href="/wiki/Modus_tollens" title="Modus tollens"><span title="A→B,   ¬B   ⊢   ¬A"><i>modus tollens</i></span></a> / <a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens"><span title="¬(A∧B),   A   ⊢   ¬B"><i>modus ponendo tollens</i></span></a></li> <li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <div class="hlist"> <ul><li><a href="/wiki/Associative_property#Propositional_logic" title="Associative property"><span title="A∨(B∨C)   =   (A∨B)∨C">Associativity</span></a></li> <li><a href="/wiki/Commutative_property#Propositional_logic" title="Commutative property"><span title="A∨B   =   B∨A">Commutativity</span></a></li> <li><a href="/wiki/Distributive_property#Propositional_logic" title="Distributive property"><span title="A∧(B∨C)   =   (A∧B)∨(A∧C)">Distributivity</span></a></li> <li><a href="/wiki/Double_negation" title="Double negation"><span title="¬¬A   =   A">Double negation</span></a></li> <li><a class="mw-selflink selflink">De Morgan's laws</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)"><span title="A→B   ⊢   ¬A∨B">Material implication</span></a></li> <li><a href="/wiki/Exportation_(logic)" title="Exportation (logic)"><span title="(A∧B)→C   ⊢   A→(B→C)">Exportation</span></a></li> <li><a href="/wiki/Tautology_(rule_of_inference)" title="Tautology (rule of inference)"><span title="A∨A   =   A">Tautology</span></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal generalization</a> / <a href="/wiki/Universal_instantiation" title="Universal instantiation">instantiation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="/wiki/Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td> </tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Demorganlaws.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Demorganlaws.svg/220px-Demorganlaws.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Demorganlaws.svg/330px-Demorganlaws.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Demorganlaws.svg/440px-Demorganlaws.svg.png 2x" data-file-width="600" data-file-height="800" /></a><figcaption>De Morgan's laws represented with <a href="/wiki/Venn_diagrams" class="mw-redirect" title="Venn diagrams">Venn diagrams</a>. In each case, the resultant set is the set of all points in any shade of blue.</figcaption></figure> <p>In <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a> and <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, <b>De Morgan's laws</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> also known as <b>De Morgan's theorem</b>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> are a pair of transformation rules that are both <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> <a href="/wiki/Rule_of_inference" title="Rule of inference">rules of inference</a>. They are named after <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a>, a 19th-century British mathematician. The rules allow the expression of <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunctions</a> and <a href="/wiki/Logical_disjunction" title="Logical disjunction">disjunctions</a> purely in terms of each other via <a href="/wiki/Logical_negation" class="mw-redirect" title="Logical negation">negation</a>. </p><p>The rules can be expressed in English as: </p> <ul><li>The negation of "A and B" is the same as "not A or not B".</li> <li>The negation of "A or B" is the same as "not A and not B".</li></ul> <p>or </p> <ul><li>The <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> of the union of two sets is the same as the intersection of their complements</li> <li>The complement of the intersection of two sets is the same as the union of their complements</li></ul> <p>or </p> <ul><li>not (A or B) = (not A) and (not B)</li> <li>not (A and B) = (not A) or (not B)</li></ul> <p>where "A or B" is an "<a href="/wiki/Inclusive_or" class="mw-redirect" title="Inclusive or">inclusive or</a>" meaning <i>at least</i> one of A or B rather than an "<a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>" that means <i>exactly</i> one of A or B. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:De_Morgan%27s_law_with_set_subtraction_operation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/De_Morgan%27s_law_with_set_subtraction_operation.png/220px-De_Morgan%27s_law_with_set_subtraction_operation.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/De_Morgan%27s_law_with_set_subtraction_operation.png/330px-De_Morgan%27s_law_with_set_subtraction_operation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/De_Morgan%27s_law_with_set_subtraction_operation.png/440px-De_Morgan%27s_law_with_set_subtraction_operation.png 2x" data-file-width="2294" data-file-height="1411" /></a><figcaption>De Morgan's law with set subtraction operation</figcaption></figure> <p>Another form of De Morgan's law is the following as seen 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 A-(B\cup C)=(A-B)\cap (A-C),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mo stretchy="false">)</mo> <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>A</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-(B\cup C)=(A-B)\cap (A-C),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88210c1cd0d3814baf9da5ddeba879e1bc342e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.15ex; height:2.843ex;" alt="{\displaystyle A-(B\cup C)=(A-B)\cap (A-C),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-(B\cap C)=(A-B)\cup (A-C).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">)</mo> <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>A</mi> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-(B\cap C)=(A-B)\cup (A-C).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0c27c73ed381d9a0b6130ad9d2d6d9b961da2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.15ex; height:2.843ex;" alt="{\displaystyle A-(B\cap C)=(A-B)\cup (A-C).}"></span></dd></dl> <p>Applications of the rules include simplification of logical <a href="/wiki/Expression_(computer_science)" title="Expression (computer science)">expressions</a> in <a href="/wiki/Computer_program" title="Computer program">computer programs</a> and digital circuit designs. De Morgan's laws are an example of a more general concept of <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">mathematical duality</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_notation">Formal notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=1" title="Edit section: Formal notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>negation of conjunction</i> rule may be written in <a href="/wiki/Sequent" title="Sequent">sequent</a> 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{aligned}\neg (P\land Q)&\vdash (\neg P\lor \neg Q),{\text{and}}\\(\neg P\lor \neg Q)&\vdash \neg (P\land Q).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <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> <mi></mi> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\neg (P\land Q)&\vdash (\neg P\lor \neg Q),{\text{and}}\\(\neg P\lor \neg Q)&\vdash \neg (P\land Q).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105b206ad8cafead0c8a26490febb205b18eee05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.396ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\neg (P\land Q)&\vdash (\neg P\lor \neg Q),{\text{and}}\\(\neg P\lor \neg Q)&\vdash \neg (P\land Q).\end{aligned}}}"></span></dd></dl> <p>The <i>negation of disjunction</i> rule may be written as: </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{aligned}\neg (P\lor Q)&\vdash (\neg P\land \neg Q),{\text{and}}\\(\neg P\land \neg Q)&\vdash \neg (P\lor Q).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <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> <mi></mi> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\neg (P\lor Q)&\vdash (\neg P\land \neg Q),{\text{and}}\\(\neg P\land \neg Q)&\vdash \neg (P\lor Q).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb98bf4f6adf54794b2428978dc53d15a028fe7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.396ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\neg (P\lor Q)&\vdash (\neg P\land \neg Q),{\text{and}}\\(\neg P\land \neg Q)&\vdash \neg (P\lor Q).\end{aligned}}}"></span></dd></dl> <p>In <a href="/wiki/Rule_of_inference" title="Rule of inference">rule form</a>: <i>negation of conjunction</i> </p> <div style="margin-left: 2em"> <p><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 {\frac {\neg (P\land Q)}{\therefore \neg P\lor \neg Q}}\qquad {\frac {\neg P\lor \neg Q}{\therefore \neg (P\land Q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>∴</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mrow> <mrow> <mo>∴</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\neg (P\land Q)}{\therefore \neg P\lor \neg Q}}\qquad {\frac {\neg P\lor \neg Q}{\therefore \neg (P\land Q)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b304636baed674246be882df946530704aab1570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.501ex; height:6.509ex;" alt="{\displaystyle {\frac {\neg (P\land Q)}{\therefore \neg P\lor \neg Q}}\qquad {\frac {\neg P\lor \neg Q}{\therefore \neg (P\land Q)}}}"></span> </p> </div> <p>and <i>negation of disjunction</i> </p> <div style="margin-left: 2em"> <p><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 {\frac {\neg (P\lor Q)}{\therefore \neg P\land \neg Q}}\qquad {\frac {\neg P\land \neg Q}{\therefore \neg (P\lor Q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>∴</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mrow> <mrow> <mo>∴</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\neg (P\lor Q)}{\therefore \neg P\land \neg Q}}\qquad {\frac {\neg P\land \neg Q}{\therefore \neg (P\lor Q)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9bd29f83aff3445d7e6258ef6e5569ba4461253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.501ex; height:6.509ex;" alt="{\displaystyle {\frac {\neg (P\lor Q)}{\therefore \neg P\land \neg Q}}\qquad {\frac {\neg P\land \neg Q}{\therefore \neg (P\lor Q)}}}"></span> </p> </div> <p>and expressed as truth-functional <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a> or <a href="/wiki/Theorem" title="Theorem">theorems</a> of propositional logic: </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{aligned}\neg (P\land Q)&\leftrightarrow (\neg P\lor \neg Q),\\\neg (P\lor Q)&\leftrightarrow (\neg P\land \neg Q).\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↔</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>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↔</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>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\neg (P\land Q)&\leftrightarrow (\neg P\lor \neg Q),\\\neg (P\lor Q)&\leftrightarrow (\neg P\land \neg Q).\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff42da93d5c3ca6d98cbf97fd887fbe428d5e60b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.615ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\neg (P\land Q)&\leftrightarrow (\neg P\lor \neg Q),\\\neg (P\lor Q)&\leftrightarrow (\neg P\land \neg Q).\\\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> are propositions expressed in some formal system. </p><p>The <b>generalized De Morgan's laws</b> provide an equivalence for negating a conjunction or disjunction involving multiple terms.<br />For a set of propositions <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_{1},P_{2},\dots ,P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1},P_{2},\dots ,P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375147ce9f59c5cba6d0103093d5efe352826752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.016ex; height:2.509ex;" alt="{\displaystyle P_{1},P_{2},\dots ,P_{n}}"></span>, the generalized De Morgan's Laws are 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{aligned}\lnot (P_{1}\land P_{2}\land \dots \land P_{n})\leftrightarrow \lnot P_{1}\lor \lnot P_{2}\lor \ldots \lor \lnot P_{n}\\\lnot (P_{1}\lor P_{2}\lor \dots \lor P_{n})\leftrightarrow \lnot P_{1}\land \lnot P_{2}\land \ldots \land \lnot P_{n}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <mo>⋯</mo> <mo>∨</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>…</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lnot (P_{1}\land P_{2}\land \dots \land P_{n})\leftrightarrow \lnot P_{1}\lor \lnot P_{2}\lor \ldots \lor \lnot P_{n}\\\lnot (P_{1}\lor P_{2}\lor \dots \lor P_{n})\leftrightarrow \lnot P_{1}\land \lnot P_{2}\land \ldots \land \lnot P_{n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867df76b887d7de40992830672c2b2f08c966763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.926ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\lnot (P_{1}\land P_{2}\land \dots \land P_{n})\leftrightarrow \lnot P_{1}\lor \lnot P_{2}\lor \ldots \lor \lnot P_{n}\\\lnot (P_{1}\lor P_{2}\lor \dots \lor P_{n})\leftrightarrow \lnot P_{1}\land \lnot P_{2}\land \ldots \land \lnot P_{n}\end{aligned}}}"></span></dd></dl> <p>These laws generalize De Morgan's original laws for negating conjunctions and disjunctions. </p> <div class="mw-heading mw-heading3"><h3 id="Substitution_form">Substitution form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=2" title="Edit section: Substitution form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De Morgan's laws are normally shown in the compact form above, with the negation of the output on the left and negation of the inputs on the right. A clearer form for substitution can be stated as: </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{aligned}(P\land Q)&\Longleftrightarrow \neg (\neg P\lor \neg Q),\\(P\lor Q)&\Longleftrightarrow \neg (\neg P\land \neg Q).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">⟺</mo> <mi mathvariant="normal">¬</mi> <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>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">⟺</mo> <mi mathvariant="normal">¬</mi> <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>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(P\land Q)&\Longleftrightarrow \neg (\neg P\lor \neg Q),\\(P\lor Q)&\Longleftrightarrow \neg (\neg P\land \neg Q).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6054f5af5a2326dd9e32a981fa400e0da5e8d414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.607ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(P\land Q)&\Longleftrightarrow \neg (\neg P\lor \neg Q),\\(P\lor Q)&\Longleftrightarrow \neg (\neg P\land \neg Q).\end{aligned}}}"></span></dd></dl> <p>This emphasizes the need to invert both the inputs and the output, as well as change the operator when doing a substitution. </p> <div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=3" title="Edit section: Set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In set theory, it is often stated as "union and intersection interchange under complementation",<sup id="cite_ref-r_l_goodstein_5-0" class="reference"><a href="#cite_note-r_l_goodstein-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> which can be formally expressed as: </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{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55ab78fcd4c3b617df4e2195d487dda13c09e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.021ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}"></span></dd></dl> <p>where: </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 {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></span> is the negation 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}"> <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>, the <a href="/wiki/Overline" title="Overline">overline</a> being written above the terms to be negated,</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 \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" 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 \cap }"></span> is the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> operator (AND),</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 \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> is the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> operator (OR).</li></ul> <div class="mw-heading mw-heading4"><h4 id="Unions_and_intersections_of_any_number_of_sets">Unions and intersections of any number of sets</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=4" title="Edit section: Unions and intersections of any number of sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generalized form is </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{aligned}{\overline {\bigcap _{i\in I}A_{i}}}&\equiv \bigcup _{i\in I}{\overline {A_{i}}},\\{\overline {\bigcup _{i\in I}A_{i}}}&\equiv \bigcap _{i\in I}{\overline {A_{i}}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {\bigcap _{i\in I}A_{i}}}&\equiv \bigcup _{i\in I}{\overline {A_{i}}},\\{\overline {\bigcup _{i\in I}A_{i}}}&\equiv \bigcap _{i\in I}{\overline {A_{i}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c78bc866cb39308e86454d85c902532f52983e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:15.75ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {\bigcap _{i\in I}A_{i}}}&\equiv \bigcup _{i\in I}{\overline {A_{i}}},\\{\overline {\bigcup _{i\in I}A_{i}}}&\equiv \bigcap _{i\in I}{\overline {A_{i}}},\end{aligned}}}"></span></dd></dl> <p>where <span class="texhtml"><i>I</i></span> is some, possibly countably or uncountably infinite, indexing set. </p><p>In set notation, De Morgan's laws can be remembered using the <a href="/wiki/Mnemonic" title="Mnemonic">mnemonic</a> "break the line, change the sign".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Boolean_algebra">Boolean algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=5" title="Edit section: Boolean algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Boolean algebra, similarly, this law which can be formally expressed as: </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{aligned}{\overline {A\land B}}&={\overline {A}}\lor {\overline {B}},\\{\overline {A\lor B}}&={\overline {A}}\land {\overline {B}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A\land B}}&={\overline {A}}\lor {\overline {B}},\\{\overline {A\lor B}}&={\overline {A}}\land {\overline {B}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75e0ddc0136d3a3049f108c31d858e1494d48bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.021ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\overline {A\land B}}&={\overline {A}}\lor {\overline {B}},\\{\overline {A\lor B}}&={\overline {A}}\land {\overline {B}},\end{aligned}}}"></span></dd></dl> <p>where: </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 {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></span> is the negation 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}"> <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>, the <a href="/wiki/Overline" title="Overline">overline</a> being written above the terms to be negated,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span> is the <a href="/wiki/Logical_conjunction" title="Logical conjunction">logical conjunction</a> operator (AND),</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 \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> is the <a href="/wiki/Logical_disjunction" title="Logical disjunction">logical disjunction</a> operator (OR).</li></ul> <p>which can be generalized to </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{aligned}{\overline {A_{1}\land A_{2}\land \ldots \land A_{n}}}={\overline {A_{1}}}\lor {\overline {A_{2}}}\lor \ldots \lor {\overline {A_{n}}},\\{\overline {A_{1}\lor A_{2}\lor \ldots \lor A_{n}}}={\overline {A_{1}}}\land {\overline {A_{2}}}\land \ldots \land {\overline {A_{n}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>…</mo> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <mo>…</mo> <mo>∨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>∧</mo> <mo>…</mo> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A_{1}\land A_{2}\land \ldots \land A_{n}}}={\overline {A_{1}}}\lor {\overline {A_{2}}}\lor \ldots \lor {\overline {A_{n}}},\\{\overline {A_{1}\lor A_{2}\lor \ldots \lor A_{n}}}={\overline {A_{1}}}\land {\overline {A_{2}}}\land \ldots \land {\overline {A_{n}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8248c9cf48f60c60bd8d74e430e3da4d1f7c5f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.011ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\overline {A_{1}\land A_{2}\land \ldots \land A_{n}}}={\overline {A_{1}}}\lor {\overline {A_{2}}}\lor \ldots \lor {\overline {A_{n}}},\\{\overline {A_{1}\lor A_{2}\lor \ldots \lor A_{n}}}={\overline {A_{1}}}\land {\overline {A_{2}}}\land \ldots \land {\overline {A_{n}}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Engineering">Engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=6" title="Edit section: Engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical</a> and <a href="/wiki/Computer_engineering" title="Computer engineering">computer engineering</a>, De Morgan's laws are commonly written as: </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 {\overline {(A\cdot B)}}\equiv ({\overline {A}}+{\overline {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {(A\cdot B)}}\equiv ({\overline {A}}+{\overline {B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e554a7cd8c85895b6be40f5e9e430dbf9e1fde7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.596ex; height:3.676ex;" alt="{\displaystyle {\overline {(A\cdot B)}}\equiv ({\overline {A}}+{\overline {B}})}"></span></dd></dl> <p>and </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 {\overline {(A+B)}}\equiv ({\overline {A}}\cdot {\overline {B}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {(A+B)}}\equiv ({\overline {A}}\cdot {\overline {B}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973cf311fb2592c7d39efdc440ea045285bdb11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.243ex; height:3.676ex;" alt="{\displaystyle {\overline {(A+B)}}\equiv ({\overline {A}}\cdot {\overline {B}}),}"></span></dd></dl> <p>where: </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 \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> is the logical AND,</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 +}"> <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 the logical OR,</li> <li>the <span style="text-decoration:overline;">overbar</span> is the logical NOT of what is underneath the overbar.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Text_searching">Text searching</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=7" title="Edit section: Text searching"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De Morgan's laws commonly apply to text searching using Boolean operators AND, OR, and NOT. Consider a set of documents containing the words "cats" and "dogs". De Morgan's laws hold that these two searches will return the same set of documents: </p> <dl><dd>Search A: NOT (cats OR dogs)</dd> <dd>Search B: (NOT cats) AND (NOT dogs)</dd></dl> <p>The corpus of documents containing "cats" or "dogs" can be represented by four documents: </p> <dl><dd>Document 1: Contains only the word "cats".</dd> <dd>Document 2: Contains only "dogs".</dd> <dd>Document 3: Contains both "cats" and "dogs".</dd> <dd>Document 4: Contains neither "cats" nor "dogs".</dd></dl> <p>To evaluate Search A, clearly the search "(cats OR dogs)" will hit on Documents 1, 2, and 3. So the negation of that search (which is Search A) will hit everything else, which is Document 4. </p><p>Evaluating Search B, the search "(NOT cats)" will hit on documents that do not contain "cats", which is Documents 2 and 4. Similarly the search "(NOT dogs)" will hit on Documents 1 and 4. Applying the AND operator to these two searches (which is Search B) will hit on the documents that are common to these two searches, which is Document 4. </p><p>A similar evaluation can be applied to show that the following two searches will both return Documents 1, 2, and 4: </p> <dl><dd>Search C: NOT (cats AND dogs),</dd> <dd>Search D: (NOT cats) OR (NOT dogs).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=8" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The laws are named after <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> (1806–1871),<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> who introduced a formal version of the laws to classical <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. De Morgan's formulation was influenced by the algebraization of logic undertaken by <a href="/wiki/George_Boole" title="George Boole">George Boole</a>, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, and was known to Greek and Medieval logicians.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, in the 14th century, <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a> wrote down the words that would result by reading the laws out.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Jean_Buridan" title="Jean Buridan">Jean Buridan</a>, in his <span title="Latin-language text"><i lang="la">Summulae de Dialectica</i></span>, also describes rules of conversion that follow the lines of De Morgan's laws.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Still, De Morgan is given credit for stating the laws in the terms of modern formal logic, and incorporating them into the language of logic. De Morgan's laws can be proved easily, and may even seem trivial.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_for_Boolean_algebra">Proof for Boolean algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=9" title="Edit section: Proof for Boolean algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De Morgan's theorem may be applied to the negation of a <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> or the negation of a <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a> in all or part of a formula. </p> <div class="mw-heading mw-heading3"><h3 id="Negation_of_a_disjunction">Negation of a disjunction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=10" title="Edit section: Negation of a disjunction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as: </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 \neg (A\lor B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\lor B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b6d731539112baade7198971d2b0267249ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.096ex; height:2.843ex;" alt="{\displaystyle \neg (A\lor B).}"></span></dd></dl> <p>In that it has been established that <i>neither</i> A nor B is true, then it must follow that both A is not true <a href="/wiki/Logical_AND" class="mw-redirect" title="Logical AND">and</a> B is not true, which may be written directly as: </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 (\neg A)\wedge (\neg B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg A)\wedge (\neg B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acc972864a161b3b7eae4c14298ec0a42f5d4bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.456ex; height:2.843ex;" alt="{\displaystyle (\neg A)\wedge (\neg B).}"></span></dd></dl> <p>If either A or B <i>were</i> true, then the disjunction of A and B would be true, making its negation false. Presented in English, this follows the logic that "since two things are both false, it is also false that either of them is true". </p><p>Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first claim. </p> <div class="mw-heading mw-heading3"><h3 id="Negation_of_a_conjunction">Negation of a conjunction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=11" title="Edit section: Negation of a conjunction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The application of De Morgan's theorem to conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: "it is false that A and B are both true", which is written as: </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 \neg (A\land B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\land B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc599f4aa696bf757a5e068835e7f9933a494ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.096ex; height:2.843ex;" alt="{\displaystyle \neg (A\land B).}"></span></dd></dl> <p>In order for this claim to be true, either or both of A or B must be false, for if they both were true, then the conjunction of A and B would be true, making its negation false. Thus, <a href="/wiki/Inclusive_or" class="mw-redirect" title="Inclusive or">one (at least) or more</a> of A and B must be false (or equivalently, one or more of "not A" and "not B" must be true). This may be written directly as, </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 (\neg A)\lor (\neg B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg A)\lor (\neg B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ade363564c4ed38650df8d3500e5d9d98f08638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.456ex; height:2.843ex;" alt="{\displaystyle (\neg A)\lor (\neg B).}"></span></dd></dl> <p>Presented in English, this follows the logic that "since it is false that two things are both true, at least one of them must be false". </p><p>Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction thus results in a true expression, and this expression is identical to the first claim. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_for_set_theory">Proof for set theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=12" title="Edit section: Proof for set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here we 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 {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></span> to denote the complement of A, as above in <a class="mw-selflink-fragment" href="#Set_theory_and_Boolean_algebra">§ Set theory and Boolean algebra</a>. The proof 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 {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6221733eae7cb566a727a5f49a3ee7955b760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.623ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}"></span> is completed in 2 steps by proving both <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 {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105d54f0ad0f5f4c3ad106f2b114eb04c03b1635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"></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 {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <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 {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728a576fb939de5791e3492f7844b61f300d4d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Part_1">Part 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=13" title="Edit section: Part 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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 x\in {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35e86dd5a5166c987b492a41fe9ca32fb4f7811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.375ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A\cap B}}}"></span>. Then, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not \in A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b650142d009e07158b9ee0a8343b30b4078bdff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.26ex; height:2.676ex;" alt="{\displaystyle x\not \in A\cap B}"></span>. </p><p>Because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=\{\,y\ |\ y\in A\wedge y\in B\,\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mo>∧</mo> <mi>y</mi> <mo>∈</mo> <mi>B</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B=\{\,y\ |\ y\in A\wedge y\in B\,\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e840b0aaad24a3a570da4969b23afad4f3f4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.333ex; height:2.843ex;" alt="{\displaystyle A\cap B=\{\,y\ |\ y\in A\wedge y\in B\,\}}"></span>, it must be 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\not \in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5cc57b95ae64cd51e1971d3cefb81c8e3f9e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.913ex; height:2.676ex;" alt="{\displaystyle x\not \in A}"></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 x\not \in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e36f5f4749e9224dfb3d2120a4e1f2850617df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.934ex; height:2.676ex;" alt="{\displaystyle x\not \in B}"></span>. </p><p>If <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\not \in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5cc57b95ae64cd51e1971d3cefb81c8e3f9e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.913ex; height:2.676ex;" alt="{\displaystyle x\not \in A}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9ffd7e111059144fe8f59ee5c50d65bede3d73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.028ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A}}}"></span>, so <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\in {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663c8fd652e9fa1bd6aa5ed639ad5d3556129b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.49ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}"></span>. </p><p>Similarly, if <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\not \in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e36f5f4749e9224dfb3d2120a4e1f2850617df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.934ex; height:2.676ex;" alt="{\displaystyle x\not \in B}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ad05753be3e2a4885e24cf07ecad42de94c363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.049ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {B}}}"></span>, so <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\in {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663c8fd652e9fa1bd6aa5ed639ad5d3556129b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.49ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}"></span>. </p><p>Thus, <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 \forall x{\Big (}x\in {\overline {A\cap B}}\implies x\in {\overline {A}}\cup {\overline {B}}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>x</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> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x{\Big (}x\in {\overline {A\cap B}}\implies x\in {\overline {A}}\cup {\overline {B}}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f52d5538546831fea40d247c2c366473c3e13829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.649ex; height:4.843ex;" alt="{\displaystyle \forall x{\Big (}x\in {\overline {A\cap B}}\implies x\in {\overline {A}}\cup {\overline {B}}{\Big )}}"></span>; </p><p>that 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 {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105d54f0ad0f5f4c3ad106f2b114eb04c03b1635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Part_2">Part 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=14" title="Edit section: Part 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To prove the reverse direction, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663c8fd652e9fa1bd6aa5ed639ad5d3556129b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.49ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}"></span>, and for contradiction assume <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\not \in {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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 x\not \in {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed59ac2e8b17223c9046c90f785ecea87f8a9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.375ex; height:3.509ex;" alt="{\displaystyle x\not \in {\overline {A\cap B}}}"></span>. </p><p>Under that assumption, it must be 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\in A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0259b3f4a3d584762f9b950f4ad35ce2a4077e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.26ex; height:2.176ex;" alt="{\displaystyle x\in A\cap B}"></span>, </p><p>so it follows 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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bcc9b2afb295d4234bc294860cd0c63bcad2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle x\in A}"></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 x\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aac01724708de4e1d41423bc64b35e9d94c9009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.934ex; height:2.176ex;" alt="{\displaystyle x\in B}"></span>, and thus <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\not \in {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc8f8479c05330db7c5564a4cc689087ce7e758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.028ex; height:3.509ex;" alt="{\displaystyle x\not \in {\overline {A}}}"></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 x\not \in {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d950b4e256abe0c82fcc56e3641932b6c44a188b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.049ex; height:3.509ex;" alt="{\displaystyle x\not \in {\overline {B}}}"></span>. </p><p>However, that means <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\not \in {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8434de87526cf0c5293099b6531a2c6ca88129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.49ex; height:3.509ex;" alt="{\displaystyle x\not \in {\overline {A}}\cup {\overline {B}}}"></span>, in contradiction to the hypothesis 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\in {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663c8fd652e9fa1bd6aa5ed639ad5d3556129b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.49ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}"></span>, </p><p>therefore, the assumption <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\not \in {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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 x\not \in {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed59ac2e8b17223c9046c90f785ecea87f8a9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.375ex; height:3.509ex;" alt="{\displaystyle x\not \in {\overline {A\cap B}}}"></span> must not be the case, meaning 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\in {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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 x\in {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35e86dd5a5166c987b492a41fe9ca32fb4f7811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.375ex; height:3.009ex;" alt="{\displaystyle x\in {\overline {A\cap B}}}"></span>. </p><p>Hence, <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 \forall x{\Big (}x\in {\overline {A}}\cup {\overline {B}}\implies x\in {\overline {A\cap B}}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>x</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x{\Big (}x\in {\overline {A}}\cup {\overline {B}}\implies x\in {\overline {A\cap B}}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a242700676b631db5d907c5c0345e07e62fbf65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.649ex; height:4.843ex;" alt="{\displaystyle \forall x{\Big (}x\in {\overline {A}}\cup {\overline {B}}\implies x\in {\overline {A\cap B}}{\Big )}}"></span>, </p><p>that 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 {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <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 {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728a576fb939de5791e3492f7844b61f300d4d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Conclusion">Conclusion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=15" title="Edit section: Conclusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <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 {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> <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 {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728a576fb939de5791e3492f7844b61f300d4d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}"></span> <i>and</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105d54f0ad0f5f4c3ad106f2b114eb04c03b1635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:3.176ex;" alt="{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6221733eae7cb566a727a5f49a3ee7955b760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.623ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}"></span>; this concludes the proof of De Morgan's law. </p><p>The other De Morgan's law, <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 {A\cup B}}={\overline {A}}\cap {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cup B}}={\overline {A}}\cap {\overline {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b63669c8a93e8ab33faecff37c72eaf8e93cb32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.623ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cup B}}={\overline {A}}\cap {\overline {B}}}"></span>, is proven similarly. </p> <div class="mw-heading mw-heading2"><h2 id="Generalising_De_Morgan_duality">Generalising De Morgan duality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=16" title="Edit section: Generalising De Morgan duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:DeMorgan_Logic_Circuit_diagram_DIN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/DeMorgan_Logic_Circuit_diagram_DIN.svg/220px-DeMorgan_Logic_Circuit_diagram_DIN.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/DeMorgan_Logic_Circuit_diagram_DIN.svg/330px-DeMorgan_Logic_Circuit_diagram_DIN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/DeMorgan_Logic_Circuit_diagram_DIN.svg/440px-DeMorgan_Logic_Circuit_diagram_DIN.svg.png 2x" data-file-width="1300" data-file-height="900" /></a><figcaption>De Morgan's Laws represented as a circuit with logic gates (<a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">International Electrotechnical Commission</a> diagrams)</figcaption></figure> <p>In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, namely the existence of <a href="/wiki/Negation_normal_form" title="Negation normal form">negation normal forms</a>: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in <a href="/wiki/Digital_circuit" class="mw-redirect" title="Digital circuit">digital circuit</a> design, where it is used to manipulate the types of <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a>, and in formal logic, where it is needed to find the <a href="/wiki/Conjunctive_normal_form" title="Conjunctive normal form">conjunctive normal form</a> and <a href="/wiki/Disjunctive_normal_form" title="Disjunctive normal form">disjunctive normal form</a> of a formula. Computer programmers use them to simplify or properly negate complicated <a href="/wiki/Conditional_(programming)" class="mw-redirect" title="Conditional (programming)">logical conditions</a>. They are also often useful in computations in elementary <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. </p><p>Let one define the dual of any propositional operator P(<i>p</i>, <i>q</i>, ...) depending on elementary propositions <i>p</i>, <i>q</i>, ... to be the 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 {\mbox{P}}^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>P</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{P}}^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bcdea71a8fb44d6cbb31906e46720fa39fc720a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.675ex; height:2.676ex;" alt="{\displaystyle {\mbox{P}}^{d}}"></span> defined by </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 {\mbox{P}}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>P</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>,</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{P}}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ab30149df8f25cca87737807e5028b04cc2e55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.874ex; height:3.176ex;" alt="{\displaystyle {\mbox{P}}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots ).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Extension_to_predicate_and_modal_logic">Extension to predicate and modal logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=17" title="Edit section: Extension to predicate and modal logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This duality can be generalised to quantifiers, so for example the <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universal quantifier</a> and <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential quantifier</a> are duals: </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 \forall x\,P(x)\equiv \neg [\exists x\,\neg P(x)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,P(x)\equiv \neg [\exists x\,\neg P(x)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cf60133e36276a8be3ee6803163417f57fb8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.28ex; height:2.843ex;" alt="{\displaystyle \forall x\,P(x)\equiv \neg [\exists x\,\neg P(x)]}"></span></dd></dl> <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 \exists x\,P(x)\equiv \neg [\forall x\,\neg P(x)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,P(x)\equiv \neg [\forall x\,\neg P(x)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcaa2aeda126c84f59493114c26dc26d43f39332" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.28ex; height:2.843ex;" alt="{\displaystyle \exists x\,P(x)\equiv \neg [\forall x\,\neg P(x)]}"></span></dd></dl> <p>To relate these quantifier dualities to the De Morgan laws, set up a <a href="/wiki/Model_theory" title="Model theory">model</a> with some small number of elements in its domain <i>D</i>, such as </p> <dl><dd><i>D</i> = {<i>a</i>, <i>b</i>, <i>c</i>}.</dd></dl> <p>Then </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 \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddd10a3e20885bd81fcf77c8d721053264ec69e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.056ex; height:2.843ex;" alt="{\displaystyle \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)}"></span></dd></dl> <p>and </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 \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1eafa2fce3c188de2b6f9a044326ad71949d024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.703ex; height:2.843ex;" alt="{\displaystyle \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).}"></span></dd></dl> <p>But, using De Morgan's laws, </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 P(a)\land P(b)\land P(c)\equiv \neg (\neg P(a)\lor \neg P(b)\lor \neg P(c))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(a)\land P(b)\land P(c)\equiv \neg (\neg P(a)\lor \neg P(b)\lor \neg P(c))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5da4e6b1170a42ab269a894bca62945cc574b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.236ex; height:2.843ex;" alt="{\displaystyle P(a)\land P(b)\land P(c)\equiv \neg (\neg P(a)\lor \neg P(b)\lor \neg P(c))}"></span></dd></dl> <p>and </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 P(a)\lor P(b)\lor P(c)\equiv \neg (\neg P(a)\land \neg P(b)\land \neg P(c)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(a)\lor P(b)\lor P(c)\equiv \neg (\neg P(a)\land \neg P(b)\land \neg P(c)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de63847c13cc8173d0458c2a7807ad955452b440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.883ex; height:2.843ex;" alt="{\displaystyle P(a)\lor P(b)\lor P(c)\equiv \neg (\neg P(a)\land \neg P(b)\land \neg P(c)),}"></span></dd></dl> <p>verifying the quantifier dualities in the model. </p><p>Then, the quantifier dualities can be extended further to <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>, relating the box ("necessarily") and diamond ("possibly") operators: </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 \Box p\equiv \neg \Diamond \neg p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi>p</mi> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mi>◊</mi> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box p\equiv \neg \Diamond \neg p,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02425729ac9695f7b65406c4aca81969c8362d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.509ex;" alt="{\displaystyle \Box p\equiv \neg \Diamond \neg p,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Diamond p\equiv \neg \Box \neg p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◊</mi> <mi>p</mi> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mi>◻</mi> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Diamond p\equiv \neg \Box \neg p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e098c6952210118b0eb598b0a96d77947ca02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.509ex;" alt="{\displaystyle \Diamond p\equiv \neg \Box \neg p.}"></span></dd></dl> <p>In its application to the <a href="/wiki/Alethic_modalities" class="mw-redirect" title="Alethic modalities">alethic modalities</a> of possibility and necessity, <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> observed this case, and in the case of <a href="/wiki/Normal_modal_logic" title="Normal modal logic">normal modal logic</a>, the relationship of these modal operators to the quantification can be understood by setting up models using <a href="/wiki/Kripke_semantics" title="Kripke semantics">Kripke semantics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_intuitionistic_logic">In intuitionistic logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=18" title="Edit section: In intuitionistic logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three out of the four implications of de Morgan's laws hold in <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. Specifically, we have </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 \neg (P\lor Q)\,\leftrightarrow \,{\big (}(\neg P)\land (\neg Q){\big )},}"> <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> <mspace width="thinmathspace" /> <mo stretchy="false">↔</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (P\lor Q)\,\leftrightarrow \,{\big (}(\neg P)\land (\neg Q){\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1969223952866067d36f69b1bcebe6b88ba517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.576ex; height:3.176ex;" alt="{\displaystyle \neg (P\lor Q)\,\leftrightarrow \,{\big (}(\neg P)\land (\neg Q){\big )},}"></span></dd></dl> <p>and </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 {\big (}(\neg P)\lor (\neg Q){\big )}\,\to \,\neg (P\land Q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}(\neg P)\lor (\neg Q){\big )}\,\to \,\neg (P\land Q).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f188a698f6453bfd78f7ac126284c3ee63ffd89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.576ex; height:3.176ex;" alt="{\displaystyle {\big (}(\neg P)\lor (\neg Q){\big )}\,\to \,\neg (P\land Q).}"></span></dd></dl> <p>The converse of the last implication does not hold in pure intuitionistic logic. That is, the failure of the joint proposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <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\land Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}"></span> cannot necessarily be resolved to the failure of either of the two <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjuncts</a>. For example, from knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow who did not show up. The latter principle is equivalent to the principle of the <a href="/wiki/Law_of_excluded_middle#Law_of_the_weak_excluded_middle" title="Law of excluded middle">weak excluded middle</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 {\mathrm {WPEM} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {WPEM} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4057add394b63f1f245ab4f395bc3877802247d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.685ex; height:2.176ex;" alt="{\displaystyle {\mathrm {WPEM} }}"></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 (\neg P)\lor \neg (\neg P).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg P)\lor \neg (\neg P).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26bd10b7de05b988e1667b5ef2f35360305614ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.99ex; height:2.843ex;" alt="{\displaystyle (\neg P)\lor \neg (\neg P).}"></span></dd></dl> <p>This weak form can be used as a foundation for an <a href="/wiki/Intermediate_logic" title="Intermediate logic">intermediate logic</a>. For a refined version of the failing law concerning existential statements, see the <a href="/wiki/Limited_principle_of_omniscience" title="Limited principle of omniscience">lesser limited principle of omniscience</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 {\mathrm {LLPO} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {LLPO} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc1dd301051b278ca3d61c44092e0c4574029da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.297ex; height:2.176ex;" alt="{\displaystyle {\mathrm {LLPO} }}"></span>, which however is different from <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 {\mathrm {WLPO} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {WLPO} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d49391e324f0967d8e1cbe3a45bf26e998a16a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.233ex; height:2.176ex;" alt="{\displaystyle {\mathrm {WLPO} }}"></span>. </p><p>The validity of the other three De Morgan's laws remains true if negation <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></span> is replaced by implication <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\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/617876031b9c07f8893f5a12a727e2a921354001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.126ex; height:2.176ex;" alt="{\displaystyle P\to C}"></span> for some arbitrary constant predicate C, meaning that the above laws are still true in <a href="/wiki/Minimal_logic" title="Minimal logic">minimal logic</a>. </p><p>Similarly to the above, the quantifier laws: </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 \forall x\,\neg P(x)\,\leftrightarrow \,\neg \exists x\,P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">↔</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,\neg P(x)\,\leftrightarrow \,\neg \exists x\,P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571bc7cb26c1023a3365fe9ba01c394cdf28d906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.276ex; height:2.843ex;" alt="{\displaystyle \forall x\,\neg P(x)\,\leftrightarrow \,\neg \exists x\,P(x)}"></span></dd></dl> <p>and </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 \exists x\,\neg P(x)\,\to \,\neg \forall x\,P(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,\neg P(x)\,\to \,\neg \forall x\,P(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad6cd03e1e2e25e5c1a4fa5f803f52b319da7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.923ex; height:2.843ex;" alt="{\displaystyle \exists x\,\neg P(x)\,\to \,\neg \forall x\,P(x).}"></span></dd></dl> <p>are tautologies even in minimal logic with negation replaced with implying a fixed <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>, while the converse of the last law does not have to be true in general. </p><p>Further, one still has </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 (P\lor Q)\,\to \,\neg {\big (}(\neg P)\land (\neg Q){\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor Q)\,\to \,\neg {\big (}(\neg P)\land (\neg Q){\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/905c278fd7550c65d6058cc4347a9eabd3ffcdfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.576ex; height:3.176ex;" alt="{\displaystyle (P\lor Q)\,\to \,\neg {\big (}(\neg P)\land (\neg Q){\big )},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\land Q)\,\to \,\neg {\big (}(\neg P)\lor (\neg Q){\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land Q)\,\to \,\neg {\big (}(\neg P)\lor (\neg Q){\big )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c232c1074f11a93801334745bbed539d67e013f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.576ex; height:3.176ex;" alt="{\displaystyle (P\land Q)\,\to \,\neg {\big (}(\neg P)\lor (\neg Q){\big )},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\,P(x)\,\to \,\neg \exists x\,\neg P(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,P(x)\,\to \,\neg \exists x\,\neg P(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d31896f096e815d1b799cfd246f143f393c7773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.923ex; height:2.843ex;" alt="{\displaystyle \forall x\,P(x)\,\to \,\neg \exists x\,\neg P(x),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\,P(x)\,\to \,\neg \forall x\,\neg P(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,P(x)\,\to \,\neg \forall x\,\neg P(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ebf0999dbaa234df86c443c4eddcd94d03aa1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.923ex; height:2.843ex;" alt="{\displaystyle \exists x\,P(x)\,\to \,\neg \forall x\,\neg P(x),}"></span></dd></dl> <p>but their inversion implies <a href="/wiki/Excluded_middle" class="mw-redirect" title="Excluded middle">excluded middle</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 {\mathrm {PEM} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {PEM} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/933a14d796125020390d99ccd8a3db5647abe2a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.176ex;" alt="{\displaystyle {\mathrm {PEM} }}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="In_computer_engineering">In computer engineering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=19" title="Edit section: In computer engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>In modern programming languages, due to the optimisation of compilers and interpreters, the performance differences between these options are negligible or completely absent.</li></ul> <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=De_Morgan%27s_laws&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Conjunction/disjunction_duality" title="Conjunction/disjunction duality">Conjunction/disjunction duality</a></li> <li><a href="/wiki/Homogeneity_(linguistics)" class="mw-redirect" title="Homogeneity (linguistics)">Homogeneity (linguistics)</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</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/List_of_set_identities_and_relations" title="List of set identities and relations">List of set identities and relations</a></li> <li><a href="/wiki/Positive_logic" class="mw-redirect" title="Positive logic">Positive logic</a></li> <li><a href="/wiki/De_Morgan_algebra" title="De Morgan algebra">De_Morgan_algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Morgan%27s_laws&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFCopiCohenMcMahon2016" class="citation book cs1">Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016). <a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/mono/10.4324/9781315510897/introduction-logic-irving-copi-carl-cohen-kenneth-mcmahon"><i>Introduction to Logic</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4324%2F9781315510897">10.4324/9781315510897</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781315510880" title="Special:BookSources/9781315510880"><bdi>9781315510880</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Logic&rft.date=2016&rft_id=info%3Adoi%2F10.4324%2F9781315510897&rft.isbn=9781315510880&rft.aulast=Copi&rft.aufirst=Irving+M.&rft.au=Cohen%2C+Carl&rft.au=McMahon%2C+Kenneth&rft_id=https%3A%2F%2Fwww.taylorfrancis.com%2Fbooks%2Fmono%2F10.4324%2F9781315510897%2Fintroduction-logic-irving-copi-carl-cohen-kenneth-mcmahon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHurley2015" class="citation cs2">Hurley, Patrick J. (2015), <i>A Concise Introduction to Logic</i> (12th ed.), Cengage Learning, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-285-19654-1" title="Special:BookSources/978-1-285-19654-1"><bdi>978-1-285-19654-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Logic&rft.edition=12th&rft.pub=Cengage+Learning&rft.date=2015&rft.isbn=978-1-285-19654-1&rft.aulast=Hurley&rft.aufirst=Patrick+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" 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="CITEREFMoore2012" class="citation book cs1">Moore, Brooke Noel (2012). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/689858599"><i>Critical thinking</i></a>. Richard Parker (10th ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-803828-0" title="Special:BookSources/978-0-07-803828-0"><bdi>978-0-07-803828-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/689858599">689858599</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Critical+thinking&rft.place=New+York&rft.edition=10th&rft.pub=McGraw-Hill&rft.date=2012&rft_id=info%3Aoclcnum%2F689858599&rft.isbn=978-0-07-803828-0&rft.aulast=Moore&rft.aufirst=Brooke+Noel&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F689858599&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/DeMorgan.html">DeMorgan's [sic] Theorem</a></span> </li> <li id="cite_note-r_l_goodstein-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-r_l_goodstein_5-0">^</a></b></span> <span class="reference-text"><i>Boolean Algebra</i> by R. L. Goodstein. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-45894-6" title="Special:BookSources/0-486-45894-6">0-486-45894-6</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=NdAjEDP5mDsC&pg=PA81"><i>2000 Solved Problems in Digital Electronics</i></a> by S. P. Bali</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080323122125/http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html">"DeMorgan's Theorems"</a>. <a href="/wiki/Middle_Tennessee_State_University" title="Middle Tennessee State University">Middle Tennessee State University</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html">the original</a> on 2008-03-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=DeMorgan%27s+Theorems&rft.pub=Middle+Tennessee+State+University&rft_id=http%3A%2F%2Fwww.mtsu.edu%2F~phys2020%2FLectures%2FL19-L25%2FL3%2FDeMorgan%2Fbody_demorgan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Bocheński's <i>History of Formal Logic</i></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">William of Ockham, <i>Summa Logicae</i>, part II, sections 32 and 33.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Jean Buridan, <i>Summula de Dialectica</i>. Trans. Gyula Klima. New Haven: Yale University Press, 2001. See especially Treatise 1, Chapter 7, Section 5. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-300-08425-0" title="Special:BookSources/0-300-08425-0">0-300-08425-0</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_H._Orr" class="citation web cs1">Robert H. Orr. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100715185655/http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/ademo.htm">"Augustus De Morgan (1806–1871)"</a>. <a href="/wiki/Indiana_University%E2%80%93Purdue_University_Indianapolis" title="Indiana University–Purdue University Indianapolis">Indiana University–Purdue University Indianapolis</a>. Archived from <a rel="nofollow" class="external text" href="http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/ademo.htm">the original</a> on 2010-07-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Augustus+De+Morgan+%281806%E2%80%931871%29&rft.pub=Indiana+University%E2%80%93Purdue+University+Indianapolis&rft.au=Robert+H.+Orr&rft_id=http%3A%2F%2Fwww.engr.iupui.edu%2F~orr%2Fwebpages%2Fcpt120%2Fmathbios%2Fademo.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWirth1995" class="citation cs2"><a href="/wiki/Niklaus_Wirth" title="Niklaus Wirth">Wirth, Niklaus</a> (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zSIzKYQ6x0AC&pg=PA16"><i>Digital Circuit Design for Computer Science Students: An Introductory Textbook</i></a>, Springer, p. 16, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540585770" title="Special:BookSources/9783540585770"><bdi>9783540585770</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Circuit+Design+for+Computer+Science+Students%3A+An+Introductory+Textbook&rft.pages=16&rft.pub=Springer&rft.date=1995&rft.isbn=9783540585770&rft.aulast=Wirth&rft.aufirst=Niklaus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzSIzKYQ6x0AC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span> </li> </ol></div></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=De_Morgan%27s_laws&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Duality_principle">"Duality principle"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Duality+principle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDuality_principle&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-de_Morgan's_Laws"><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/deMorgansLaws.html">"de Morgan's Laws"</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=de+Morgan%27s+Laws&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FdeMorgansLaws.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADe+Morgan%27s+laws" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/DeMorgansLaws">de Morgan's laws</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/dual-log/">Duality in Logic and Language</a>, <i>Internet Encyclopedia of Philosophy</i>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Well-formed formula</a></li> <li><a href="/wiki/Idempotency_of_entailment" title="Idempotency of entailment">Idempotency of entailment</a></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">Problem of multiple generality</a></li> <li><a href="/wiki/Associative_property" title="Associative property">Associativity</a></li> <li><a href="/wiki/Distributive_property" title="Distributive property">Distribution</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logic.svg" class="mw-file-description"><img alt="Law of noncontradiction" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/75px-Logic.svg.png" decoding="async" width="75" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/113px-Logic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/150px-Logic.svg.png 2x" data-file-width="85" data-file-height="28" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical logics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Term_logic" title="Term logic">Term</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Principles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commutativity_of_conjunction" title="Commutativity of conjunction">Commutativity of conjunction</a></li> <li><a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">Excluded middle</a></li> <li><a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">Bivalence</a></li> <li><a href="/wiki/Law_of_noncontradiction" title="Law of noncontradiction">Noncontradiction</a></li> <li><a href="/wiki/Monotonicity_of_entailment" title="Monotonicity of entailment">Monotonicity of entailment</a></li> <li><a href="/wiki/Principle_of_explosion" title="Principle of explosion">Explosion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Rules</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">De Morgan's laws</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)">Material implication</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></li> <li><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></li> <li><a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens">modus ponendo tollens</a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a></li> <li><a href="/wiki/Destructive_dilemma" title="Destructive dilemma">Destructive dilemma</a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism">Disjunctive syllogism</a></li> <li><a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">Hypothetical syllogism</a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)">Absorption</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Introduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation</a></li> <li><a href="/wiki/Double_negation_introduction" class="mw-redirect" title="Double negation introduction">Double negation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential</a></li> <li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal</a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction">Biconditional</a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction">Conjunction</a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction">Disjunction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Elimination</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Double_negation_elimination" class="mw-redirect" title="Double negation elimination">Double negation</a></li> <li><a href="/wiki/Existential_instantiation" title="Existential instantiation">Existential</a></li> <li><a href="/wiki/Universal_instantiation" title="Universal instantiation">Universal</a></li> <li><a href="/wiki/Biconditional_elimination" title="Biconditional elimination">Biconditional</a></li> <li><a href="/wiki/Conjunction_elimination" title="Conjunction elimination">Conjunction</a></li> <li><a href="/wiki/Disjunction_elimination" title="Disjunction elimination">Disjunction</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a></li> <li><a href="/wiki/George_Boole" title="George Boole">George Boole</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a></li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Hugh_MacColl" title="Hugh MacColl">Hugh MacColl</a></li> <li><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a></li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a></li> <li><a href="/wiki/Henry_M._Sheffer" title="Henry M. Sheffer">Henry M. Sheffer</a></li> <li><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a></li> <li><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a></li> <li><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li> <li><a href="/wiki/Function_and_Concept" title="Function and Concept">Function and Concept</a></li> <li><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></li> <li><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li> <li><a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Set_theory" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" 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class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Venn_diagram" title="Venn diagram"><img alt="Venn diagram of set intersection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/100px-Venn_A_intersect_B.svg.png" decoding="async" width="100" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a class="mw-selflink selflink">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a 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De Morgan's laws - Wikipedia