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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algèbre_de_Boole_des_valeurs_de_vérité"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Algèbre de Boole des valeurs de vérité</span> </div> </a> <button aria-controls="toc-Algèbre_de_Boole_des_valeurs_de_vérité-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>Afficher / masquer la sous-section Algèbre de Boole des valeurs de vérité</span> </button> <ul id="toc-Algèbre_de_Boole_des_valeurs_de_vérité-sublist" class="vector-toc-list"> <li id="toc-Conjonction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conjonction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Conjonction</span> </div> </a> <ul id="toc-Conjonction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Disjonction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Disjonction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Disjonction</span> </div> </a> <ul id="toc-Disjonction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Négation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Négation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Négation</span> </div> </a> <ul id="toc-Négation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Propriétés</span> </div> </a> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> <li id="toc-Propriétés_des_opérateurs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Propriétés_des_opérateurs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>Propriétés des opérateurs</span> </div> </a> <ul id="toc-Propriétés_des_opérateurs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.2</span> <span>Structure</span> </div> </a> <ul id="toc-Structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Priorité" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Priorité"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.3</span> <span>Priorité</span> </div> </a> <ul id="toc-Priorité-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lois_de_De_Morgan" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lois_de_De_Morgan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.4</span> <span>Lois de De Morgan</span> </div> </a> <ul id="toc-Lois_de_De_Morgan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Fonctions_logiques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fonctions_logiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Fonctions logiques</span> </div> </a> <button aria-controls="toc-Fonctions_logiques-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>Afficher / masquer la sous-section Fonctions logiques</span> </button> <ul id="toc-Fonctions_logiques-sublist" class="vector-toc-list"> <li id="toc-Fonctions_logiques_fondamentales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fonctions_logiques_fondamentales"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fonctions logiques fondamentales</span> </div> </a> <ul id="toc-Fonctions_logiques_fondamentales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonctions_logiques_composées" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fonctions_logiques_composées"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Fonctions logiques composées</span> </div> </a> <ul id="toc-Fonctions_logiques_composées-sublist" class="vector-toc-list"> <li id="toc-Disjonction_exclusive" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Disjonction_exclusive"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Disjonction exclusive</span> </div> </a> <ul id="toc-Disjonction_exclusive-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Équivalence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Équivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Équivalence</span> </div> </a> <ul id="toc-Équivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implication" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Implication"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Implication</span> </div> </a> <ul id="toc-Implication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inhibition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inhibition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Inhibition</span> </div> </a> <ul id="toc-Inhibition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exemple_de_fonctions_logiques_à_trois_ou_quatre_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemple_de_fonctions_logiques_à_trois_ou_quatre_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Exemple de fonctions logiques à trois ou quatre variables</span> </div> </a> <ul id="toc-Exemple_de_fonctions_logiques_à_trois_ou_quatre_variables-sublist" class="vector-toc-list"> <li id="toc-Fonction_logique_à_trois_variables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fonction_logique_à_trois_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Fonction logique à trois variables</span> </div> </a> <ul id="toc-Fonction_logique_à_trois_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonction_logique_à_quatre_variables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fonction_logique_à_quatre_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Fonction logique à quatre variables</span> </div> </a> <ul id="toc-Fonction_logique_à_quatre_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Factorisation_d'une_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Factorisation_d'une_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Factorisation d'une expression</span> </div> </a> <ul id="toc-Factorisation_d'une_expression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arbre_d'expression" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arbre_d'expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Arbre d'expression</span> </div> </a> <ul id="toc-Arbre_d'expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Annexes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Annexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Annexes</span> </div> </a> <button aria-controls="toc-Annexes-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>Afficher / masquer la sous-section Annexes</span> </button> <ul id="toc-Annexes-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Algèbre de Boole (logique)</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="Aller à un article dans une autre langue. Disponible en 66 langues." > <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-66" 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">66 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Boolse_algebra" title="Boolse algebra – afrikaans" lang="af" hreflang="af" data-title="Boolse algebra" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%A8%D9%88%D9%84" title="جبر بول – arabe" lang="ar" hreflang="ar" data-title="جبر بول" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81lxebra_de_Boole" title="Álxebra de Boole – asturien" lang="ast" hreflang="ast" data-title="Álxebra de Boole" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bul_c%C9%99bri_(m%C9%99ntiqi)" title="Bul cəbri (məntiqi) – azerbaïdjanais" lang="az" hreflang="az" data-title="Bul cəbri (məntiqi)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A8%D9%88%D9%84_%D8%AC%D8%A8%D8%B1%DB%8C" title="بول جبری – South Azerbaijani" lang="azb" hreflang="azb" data-title="بول جبری" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D2%BB%D1%8B" title="Логика алгебраһы – bachkir" lang="ba" hreflang="ba" data-title="Логика алгебраһы" data-language-autonym="Башҡортса" data-language-local-name="bachkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булева алгебра – bulgare" lang="bg" hreflang="bg" data-title="Булева алгебра" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%81%E0%A6%B2%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="বুলিয়ান বীজগণিত – bengali" lang="bn" hreflang="bn" data-title="বুলিয়ান বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – bosniaque" lang="bs" hreflang="bs" data-title="Booleova algebra" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_de_Boole" title="Àlgebra de Boole – catalan" lang="ca" hreflang="ca" data-title="Àlgebra de Boole" 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/Booleova_algebra" title="Booleova algebra – tchèque" lang="cs" hreflang="cs" data-title="Booleova algebra" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D0%B0%D0%BD%C4%83%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8" title="Каланăлăхсен алгебри – tchouvache" lang="cv" hreflang="cv" data-title="Каланăлăхсен алгебри" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra – danois" lang="da" hreflang="da" data-title="Boolesk algebra" data-language-autonym="Dansk" data-language-local-name="danois" 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/Boolesche_Algebra" title="Boolesche Algebra – allemand" lang="de" hreflang="de" data-title="Boolesche Algebra" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1_%CE%9C%CF%80%CE%BF%CF%85%CE%BB" title="Άλγεβρα Μπουλ – grec" lang="el" hreflang="el" data-title="Άλγεβρα Μπουλ" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Boolean_algebra" title="Boolean algebra – anglais" lang="en" hreflang="en" data-title="Boolean algebra" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Bulea_algebro" title="Bulea algebro – espéranto" lang="eo" hreflang="eo" data-title="Bulea algebro" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_de_Boole" title="Álgebra de Boole – espagnol" lang="es" hreflang="es" data-title="Álgebra de Boole" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Boole%27i_algebra" title="Boole'i algebra – estonien" lang="et" hreflang="et" data-title="Boole'i algebra" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Booleren_aljebra" title="Booleren aljebra – basque" lang="eu" hreflang="eu" data-title="Booleren aljebra" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%A8%D9%88%D9%84%DB%8C" title="جبر بولی – persan" lang="fa" hreflang="fa" data-title="جبر بولی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Boolen_algebra" title="Boolen algebra – finnois" lang="fi" hreflang="fi" data-title="Boolen algebra" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ailg%C3%A9abar_Boole" title="Ailgéabar Boole – irlandais" lang="ga" hreflang="ga" data-title="Ailgéabar Boole" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_de_Boole" title="Álxebra de Boole – galicien" lang="gl" hreflang="gl" data-title="Álxebra de Boole" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%95%D7%9C%D7%99%D7%90%D7%A0%D7%99%D7%AA" title="אלגברה בוליאנית – hébreu" lang="he" hreflang="he" data-title="אלגברה בוליאנית" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%82%E0%A4%B2%E0%A5%80%E0%A4%AF_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4_(%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="बूलीय बीजगणित (तर्कशास्त्र) – hindi" lang="hi" hreflang="hi" data-title="बूलीय बीजगणित (तर्कशास्त्र)" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – croate" lang="hr" hreflang="hr" data-title="Booleova algebra" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Boole-algebra_(informatika)" title="Boole-algebra (informatika) – hongrois" lang="hu" hreflang="hu" data-title="Boole-algebra (informatika)" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D6%82%D5%AC%D5%B5%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Բուլյան հանրահաշիվ – arménien" lang="hy" hreflang="hy" data-title="Բուլյան հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_Boole" title="Aljabar Boole – indonésien" lang="id" hreflang="id" data-title="Aljabar Boole" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Booleana_algebro" title="Booleana algebro – ido" lang="io" hreflang="io" data-title="Booleana algebro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_di_Boole" title="Algebra di Boole – italien" lang="it" hreflang="it" data-title="Algebra di Boole" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Boolean_Logic" title="Boolean Logic – kikuyu" lang="ki" hreflang="ki" data-title="Boolean Logic" data-language-autonym="Gĩkũyũ" data-language-local-name="kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%82%E0%B2%B2%E0%B2%BF%E0%B2%AF%E0%B2%A8%E0%B3%8D_%E0%B2%AC%E0%B3%80%E0%B2%9C%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಬೂಲಿಯನ್ ಬೀಜಗಣಿತ – kannada" lang="kn" hreflang="kn" data-title="ಬೂಲಿಯನ್ ಬೀಜಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%88_%EB%85%BC%EB%A6%AC" title="불 논리 – coréen" lang="ko" hreflang="ko" data-title="불 논리" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Mantiq%C3%AA_B%C3%BBl%C3%AE" title="Mantiqê Bûlî – kurde" lang="ku" hreflang="ku" data-title="Mantiqê Bûlî" data-language-autonym="Kurdî" data-language-local-name="kurde" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D1%82%D1%8B%D0%BB%D1%8B%D1%88%D1%82%D0%B0%D1%80_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%81%D1%8B" title="Айтылыштар алгебрасы – kirghize" lang="ky" hreflang="ky" data-title="Айтылыштар алгебрасы" data-language-autonym="Кыргызча" data-language-local-name="kirghize" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_Booleana_(logica)" title="Algebra Booleana (logica) – latin" lang="la" hreflang="la" data-title="Algebra Booleana (logica)" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/B%C5%ABlio_algebra" title="Būlio algebra – lituanien" lang="lt" hreflang="lt" data-title="Būlio algebra" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/B%C5%ABla_algebra" title="Būla algebra – letton" lang="lv" hreflang="lv" data-title="Būla algebra" data-language-autonym="Latviešu" data-language-local-name="letton" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булова алгебра – macédonien" lang="mk" hreflang="mk" data-title="Булова алгебра" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/%C3%81lgebra_de_Boole" title="Álgebra de Boole – mirandais" lang="mwl" hreflang="mwl" data-title="Álgebra de Boole" data-language-autonym="Mirandés" data-language-local-name="mirandais" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%98%E1%80%B0%E1%80%9C%E1%80%AE%E1%80%9A%E1%80%94%E1%80%BA%E1%80%A1%E1%80%80%E1%80%B9%E1%80%81%E1%80%9B%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ဘူလီယန်အက္ခရာသင်္ချာ – birman" lang="my" hreflang="my" data-title="ဘူလီယန်အက္ခရာသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Booleaanse_algebra" title="Booleaanse algebra – néerlandais" lang="nl" hreflang="nl" data-title="Booleaanse algebra" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Boolsk_algebra" title="Boolsk algebra – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Boolsk algebra" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Boolsk_algebra" title="Boolsk algebra – norvégien bokmål" lang="nb" hreflang="nb" data-title="Boolsk algebra" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/%C3%80lgebra_%C3%ABd_Boole" title="Àlgebra ëd Boole – piémontais" lang="pms" hreflang="pms" data-title="Àlgebra ëd Boole" data-language-autonym="Piemontèis" data-language-local-name="piémontais" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_booliana" title="Álgebra booliana – portugais" lang="pt" hreflang="pt" data-title="Álgebra booliana" data-language-autonym="Português" data-language-local-name="portugais" 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/Algebr%C4%83_boolean%C4%83" title="Algebră booleană – roumain" lang="ro" hreflang="ro" data-title="Algebră booleană" data-language-autonym="Română" data-language-local-name="roumain" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B8" title="Алгебра логики – russe" lang="ru" hreflang="ru" data-title="Алгебра логики" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bulova_algebra" title="Bulova algebra – serbo-croate" lang="sh" hreflang="sh" data-title="Bulova algebra" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Boolean_algebra" title="Boolean algebra – Simple English" lang="en-simple" hreflang="en-simple" data-title="Boolean algebra" 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/Boolova_algebra" title="Boolova algebra – slovaque" lang="sk" hreflang="sk" data-title="Boolova algebra" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – slovène" lang="sl" hreflang="sl" data-title="Booleova algebra" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булова алгебра – serbe" lang="sr" hreflang="sr" data-title="Булова алгебра" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra – suédois" lang="sv" hreflang="sv" data-title="Boolesk algebra" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%82%E0%AE%B2%E0%AE%BF%E0%AE%AF_%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="பூலிய இயற்கணிதம் – tamoul" lang="ta" hreflang="ta" data-title="பூலிய இயற்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8_%D0%BC%D0%B0%D0%BD%D1%82%D0%B8%D2%9B" title="Алгебраи мантиқ – tadjik" lang="tg" hreflang="tg" data-title="Алгебраи мантиқ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadjik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B5%E0%B8%8A%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%9A%E0%B8%B9%E0%B8%A5" title="พีชคณิตแบบบูล – thaï" lang="th" hreflang="th" data-title="พีชคณิตแบบบูล" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alhebrang_Boolean" title="Alhebrang Boolean – tagalog" lang="tl" hreflang="tl" data-title="Alhebrang Boolean" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boole_cebiri" title="Boole cebiri – turc" lang="tr" hreflang="tr" data-title="Boole cebiri" data-language-autonym="Türkçe" data-language-local-name="turc" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B8" title="Алгебра логіки – ukrainien" lang="uk" hreflang="uk" data-title="Алгебра логіки" data-language-autonym="Українська" data-language-local-name="ukrainien" 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/%C4%90%E1%BA%A1i_s%E1%BB%91_Boole" title="Đại số Boole – vietnamien" lang="vi" hreflang="vi" data-title="Đại số Boole" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数 – wu" lang="wuu" hreflang="wuu" data-title="逻辑代数" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数 – chinois" lang="zh" hreflang="zh" data-title="逻辑代数" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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Cliquez ici pour en savoir plus.</figcaption></figure><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Fichier:2017-fr.wp-orange-source.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2017-fr.wp-orange-source.svg/45px-2017-fr.wp-orange-source.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2017-fr.wp-orange-source.svg/68px-2017-fr.wp-orange-source.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/2017-fr.wp-orange-source.svg/90px-2017-fr.wp-orange-source.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Certaines informations figurant dans cet article ou cette section devraient être mieux reliées aux sources mentionnées dans les sections « Bibliographie », « Sources » ou « Liens externes »</strong> <small>(<time class="nowrap" datetime="2021-02" data-sort-value="2021-02">février 2021</time>).</small> </p><p>Vous pouvez améliorer la <a href="/wiki/Wikip%C3%A9dia:V%C3%A9rifiabilit%C3%A9" title="Wikipédia:Vérifiabilité">vérifiabilité</a> en <a href="/wiki/Mod%C3%A8le:Sources_%C3%A0_lier/Explication" title="Modèle:Sources à lier/Explication">associant ces informations à des références</a> à l'aide d'<a href="/wiki/Aide:Note" title="Aide:Note">appels de notes</a>. </p> </div></div> <div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Alg%C3%A8bre_de_Boole" class="mw-disambig" title="Algèbre de Boole">Algèbre de Boole</a>. </p> </div></div> <p>L'<b>algèbre de Boole</b>, ou calcul booléen, est la partie des <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a> qui s'intéresse à une approche <a href="/wiki/Alg%C3%A8bre" title="Algèbre">algébrique</a> de la <a href="/wiki/Logique" title="Logique">logique</a>, vue en termes de <a href="/wiki/Variable_(math%C3%A9matiques)" title="Variable (mathématiques)">variables</a>, d'<a href="/wiki/Op%C3%A9rateur_(math%C3%A9matiques)" title="Opérateur (mathématiques)">opérateurs</a> et de <a href="/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques)">fonctions</a> sur les variables logiques, ce qui permet d'utiliser des techniques algébriques pour traiter les expressions à deux valeurs du <a href="/wiki/Calcul_des_propositions" title="Calcul des propositions">calcul des propositions</a>. Elle fut lancée en 1854 par le <a href="/wiki/Math%C3%A9maticien" title="Mathématicien">mathématicien</a> <a href="/wiki/Britanniques" title="Britanniques">britannique</a> <a href="/wiki/George_Boole" title="George Boole">George Boole</a>. L'algèbre de Boole trouve de nombreuses applications en <a href="/wiki/Informatique" title="Informatique">informatique</a> et dans la <a href="/wiki/Conception_de_circuits_int%C3%A9gr%C3%A9s" title="Conception de circuits intégrés">conception</a> des <a href="/wiki/Circuits_%C3%A9lectroniques" class="mw-redirect" title="Circuits électroniques">circuits électroniques</a>. </p><p>Elle fut utilisée la première fois pour les circuits de <a href="/wiki/Commutateur_t%C3%A9l%C3%A9phonique" title="Commutateur téléphonique">commutation téléphonique</a> par <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Exemple_introductif">Exemple introductif</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=1" title="Modifier la section : Exemple introductif" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=1" title="Modifier le code source de la section : Exemple introductif"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'algèbre de Boole des <a href="/wiki/Fonction_logique" title="Fonction logique">fonctions logiques</a> permet de modéliser des <a href="/wiki/Logique" title="Logique">raisonnements logiques</a>, en exprimant un « état » en fonction de conditions. Par exemple, si nous étudions l'expression Communication et l'expression Décrocher : </p> <ul><li>Communication = Émetteur ET Récepteur</li></ul> <dl><dd><i>Communication</i> serait « VRAI » si à la fois les variables <i>Émetteur ET Récepteur</i> étaient actifs (c'est une fonction logique dépendant des variables <i>Émetteur</i> et <i>Récepteur</i>)</dd></dl> <ul><li>Décrocher = (Sonnerie ET Décision de répondre) OU Décision d'appeler</li></ul> <dl><dd><i>Décrocher</i> serait « VRAI » soit si à la fois on entend la <i>sonnerie ET</i> l'on <i>décide de répondre</i>, soit (<i>OU</i>) si simplement l'on <i>décide d'appeler</i>.</dd></dl> <p>L'algèbre de Boole étant un domaine commun à trois disciplines, on rencontre des notations différentes pour désigner un même objet. Dans le reste de l'article, on indiquera les diverses notations, mais on en privilégiera une pour conserver une certaine homogénéité. </p> <div class="mw-heading mw-heading2"><h2 id="Algèbre_de_Boole_des_valeurs_de_vérité"><span id="Alg.C3.A8bre_de_Boole_des_valeurs_de_v.C3.A9rit.C3.A9"></span>Algèbre de Boole des valeurs de vérité</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=2" title="Modifier la section : Algèbre de Boole des valeurs de vérité" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=2" title="Modifier le code source de la section : Algèbre de Boole des valeurs de vérité"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On appelle <i>B</i> l'<a href="/wiki/Ensemble" title="Ensemble">ensemble</a> constitué de deux éléments appelés <b>valeurs de vérité</b> {VRAI, FAUX}. Cet ensemble est aussi noté </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 B=\{1,0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{1,0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7bff627b9ffe227e4e6669e450c7857e31a149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.546ex; height:2.843ex;" alt="{\displaystyle B=\{1,0\}}"></span></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 B=\{\top ,\perp \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">⊤</mi> <mo>,</mo> <mo>⊥</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{\top ,\perp \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2bc328ba1d0c024f9c7e3c29ad1043be5bae31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.838ex; height:2.843ex;" alt="{\displaystyle B=\{\top ,\perp \}}"></span>.</li></ul> <p>avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span> pour <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \perp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb90d6db42aa12f9e2f31176a4ed4e741c69eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \perp }"></span> pour <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. </p><p>On privilégiera dans la suite la notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{1,0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{1,0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7bff627b9ffe227e4e6669e450c7857e31a149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.546ex; height:2.843ex;" alt="{\displaystyle B=\{1,0\}}"></span>. </p><p>Sur cet ensemble on peut définir deux <a href="/wiki/Loi_de_composition" title="Loi de composition">lois</a> (ou opérations) binaires, les lois ET et OU, et une opération unaire, la négation (ou le complémentaire). </p><p>Pour l'ensemble des exemples et propriétés suivantes, <span class="mwe-math-element"><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,c\}\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b,c\}\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97235fbb134ba5a2faf11d5ff2ddb6db31a930f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.489ex; height:2.843ex;" alt="{\displaystyle \{a,b,c\}\subset B}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Conjonction">Conjonction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=3" title="Modifier la section : Conjonction" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=3" title="Modifier le code source de la section : Conjonction"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Articles connexes : <a href="/wiki/Fonction_ET" title="Fonction ET">Fonction ET</a> et <a href="/wiki/Conjonction_logique" title="Conjonction logique">Conjonction logique</a>.</div></div> <table align="right" class="wikitable" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de la loi <b>ET</b> </td></tr> <tr style="text-align:center"> <th scope="col" width="50">a→<br />b↓</th> <th width="50"><b>0</b></th> <th width="50"><b>1</b> </th></tr> <tr style="text-align:center"> <th scope="row">0 </th> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <td>0</td> <td>1 </td></tr></tbody></table> <p>Elle est définie de la manière suivante : <i>a</i> ET <i>b</i> est VRAI si et seulement si <i>a</i> est VRAI et <i>b</i> est VRAI. </p><p>Cette loi est aussi notée : </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> <mspace width="thinmathspace" /> </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/6db7ee9d0e42ec5a541d8e771e19bace05b0cb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.034ex; height:1.176ex;" alt="{\displaystyle \cdot \,}"></span> : <a href="/wiki/Op%C3%A9rateur_point" class="mw-redirect" title="Opérateur point">Opérateur point</a> ;</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 \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> ;</li> <li>« & » ou « && » : cette <a href="/wiki/Mise_en_%C5%93uvre" title="Mise en œuvre">implémentation</a> fait partie de plusieurs <a href="/wiki/Langage_de_programmation" title="Langage de programmation">langages de programmation</a> tels que <a href="/wiki/Perl_(langage)" title="Perl (langage)">Perl</a>, <a href="/wiki/C_(langage)" title="C (langage)">C</a>, <a href="/wiki/PHP" title="PHP">PHP</a>, <a href="/wiki/Swift_(langage_d%27Apple)" title="Swift (langage d'Apple)">Swift</a>, <a href="/wiki/Go_(langage)" title="Go (langage)">Golang</a>. ;</li> <li>« and » ou « <i>AND</i> » : la plupart des langages de programmation, par exemple <a href="/wiki/Ada_(langage)" title="Ada (langage)">Ada</a>, <a href="/wiki/Pascal_(langage)" title="Pascal (langage)">Pascal</a>, Perl, <a href="/wiki/Python_(langage)" title="Python (langage)">Python</a>, <a href="/wiki/PHP" title="PHP">PHP</a> proposent cette fonction ;</li> <li>« ∧ » : utilisée dans plusieurs <a href="/wiki/Notation_alg%C3%A9brique" title="Notation algébrique">notations algébriques</a> et en <a href="/wiki/APL_(langage)" title="APL (langage)">APL</a> ;</li> <li>« * » ; le symbole d'une <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> ordinaire est utilisée dans quelques langages ne disposant pas de fonction adaptée.</li></ul> <p>On privilégiera dans la suite la notation « <span class="mwe-math-element"><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> ». </p><p>La table de cette loi (analogue à une <a href="/wiki/Table_d%27addition" title="Table d'addition">table d'addition</a> ou <a href="/wiki/Table_de_multiplication" title="Table de multiplication">de multiplication</a>) n'est pas une <a href="/wiki/Table_de_v%C3%A9rit%C3%A9" title="Table de vérité">table de vérité</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Disjonction">Disjonction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=4" title="Modifier la section : Disjonction" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=4" title="Modifier le code source de la section : Disjonction"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de la loi <b>OU</b> </td></tr> <tr style="text-align:center"> <th scope="col" width="50">a→<br />b↓</th> <th width="50"><b>0</b></th> <th width="50"><b>1</b> </th></tr> <tr style="text-align:center"> <th scope="row">0 </th> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <td>1</td> <td>1 </td></tr></tbody></table> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Articles connexes : <a href="/wiki/Fonction_OU" title="Fonction OU">Fonction OU</a> et <a href="/wiki/Disjonction_logique" title="Disjonction logique">Disjonction logique</a>.</div></div> <p>Elle est définie de la manière suivante : <i>a</i> OU <i>b</i> est VRAI si et seulement si <i>a</i> est VRAI ou <i>b</i> est VRAI. (En particulier, si <i>a</i> est vrai et que <i>b</i> est vrai aussi, alors <i>a</i> OU <i>b</i> est vrai.) </p><p>Cette loi est aussi notée : </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span></li> <li>« ∨ » (« <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> ») en mathématiques (et en logique mathématique) ou en APL ;</li> <li>«|» ou «||» dans certains langages de programmation ;</li> <li>en toutes lettres « or » ou « OR » en logique ou dans certains langages de programmation.</li></ul> <p>On privilégiera dans la suite la notation <span class="mwe-math-element"><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> mais on prendra garde du fait que cette loi n'est pas l'addition usuelle dans <a href="/wiki/Anneau_%E2%84%A4/n%E2%84%A4" title="Anneau ℤ/nℤ"><b>Z</b>/2<b>Z</b></a>. C'est pourquoi, en mathématiques et en <a href="/wiki/Logique_math%C3%A9matique" title="Logique mathématique">logique mathématique</a>, la notation <span class="mwe-math-element"><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> n'est pas utilisée pour désigner le « ou inclusif » : elle est réservée au « <a href="#Disjonction_exclusive">ou exclusif</a> », opération qui (jointe au « et ») fait de toute <a href="/wiki/Alg%C3%A8bre_de_Boole_(structure)" title="Algèbre de Boole (structure)">algèbre de Boole</a> un <a href="/wiki/Anneau_de_Boole" title="Anneau de Boole">anneau de Boole</a>, en particulier une <b>Z</b>/2<b>Z</b>-<a href="/wiki/Alg%C3%A8bre_sur_un_corps" title="Algèbre sur un corps">algèbre</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Négation"><span id="N.C3.A9gation"></span>Négation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=5" title="Modifier la section : Négation" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=5" title="Modifier le code source de la section : Négation"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Articles connexes : <a href="/wiki/Fonction_NON" title="Fonction NON">Fonction NON</a> et <a href="/wiki/N%C3%A9gation_logique" title="Négation logique">Négation logique</a>.</div></div> <p>La négation de <i>a</i> est VRAIE si et seulement si <i>a</i> est FAUX. </p><p>La négation de a est notée : </p> <ul><li>non-a, non a, not a</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 {\bar {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}}"></span></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 a/}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </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/5d795394782bd1e6b9aba3fcdf4a257dfbc05020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.392ex; height:2.843ex;" alt="{\displaystyle a/}"></span></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 \neg a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e72f6b2a9120b875c42a17235dbf8d417e9abbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \neg a}"></span></li> <li>« ~a » dans quelques notations algébriques, en APL et dans quelques langages d'interrogation de <a href="/wiki/Base_de_donn%C3%A9es" title="Base de données">bases de données</a> (<a href="/wiki/Structured_Query_Language" title="Structured Query Language">SQL</a>…) ;</li> <li>« ! » dans quelques langages de programmation (C, <a href="/wiki/Langage_C%2B%2B" class="mw-redirect" title="Langage C++">C++</a>…) ;</li> <li>« 1- » dans quelques langages ne disposant pas de fonctions adaptées (<a href="/wiki/Traitement_par_lots" title="Traitement par lots">Batch</a>…) <i>(puisque 1-0=1 et 1-1=0).</i></li></ul> <p>On privilégiera dans la suite la notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}}"></span>. </p><p>On obtient alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {0}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {0}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925d8ce1e242655c493be977bf7f912598e9ef38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.509ex;" alt="{\displaystyle {\bar {0}}=1}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {1}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {1}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a695069be8579e800636109ecc6b0e3a9f40d468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.509ex;" alt="{\displaystyle {\bar {1}}=0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=6" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=6" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Propriétés_des_opérateurs"><span id="Propri.C3.A9t.C3.A9s_des_op.C3.A9rateurs"></span>Propriétés des opérateurs</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=7" title="Modifier la section : Propriétés des opérateurs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=7" title="Modifier le code source de la section : Propriétés des opérateurs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les opérateurs sont concernés par plusieurs propriétés communes : </p> <ul><li><a href="/wiki/Associativit%C3%A9" title="Associativité">associativité</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c=a+(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)+c=a+(b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b7b8d31d5845966e6abdbb030c73f343c17d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a+b)+c=a+(b+c)}"></span> , qui est parfois écrit pour cette raison : <span class="mwe-math-element"><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+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1665d6bc61ca933b6a448479992cb3b606561b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.915ex; height:2.343ex;" alt="{\displaystyle a+b+c}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e48cd711b9a1eb1c3a2372ba01fa48ca7e262a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.902ex; height:2.843ex;" alt="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}"></span>, qui est parfois écrit pour cette raison : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\cdot c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece943b26af57e45033fce72c97d57f8b2c9a084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.592ex; height:2.176ex;" alt="{\displaystyle a\cdot b\cdot c}"></span> ;</li> <li><a href="/wiki/Loi_commutative" title="Loi commutative">commutativité</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7dede7493e0231b3ad6ff9b54f4eae954108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.911ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a}"></span> ;</li> <li><a href="/wiki/Distributivit%C3%A9" title="Distributivité">distributivité</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0969d65db9f1f1097aa4f72bcddac8c46f1ca6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}"></span> et <span class="mwe-math-element"><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\cdot c)=(a+b)\cdot (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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b\cdot c)=(a+b)\cdot (a+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a04ef41f73d64d0d624052dc17f372da19557f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.104ex; height:2.843ex;" alt="{\displaystyle a+(b\cdot c)=(a+b)\cdot (a+c)}"></span> ;</li> <li><a href="/wiki/Idempotence" title="Idempotence">idempotence</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7652b8688d5cebb9269afb49791644a2917e3eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.628ex; height:2.176ex;" alt="{\displaystyle a+a=a}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5700edc1e0a1d972ccef6badad306db6a97ed8a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.467ex; height:1.676ex;" alt="{\displaystyle a\cdot a=a}"></span> .</li></ul> <p>Par ailleurs, chaque opérateur possède un <a href="/wiki/%C3%89l%C3%A9ment_neutre" title="Élément neutre">élément neutre</a> et un <a href="/wiki/%C3%89l%C3%A9ment_absorbant" title="Élément absorbant">élément absorbant</a> : </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=0+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=0+a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649814bd8a1a695a94e5cd97b36b480264009912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.892ex; height:2.343ex;" alt="{\displaystyle a+0=0+a=a}"></span> ;</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 a+1=1+a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+1=1+a=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec481898a01e8e5c7fd1efd387adde4c02c5291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.825ex; height:2.343ex;" alt="{\displaystyle a+1=1+a=1}"></span> ;</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 a\cdot 1=1\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 1=1\cdot a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b73fdbfb3f6e3c6add8faaf5fc6aa331b88bac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.569ex; height:2.176ex;" alt="{\displaystyle a\cdot 1=1\cdot a=a}"></span> ;</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 a\cdot 0=0\cdot a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0\cdot a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8639b75222970d40fa175a86203e1c64d8bdfcb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.502ex; height:2.176ex;" alt="{\displaystyle a\cdot 0=0\cdot a=0}"></span> ;</li></ul> <p>Des simplifications sont possibles comme : </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 a+a\cdot b=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+a\cdot b=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70480d380a4765333e8c0ba8a9011ad509df800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.305ex; height:2.343ex;" alt="{\displaystyle a+a\cdot b=a}"></span> ;</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 a\cdot (a+b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (a+b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccc5bad6c622eba4d4e721eb4758f32d52feb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.114ex; height:2.843ex;" alt="{\displaystyle a\cdot (a+b)=a}"></span> ;</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 a+{\overline {a}}\cdot b=a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\overline {a}}\cdot b=a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf8699c2950ebcb8e3d93c0fb319fe0da0429bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.258ex; height:2.509ex;" alt="{\displaystyle a+{\overline {a}}\cdot b=a+b}"></span> ;</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 a\cdot ({\overline {a}}+b)=a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot ({\overline {a}}+b)=a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0d77e94b81037fdc636a4cc5fc6285b4683ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.906ex; height:2.843ex;" alt="{\displaystyle a\cdot ({\overline {a}}+b)=a\cdot b}"></span>.</li></ul> <p>le <a href="/wiki/Th%C3%A9or%C3%A8me_du_consensus" title="Théorème du consensus">théorème du consensus</a> s'applique aux opérateurs de l'algèbre de Boole : </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 a\cdot b+{\overline {a}}\cdot c=a\cdot b+{\overline {a}}\cdot c+b\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b+{\overline {a}}\cdot c=a\cdot b+{\overline {a}}\cdot c+b\cdot c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc586b19bd1855f94289961e3b17ebcc29d6fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.178ex; height:2.509ex;" alt="{\displaystyle a\cdot b+{\overline {a}}\cdot c=a\cdot b+{\overline {a}}\cdot c+b\cdot c}"></span>.</li></ul> <p>Enfin, ils suivent le <a href="/wiki/Principe_de_compl%C3%A9mentarit%C3%A9" title="Principe de complémentarité">principe de complémentarité</a> : </p> <ul><li><a href="/wiki/Involution_(math%C3%A9matiques)" title="Involution (mathématiques)">involution</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\overline {\overline {a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\overline {\overline {a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f183f60f68d142dd8a0241dcdf0baad80b5791b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.788ex; height:3.176ex;" alt="{\displaystyle a={\overline {\overline {a}}}}"></span> (<abbr class="abbr" title="Exemple">ex. :</abbr> la proposition "La lumière est allumée" équivaut à "la lumière n'est pas non allumée" ou, dit autrement, "la lumière n'est pas éteinte").</li> <li><a href="/wiki/Principe_du_tiers_exclu" title="Principe du tiers exclu">tiers exclu</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\overline {a}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\overline {a}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e56552cfc7998c07e4cf7c90baa0aeb6d9bd24ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.676ex; height:2.509ex;" alt="{\displaystyle a+{\overline {a}}=1}"></span> (la proposition "lumière allumée" OU "lumière non allumée" est toujours VRAI.).</li> <li><a href="/wiki/Contradiction" title="Contradiction">contradiction</a> ou antilogie : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot {\overline {a}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {\overline {a}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4aebda3b1e955a648f186fa27355a68e7666fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.515ex; height:2.343ex;" alt="{\displaystyle a\cdot {\overline {a}}=0}"></span> (la proposition "lumière allumée" ET "lumière non allumée" est toujours FAUX.).</li></ul> <div class="mw-heading mw-heading4"><h4 id="Structure">Structure</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=8" title="Modifier la section : Structure" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=8" title="Modifier le code source de la section : Structure"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On retrouve alors toutes les propriétés qui confèrent à <i>B</i> une <a href="/wiki/Alg%C3%A8bre_de_Boole_(structure)" title="Algèbre de Boole (structure)">structure d'algèbre de Boole</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Priorité"><span id="Priorit.C3.A9"></span>Priorité</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=9" title="Modifier la section : Priorité" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=9" title="Modifier le code source de la section : Priorité"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour alléger les écritures, il est d'usage que les opérations booléennes soient soumises aux mêmes règles de priorité que les opérations arithmétiques usuelles : la fonction ET (multiplication logique) est donc prioritaire par rapport à la fonction OU (somme logique). Ainsi : </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 1+b\cdot 0=1+0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+b\cdot 0=1+0=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cced5c031320c62f256b542a1eaa8b9ef5aa62a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.367ex; height:2.343ex;" alt="{\displaystyle 1+b\cdot 0=1+0=1}"></span></dd></dl> <p>Il reste possible de placer des parenthèses dans les calculs pour changer l'ordre de priorité des opérations. </p> <div class="mw-heading mw-heading4"><h4 id="Lois_de_De_Morgan">Lois de De Morgan</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=10" title="Modifier la section : Lois de De Morgan" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=10" title="Modifier le code source de la section : Lois de De Morgan"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table width="100%"> <tbody><tr> <td align="center"> <dl><dt>Première loi de De Morgan (négation de la disjonction)</dt> <dd>s'exprime par l'égalité suivante <span class="mwe-math-element"><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}}={\overline {a}}\cdot {\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+b}}={\overline {a}}\cdot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39942e484fa8b8ccb9f95ca2ef0ddb495c137366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.418ex; height:3.176ex;" alt="{\displaystyle {\overline {a+b}}={\overline {a}}\cdot {\overline {b}}}"></span></dd></dl> <table class="wikitable" width="150" cellspacing="0" border="1"> <caption style="background:#b3e2d1">Table de vérité/Table de fonctionnement </caption> <tbody><tr> <th scope="col">a </th> <th scope="col">b </th> <th scope="col"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> </th> <th scope="col"><span class="mwe-math-element"><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}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a+b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee7312d9cb3e91767585d632f23f7f732c50a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.183ex; height:3.176ex;" alt="{\displaystyle {\overline {a+b}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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/032e261791bd07a59cf1419352fc66f7901d4b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.345ex; height:2.343ex;" alt="{\displaystyle {\overline {a}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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 {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5308b10c1aa9c247a3f11bb6e5639515082749e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.113ex; height:3.009ex;" alt="{\displaystyle {\overline {b}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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 {\overline {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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a}}\cdot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b244c3d998bb9ef49a9e91c73723852919fb30ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.136ex; height:3.009ex;" alt="{\displaystyle {\overline {a}}\cdot {\overline {b}}}"></span> </th></tr> <tr style="text-align:center"> <th scope="row">0 </th> <th scope="row">0 </th> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <th scope="row">0 </th> <th scope="row">1 </th> <td>1</td> <td>0</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <th scope="row">0 </th> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td>0 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <th scope="row">1 </th> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>0 </td></tr></tbody></table> <p>Dans les deux cas, l'expression ne sera VRAIE<br />que si <i>a</i> et <i>b</i> sont <i>fausses</i>. </p> </td> <td align="center"> <dl><dt>Deuxième loi de De Morgan (négation de la conjonction)</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {a\cdot b}}={\overline {a}}+{\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\cdot b}}={\overline {a}}+{\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c015ce1c327c32971fb3cf6d60c91e351227c239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.418ex; height:3.176ex;" alt="{\displaystyle {\overline {a\cdot b}}={\overline {a}}+{\overline {b}}}"></span></dd></dl> <table class="wikitable" width="150" cellspacing="0" border="1"> <caption style="background:#b3e2d1">Table de vérité/Table de fonctionnement </caption> <tbody><tr> <th scope="col">a </th> <th scope="col">b </th> <th scope="col"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}"></span> </th> <th scope="col"><span class="mwe-math-element"><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}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a\cdot b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d370988c1a2bbc0a4d5d7b5c82b12df85380fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.021ex; height:3.009ex;" alt="{\displaystyle {\overline {a\cdot b}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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/032e261791bd07a59cf1419352fc66f7901d4b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.345ex; height:2.343ex;" alt="{\displaystyle {\overline {a}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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 {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5308b10c1aa9c247a3f11bb6e5639515082749e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.113ex; height:3.009ex;" alt="{\displaystyle {\overline {b}}}"></span> </th> <th scope="col"><span class="mwe-math-element"><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}}+{\overline {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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a}}+{\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36415de0e224fa59046f75d02192066a15f8ec9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.298ex; height:3.176ex;" alt="{\displaystyle {\overline {a}}+{\overline {b}}}"></span> </th></tr> <tr style="text-align:center"> <th scope="row">0 </th> <th scope="row">0 </th> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <th scope="row">0 </th> <th scope="row">1 </th> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <th scope="row">0 </th> <td>0</td> <td>1</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <th scope="row">1 </th> <th scope="row">1 </th> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>0 </td></tr></tbody></table> <p>Dans les deux cas, l'expression ne sera FAUSSE<br />que si <i>a</i> et <i>b</i> sont <i>vraies</i>. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Fonctions_logiques">Fonctions logiques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=11" title="Modifier la section : Fonctions logiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=11" title="Modifier le code source de la section : Fonctions logiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Fonction_logique" title="Fonction logique">Fonction logique</a>.</div></div> <p>Mathématiquement, une <b>fonction logique</b> ou opérateur logique est une application de <i>B<sup>n</sup></i> dans <i>B</i>. </p><p>En <a href="/wiki/%C3%89lectronique_(technique)" title="Électronique (technique)">électronique</a>, une fonction logique est une <i><a href="/wiki/Bo%C3%AEte_noire_(syst%C3%A8me)" title="Boîte noire (système)">boîte noire</a></i> qui reçoit en entrée un certain nombre de variables logiques et qui rend en sortie une variable logique dépendant des variables d'entrée. L'article <a href="/wiki/Fonction_logique" title="Fonction logique">fonction logique</a> précise comment construire les boîtes noires de quelques fonctions fondamentales. </p><p>Une <a href="/wiki/Table_de_v%C3%A9rit%C3%A9" title="Table de vérité">table de vérité</a> permet de préciser l'état de la sortie en fonction des états des entrées. Elle caractérise la fonction logique. </p><p>Toute table de vérité, et donc toute fonction logique, peut se décrire à l'aide des trois opérations de base : </p> <ul><li><a href="/wiki/Disjonction_logique" title="Disjonction logique">disjonction</a> (OU) ;</li> <li><a href="/wiki/Conjonction_logique" title="Conjonction logique">conjonction</a> (ET) ;</li> <li><a href="/wiki/N%C3%A9gation_logique" title="Négation logique">négation</a> (NON).</li></ul> <p>On démontre aussi qu'il existe exactement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc9417763ad5d68870290ddaa2ca025ffdaf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{2^{n}}}"></span> fonctions logiques de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> paramètres. Il suffit en effet de considérer toutes les tables de vérités possibles, ou de considérer le développement d'une fonction de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> paramètres. </p> <div class="mw-heading mw-heading3"><h3 id="Fonctions_logiques_fondamentales">Fonctions logiques fondamentales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=12" title="Modifier la section : Fonctions logiques fondamentales" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=12" title="Modifier le code source de la section : Fonctions logiques fondamentales"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table cellspacing="5"> <tbody><tr valign="top"> <td> <p>Elles sont issues des trois opérations de base et définissent alors </p> <ul><li>une fonction de <i>B</i> dans <i>B</i> : le complémentaire ou inversion</li> <li>deux fonctions de <i>B<sup>2</sup></i> dans <i>B</i> qui sont la somme (OU) et le produit (ET)</li></ul> </td> <td align="center" width="12%"> <table class="wikitable" width="100" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="2">Table de vérité de l'inverse </td></tr> <tr style="text-align:center"> <td><b>a</b></td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}}"></span></b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0 </td></tr></tbody></table> </td> <td align="center" width="17%"> <table class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité de la somme </td></tr> <tr style="text-align:center"> <td><b>a</b></td> <td><b>b</b></td> <td><b>a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58a21a9365b468c60330b9dda587fc1da5fb43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.195ex; height:2.176ex;" alt="{\displaystyle +\,}"></span> b</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> </td> <td align="center" width="17%"> <table class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité du produit </td></tr> <tr style="text-align:center"> <td><b>a</b></td> <td><b>b</b></td> <td><b>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 \cdot \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mspace width="thinmathspace" /> </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/6db7ee9d0e42ec5a541d8e771e19bace05b0cb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.034ex; height:1.176ex;" alt="{\displaystyle \cdot \,}"></span> b</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Fonctions_logiques_composées"><span id="Fonctions_logiques_compos.C3.A9es"></span>Fonctions logiques composées</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=13" title="Modifier la section : Fonctions logiques composées" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=13" title="Modifier le code source de la section : Fonctions logiques composées"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ce sont les fonctions logiques à deux variables. Parmi celles-ci, on en dénombre certaines suffisamment intéressantes pour qu'on leur donne un nom. </p> <div class="mw-heading mw-heading4"><h4 id="Disjonction_exclusive">Disjonction exclusive</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=14" title="Modifier la section : Disjonction exclusive" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=14" title="Modifier le code source de la section : Disjonction exclusive"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité de XOR </td></tr> <tr style="text-align:center"> <td width="40"><b>a</b></td> <td width="40"><b>b</b></td> <td width="70"><b>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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> b</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>0 </td></tr></tbody></table> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article connexe : <a href="/wiki/OU_exclusif" class="mw-redirect" title="OU exclusif">OU exclusif</a>.</div></div> <p>Le OU étudié jusqu'à présent doit se comprendre de la manière suivante : « l'un ou l'autre ou les deux ». Il est également appelé « OU inclusif ». Le <a href="/wiki/Fonction_OU_exclusif" title="Fonction OU exclusif">OU exclusif</a> (ou XOR pour ' e<b>X</b>clusive <b>OR'</b>) s'entend comme : « l'un ou l'autre, <i>mais pas</i> les deux ». </p> <dl><dt>a XOR b</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ \operatorname {XOR} \ b=a\oplus b=(a+b).{\overline {(a\cdot b)}}=a{\bar {b}}+{\bar {a}}b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>XOR</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>⊕</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> <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> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ \operatorname {XOR} \ b=a\oplus b=(a+b).{\overline {(a\cdot b)}}=a{\bar {b}}+{\bar {a}}b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76a3ae150c191860476b22933d98f7b870a85702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.99ex; height:3.676ex;" alt="{\displaystyle a\ \operatorname {XOR} \ b=a\oplus b=(a+b).{\overline {(a\cdot b)}}=a{\bar {b}}+{\bar {a}}b}"></span></dd></dl> <p>On peut également le définir avec un <a href="/wiki/Modulo_(informatique)" class="mw-redirect" title="Modulo (informatique)">modulo</a> sur une <a href="/wiki/Somme_(arithm%C3%A9tique)" title="Somme (arithmétique)">somme</a> ordinaire : </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\ \operatorname {XOR} \ b=(a+b)\ {\bmod {\ }}2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>XOR</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ \operatorname {XOR} \ b=(a+b)\ {\bmod {\ }}2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66288629d475cd9282a517b18b10e2990ced668f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.405ex; height:2.843ex;" alt="{\displaystyle a\ \operatorname {XOR} \ b=(a+b)\ {\bmod {\ }}2}"></span></dd></dl> <p>Le « ou exclusif » est parfois noté par le signe arithmétique <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cc3d8d8c60120bc2f905bae4d5e10d8ad6a3f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \neq }"></span> (<i>différent de</i>). Fonctionnellement, on utilise aussi un + entouré :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\oplus b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊕</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\oplus b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae14b07fb59ea003d756dcb6a5d74943821d9cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a\oplus b}"></span>. </p><p><b>Propriété</b> - Toute table de vérité, toute fonction logique, peut se décrire à l'aide de la constante 1 et des deux opérations : disjonction exclusive et conjonction, car :<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}=a\oplus \ 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>⊕</mo> <mtext> </mtext> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}=a\oplus \ 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb2e3e84f5bbca60e82ac934743e9e8d4649bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.142ex; height:2.343ex;" alt="{\displaystyle {\bar {a}}=a\oplus \ 1}"></span>, et <span class="mwe-math-element"><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=a\oplus \ b\oplus \ a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>⊕</mo> <mtext> </mtext> <mi>b</mi> <mo>⊕</mo> <mtext> </mtext> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=a\oplus \ b\oplus \ a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89da3bb913b6146ae28091a900187cc3bf8ee55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.142ex; height:2.343ex;" alt="{\displaystyle a+b=a\oplus \ b\oplus \ a\cdot b}"></span></span> </p> <div class="mw-heading mw-heading4"><h4 id="Équivalence"><span id=".C3.89quivalence"></span>Équivalence</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=15" title="Modifier la section : Équivalence" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=15" title="Modifier le code source de la section : Équivalence"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité de EQV </td></tr> <tr style="text-align:center"> <td width="40"><b>a</b></td> <td width="40"><b>b</b></td> <td width="70"><b>a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></span> b</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>L'<a href="/wiki/%C3%89quivalence_logique" title="Équivalence logique">équivalence</a> (notée EQV ou XNOR) est vraie si les deux entrées ont la même valeur et fausse sinon. C'est la négation du « ou exclusif ». </p> <dl><dt>L'<i>équivalence</i> peut s'écrire</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ \operatorname {EQV} \ b=a\odot b={\overline {a\oplus b}}={\overline {a{\overline {b}}+{\overline {a}}b}}=(ab)+({\overline {a}}\cdot {\overline {b}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>EQV</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mo>⊕</mo> <mi>b</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>b</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <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 a\ \operatorname {EQV} \ b=a\odot b={\overline {a\oplus b}}={\overline {a{\overline {b}}+{\overline {a}}b}}=(ab)+({\overline {a}}\cdot {\overline {b}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b31fec55cd746ff0866dc7e7e597e46e3f484f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.404ex; height:4.176ex;" alt="{\displaystyle a\ \operatorname {EQV} \ b=a\odot b={\overline {a\oplus b}}={\overline {a{\overline {b}}+{\overline {a}}b}}=(ab)+({\overline {a}}\cdot {\overline {b}})}"></span></dd></dl> <p>L'équivalence est souvent notée par le signe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></span>. Elle peut aussi être notée « == » dans certains langages (C++, PHP…) et « ⊙ » en électronique. </p> <div class="clear" style="clear:both;"></div> <div class="mw-heading mw-heading4"><h4 id="Implication">Implication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=16" title="Modifier la section : Implication" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=16" title="Modifier le code source de la section : Implication"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité de IMP </td></tr> <tr style="text-align:center"> <td width="40"><b>a</b></td> <td width="40"><b>b</b></td> <td width="70"><b>a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span> b</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Implication_(logique)" title="Implication (logique)">implication (logique)</a>.</div></div> <dl><dt>L'<i>implication</i> (notée <i>IMP</i>) s'écrit de la manière suivante</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ \operatorname {IMP} \ b={\overline {a}}+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>IMP</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ \operatorname {IMP} \ b={\overline {a}}+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7ba2fc4b9aad9419956a3324ebf3ac9dd304e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.997ex; height:2.509ex;" alt="{\displaystyle a\ \operatorname {IMP} \ b={\overline {a}}+b}"></span></dd></dl> <p>Cette opération n'est pas <a href="#Propriétés_des_opérateurs">commutative</a>. a est une condition <b>suffisante</b> pour b, qui, elle, est une condition <b>nécessaire</b> pour a. </p><p>Mais </p><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 a\ \operatorname {IMP} \ b={\overline {b}}\ \operatorname {IMP} \ {\overline {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>IMP</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mi>IMP</mi> <mo>⁡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ \operatorname {IMP} \ b={\overline {b}}\ \operatorname {IMP} \ {\overline {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bdfe0b8513bf7de3b3b5a1e2f62a2b8bc844d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.761ex; height:3.009ex;" alt="{\displaystyle a\ \operatorname {IMP} \ b={\overline {b}}\ \operatorname {IMP} \ {\overline {a}}}"></span> </p> <div style="margin:0.5em 2em;"><strong>Illustration :</strong> <div style="padding-left:2em; border-left:1px dotted #999;"> <p>De l'affirmation <i>« SI j'habite en Allemagne, ALORS j'habite en Europe. »</i>, on peut déduire <i>« SI je n'habite pas en Europe, ALORS je n'habite pas en Allemagne. »</i> mais pas <i>« SI je n'habite pas en Allemagne, ALORS je n'habite pas en Europe. »</i> car je peux habiter en Europe ailleurs qu'en Allemagne, sans contredire l'énoncé initial. </p> </div></div> <div class="mw-heading mw-heading4"><h4 id="Inhibition">Inhibition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=17" title="Modifier la section : Inhibition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=17" title="Modifier le code source de la section : Inhibition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable" width="150" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="3">Table de vérité de INH </td></tr> <tr style="text-align:center"> <td width="40"><b>a</b></td> <td width="40"><b>b</b></td> <td width="70"><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.{\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</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 a.{\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3682495191473304d83171ab4349d8b68d06d4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.376ex; height:3.009ex;" alt="{\displaystyle a.{\overline {b}}}"></span></b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>0 </td></tr></tbody></table> <dl><dt>L'<i>inhibition</i> (notée <i>INH</i>) se compose comme suit</dt> <dd></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\ \operatorname {INH} \ b=a.{\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>INH</mi> <mo>⁡</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mi>a</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 a\ \operatorname {INH} \ b=a.{\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4439bcd3a6de369c58c8c84140abfa0110659c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.963ex; height:3.009ex;" alt="{\displaystyle a\ \operatorname {INH} \ b=a.{\overline {b}}}"></span></dd></dl> <p>Si <i>a</i> est VRAI, l'expression vaut VRAI, SAUF si b est VRAI. </p><p>Cette opération n'est pas commutative. </p> <div class="clear" style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Exemple_de_fonctions_logiques_à_trois_ou_quatre_variables"><span id="Exemple_de_fonctions_logiques_.C3.A0_trois_ou_quatre_variables"></span>Exemple de fonctions logiques à trois ou quatre variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=18" title="Modifier la section : Exemple de fonctions logiques à trois ou quatre variables" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=18" title="Modifier le code source de la section : Exemple de fonctions logiques à trois ou quatre variables"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Fonction_logique_à_trois_variables"><span id="Fonction_logique_.C3.A0_trois_variables"></span>Fonction logique à trois variables</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=19" title="Modifier la section : Fonction logique à trois variables" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=19" title="Modifier le code source de la section : Fonction logique à trois variables"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="43">Table de vérité de décrocher </td></tr> <tr style="text-align:center"> <td width="35"><b>a</b></td> <td width="35"><b>b</b></td> <td width="35"><b>c</b></td> <td width="70"><b>d</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center;;color:red"> <td>1</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>L'égalité <span style="display: block; margin-left:1.6em;">Décrocher = (Sonnerie ET Décision de répondre) OU Décision d'appeler</span> traduit la situation pratique suivante : On <i>décroche</i> un téléphone quand on <i>décide d'appeler</i> quelqu'un ou quand le <i>téléphone sonne</i> et qu'on <i>décide de répondre</i>. </p><p>Elle est constituée de trois variables : </p> <ul><li>a = « Sonnerie » ;</li> <li>b = « Décision de répondre » ;</li> <li>c = « Décision d'appeler ».</li></ul> <p>la variable d = « Décrocher » est fonction logique des 3 précédentes et peut s'écrire <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{=}a\cdot b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{=}a\cdot b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4041b12656121f32da78d3e12234d1102178c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.778ex; height:2.343ex;" alt="{\displaystyle d{=}a\cdot b+c}"></span></span> </p><p>La table de vérité de cette fonction d est alors la suivante (à droite) : </p> <div class="clear" style="clear:both;"></div> <table align="right" class="wikitable" cellspacing="0" border="1"> <tbody><tr style="background:#b3e2d1;text-align:center"> <td colspan="43">Table de vérité de décrocher 2 </td></tr> <tr style="text-align:center"> <td width="35"><b>a</b></td> <td width="35"><b>b</b></td> <td width="35"><b>c</b></td> <td width="70"><b>d2</b> </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center"> <td>0</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr style="text-align:center;color:red"> <td>1</td> <td>0</td> <td>1</td> <td>0 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>0</td> <td>1 </td></tr> <tr style="text-align:center"> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>La table indique une situation absurde : quand on décide d'appeler quelqu'un et que le téléphone sonne sans qu'on ait envie de répondre, on décrocherait quand même. Une modification de la table comme ci-contre corrigerait cette absurdité. Cette table correspond à une fonction logique Décrocher d2 ou d2 qu'il est possible de déterminer et simplifier en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9efac08d52e59d3421fb754075570d61cdd4a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.139ex; height:2.343ex;" alt="{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Fonction_logique_à_quatre_variables"><span id="Fonction_logique_.C3.A0_quatre_variables"></span>Fonction logique à quatre variables</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=20" title="Modifier la section : Fonction logique à quatre variables" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=20" title="Modifier le code source de la section : Fonction logique à quatre variables"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un bon élève s'interroge s'il est sage de sortir un soir. Il doit décider en fonction de quatre variables : </p> <ul><li>a = il a assez d'argent ;</li> <li>b = il a fini ses devoirs ;</li> <li>c = le transport en commun est en grève ;</li> <li>d = l'automobile de son père est disponible.</li></ul> <p>Cet élève pourra sortir si : </p> <ul><li>il a assez d'argent, a = vrai ;</li> <li>il a fini ses devoirs, donc b = vrai ;</li> <li>le transport en commun n'est pas en grève, donc c = faux ;</li> <li>ou si l'automobile de son père est disponible, donc d = vrai.</li></ul> <p>L'expression logique de sortir en fonction de l'état des variables a, b, c et d peut donc s'écrire ainsi :<span style="display: block; margin-left:1.6em;">Sortir =<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\cdot ({\bar {c}}+d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\cdot ({\bar {c}}+d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce2a64534f83e56881b4feede2f0bf5ec4296a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.743ex; height:2.843ex;" alt="{\displaystyle a\cdot b\cdot ({\bar {c}}+d)}"></span></span> </p> <div class="mw-heading mw-heading3"><h3 id="Factorisation_d'une_expression"><span id="Factorisation_d.27une_expression"></span>Factorisation d'une expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=21" title="Modifier la section : Factorisation d'une expression" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=21" title="Modifier le code source de la section : Factorisation d'une expression"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une fonction logique peut être déterminée </p> <ul><li>soit sous forme d'une expression faisant intervenir les 3 opérations (<span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58a21a9365b468c60330b9dda587fc1da5fb43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.195ex; height:2.176ex;" alt="{\displaystyle +\,}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mspace width="thinmathspace" /> </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/6db7ee9d0e42ec5a541d8e771e19bace05b0cb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.034ex; height:1.176ex;" alt="{\displaystyle \cdot \,}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc430b1c925f47e505de5f4173bfe053e4bf265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0; margin-bottom: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle {\bar {}}\,}"></span>)</li> <li>soit sous forme de sa table de vérité. Dans ce cas il sera toujours possible d'effectuer un développement pour écrire cette fonction comme une <a href="/wiki/Forme_normale_conjonctive" title="Forme normale conjonctive">somme de produits</a>.</li></ul> <p>Exemple : Dans l'exemple de "Décrocher 2", la lecture de la table montre que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfd9ff222fc84a771ef457d4d0f5ea143ee8fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.176ex;" alt="{\displaystyle d2}"></span> égale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> quand <span class="mwe-math-element"><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,c)=(0,0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)=(0,0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61db83d5fe55303a5e1d9f6bf0c872af99d6e6e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.574ex; height:2.843ex;" alt="{\displaystyle (a,b,c)=(0,0,1)}"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c2df6cd05d09c31ff612e7b003514d067dc03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,1,1)}"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5568de79a98cdcb48821cd81b670c095fa4c8b6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,1,0)}"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efeac0c817a90342bab7878eb40cb33dd6facbbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,1,1)}"></span>.<span style="display: block; margin-left:1.6em;">Cela permet de définir d2 par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d2={\bar {a}}\cdot {\bar {b}}\cdot c+{\bar {a}}\cdot b\cdot c+a\cdot b\cdot {\bar {c}}+a\cdot b\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d2={\bar {a}}\cdot {\bar {b}}\cdot c+{\bar {a}}\cdot b\cdot c+a\cdot b\cdot {\bar {c}}+a\cdot b\cdot c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aeb5852d150231bfa4988ae3721df18a59072ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:40.817ex; height:2.676ex;" alt="{\displaystyle d2={\bar {a}}\cdot {\bar {b}}\cdot c+{\bar {a}}\cdot b\cdot c+a\cdot b\cdot {\bar {c}}+a\cdot b\cdot c}"></span></span> </p><p>Il est possible de trouver une expression minimisant le nombre de termes et le nombre de lettres dans chaque terme. C'est l'objectif de certaines techniques comme la <a href="/wiki/M%C3%A9thode_de_Quine-Mc_Cluskey" title="Méthode de Quine-Mc Cluskey">méthode de Quine-Mc Cluskey</a>, les <a href="/wiki/Table_de_Karnaugh" title="Table de Karnaugh">diagrammes de Karnaugh</a>, la <a href="/wiki/M%C3%A9thode_des_consensus" title="Méthode des consensus">méthode des consensus</a>, la double dualisation… </p><p>Exemple (suite) : la somme précédente peut être réduite par factorisation des deux premiers termes par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}.c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>.</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}.c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122a32fc2d9dd1db0891480a22c389a0b09943ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.271ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}.c}"></span> et factorisation des deux derniers termes par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114616ecaff7b792a007116e67313587c3944e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.294ex; height:2.176ex;" alt="{\displaystyle a\cdot b\,}"></span><span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9efac08d52e59d3421fb754075570d61cdd4a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.139ex; height:2.343ex;" alt="{\displaystyle d2={\bar {a}}\cdot c+a\cdot b}"></span></span> </p> <div class="mw-heading mw-heading2"><h2 id="Arbre_d'expression"><span id="Arbre_d.27expression"></span>Arbre d'expression</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=22" title="Modifier la section : Arbre d'expression" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=22" title="Modifier le code source de la section : Arbre d'expression"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les expressions logiques sont souvent représentées en <a href="/wiki/Informatique" title="Informatique">informatique</a> sous forme d'<a href="/wiki/Arborescence" title="Arborescence">arborescence</a>. </p><p>À un premier sommet (racine) sont rattachés différents sous-arbres (ou branches). Les sommets sans issue sont appelés feuilles. </p><p>Chaque sommet interne correspond à un sélecteur booléen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x,y,z)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(x,y,z)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a61bcd842393937f74b980bb921fcf0c6396e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.403ex; height:2.843ex;" alt="{\displaystyle S(x,y,z)=}"></span> « si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> sinon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> », qui ramène une question <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> à deux sous-questions plus simples, éventuellement réduites à 1/vrai ou 0/faux. </p><p>L'évaluation d'une fonction f dépendant d'une variable q choisie pour la première question est alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=S(q,f(q=1),f(q=0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=S(q,f(q=1),f(q=0))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/659bc720f3a33bbf1384497cfb08e542bf87cbeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.66ex; height:2.843ex;" alt="{\displaystyle f=S(q,f(q=1),f(q=0))}"></span>, qui ramène à deux expressions indépendantes de <span class="mwe-math-element"><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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>. </p><p>Soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=a\cdot b+a\cdot d\cdot f+c\cdot d+e\cdot f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>d</mi> <mo>⋅</mo> <mi>f</mi> <mo>+</mo> <mi>c</mi> <mo>⋅</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=a\cdot b+a\cdot d\cdot f+c\cdot d+e\cdot f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3dc60eb0724e97d3bf633e7671accf21fa8046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.83ex; height:2.509ex;" alt="{\displaystyle f=a\cdot b+a\cdot d\cdot f+c\cdot d+e\cdot f}"></span> ; on peut écrire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=S(a,f(a=1),f(a=0))=S(a,b+c\cdot d+d\cdot f+e\cdot f,c\cdot d+e\cdot f)=S(a,S(b,1,d\cdot f+c\cdot d+e\cdot f),S(c,d+e\cdot f,e\cdot f))\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>⋅</mo> <mi>d</mi> <mo>+</mo> <mi>d</mi> <mo>⋅</mo> <mi>f</mi> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo>,</mo> <mi>c</mi> <mo>⋅</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>⋅</mo> <mi>f</mi> <mo>+</mo> <mi>c</mi> <mo>⋅</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo>,</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=S(a,f(a=1),f(a=0))=S(a,b+c\cdot d+d\cdot f+e\cdot f,c\cdot d+e\cdot f)=S(a,S(b,1,d\cdot f+c\cdot d+e\cdot f),S(c,d+e\cdot f,e\cdot f))\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4415a657d95c0c483eb98f75aa7735a607ee89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:126.935ex; height:2.843ex;" alt="{\displaystyle f=S(a,f(a=1),f(a=0))=S(a,b+c\cdot d+d\cdot f+e\cdot f,c\cdot d+e\cdot f)=S(a,S(b,1,d\cdot f+c\cdot d+e\cdot f),S(c,d+e\cdot f,e\cdot f))\dots }"></span> </p><p>Les arbres dépendant de l'expression et de l'ordre des questions, pour une même expression certains questionnaires seront plus simples que d'autres. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=23" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=23" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Respectivement (E4.1) et (E4.2) dans <span class="ouvrage" id="Jaume2020"><span class="ouvrage" id="M._Jaume2020">M. Jaume <i><abbr class="abbr" title="et alii (et d’autres)">et al.</abbr></i>, <cite class="italique">Logique pour l'informatique</cite>, <a href="/wiki/%C3%89ditions_Ellipses" title="Éditions Ellipses">Ellipses</a>, <time>2020</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=nxlEEAAAQBAJ&pg=PA76">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 76<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logique+pour+l%27informatique&rft.pub=Ellipses&rft.aulast=Jaume&rft.aufirst=M.&rft.date=2020&rft.pages=76&rft_id=%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnxlEEAAAQBAJ%26pg%3DPA76&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AAlg%C3%A8bre+de+Boole+%28logique%29"></span></span></span>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Annexes">Annexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=24" title="Modifier la section : Annexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=24" title="Modifier le code source de la section : Annexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/Logique_s%C3%A9quentielle" class="extiw" title="v:Logique séquentielle">Logique séquentielle</a>, <span class="nowrap">sur <span class="project">Wikiversity</span></span></li><li class="wikibooks"><a href="https://fr.wikibooks.org/wiki/%C3%89lectronique_num%C3%A9rique_:_logique" class="extiw" title="b:Électronique numérique : logique">Électronique numérique : logique</a>, <span class="nowrap">sur <span class="project">Wikibooks</span></span></li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=25" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=25" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Liste_de_sujets_relatifs_%C3%A0_l%27alg%C3%A8bre_de_Boole" title="Liste de sujets relatifs à l'algèbre de Boole">Liste de sujets relatifs à l'algèbre de Boole</a>.</div></div> <ul><li><a href="/wiki/Op%C3%A9ration_bit_%C3%A0_bit" title="Opération bit à bit">Opération bit à bit</a></li> <li><a href="/wiki/Les_lois_de_la_pens%C3%A9e" class="mw-redirect" title="Les lois de la pensée">Les lois de la pensée</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&veaction=edit&section=26" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_de_Boole_(logique)&action=edit&section=26" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Kuntzmann1968"><span class="ouvrage" id="J._Kuntzmann1968">J. Kuntzmann, <cite class="italique">Algèbre de Boole</cite>, Paris, <a href="/wiki/%C3%89ditions_Dunod" title="Éditions Dunod">Dunod</a>, <time>1968</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre+de+Boole&rft.place=Paris&rft.pub=Dunod&rft.aulast=Kuntzmann&rft.aufirst=J.&rft.date=1968&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AAlg%C3%A8bre+de+Boole+%28logique%29"></span></span></span></li> <li><span class="ouvrage" id="Permingeat1988"><span class="ouvrage" id="N._Permingeat1988">N. Permingeat, <cite class="italique">Algèbre de Boole : Théorie, méthodes de calcul, applications, avec exercices</cite>, Paris, Dunod, <time>1988</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre+de+Boole+%3A+Th%C3%A9orie%2C+m%C3%A9thodes+de+calcul%2C+applications%2C+avec+exercices&rft.place=Paris&rft.pub=Dunod&rft.aulast=Permingeat&rft.aufirst=N.&rft.date=1988&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AAlg%C3%A8bre+de+Boole+%28logique%29"></span></span></span></li> <li><span class="ouvrage" id="Picard1965"><span class="ouvrage" id="Claude_François_Picard1965">Claude François <span class="nom_auteur">Picard</span>, <cite class="italique">Théorie des questionnaires</cite>, Paris, <a href="/wiki/Gauthier-Villars" title="Gauthier-Villars">Gauthier-Villars</a>, <time>1965</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+questionnaires&rft.place=Paris&rft.pub=Gauthier-Villars&rft.aulast=Picard&rft.aufirst=Claude+Fran%C3%A7ois&rft.date=1965&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AAlg%C3%A8bre+de+Boole+%28logique%29"></span></span></span></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint collapsed" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Op%C3%A9rations_binaires" title="Modèle:Palette Opérations binaires"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Op%C3%A9rations_binaires&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Op%C3%A9ration_binaire" title="Opération binaire">Opérations binaires</a></div></th> </tr> <tr> <td class="navbox-list" style="" colspan="2"><table class="navbox-columns-table" style="text-align:left;margin:0px;width:100%;"><tbody><tr><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;"><b>Numériques</b></td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;"><b>En ensemble ordonné</b></td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;"><b>Structurelles</b></td><td class="navbox-abovebelow" colspan="1" style="background-color: #DFE0FF; text-align: center;"><b>Autres</b></td></tr><tr style="vertical-align:top;"><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <p><b>Élémentaires</b><br /><span class="mwe-math-element"><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> <a href="/wiki/Addition" title="Addition">Addition</a><br /><span class="mwe-math-element"><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/04bd52ce670743d3b61bec928a7ec9f47309eb36" 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> <a href="/wiki/Soustraction" title="Soustraction">Soustraction</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \div }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>÷</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \div }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837b35ee5d25b5ce7b07f292c27cc90533dd9fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \div }"></span> <a href="/wiki/Division" title="Division">Division</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97daf05d8be90237eb4d2e4a46d3733ca829bd1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:2.009ex;" alt="{\displaystyle {\hat {}}}"></span> <a href="/wiki/Puissance_d%27un_nombre" title="Puissance d'un nombre">Puissance</a> </p><p><b>Arithmétiques</b><br /><span class="mwe-math-element"><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 {div} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2d000acb2863a923fa83e5bee0635410b909bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.167ex; height:2.176ex;" alt="{\displaystyle \mathrm {div} }"></span> <a href="/wiki/Division_euclidienne" title="Division euclidienne">Quotient euclidien</a><br /><span class="mwe-math-element"><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 {mod} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {mod} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a50b3797e46642fd900093e002af888607a78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.391ex; height:2.176ex;" alt="{\displaystyle \mathrm {mod} }"></span> <a href="/wiki/Reste" title="Reste">Reste euclidien</a><br /><span class="mwe-math-element"><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 {pgcd} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pgcd} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a99af40437989ee3ff61087ef6784068cc7d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.78ex; height:2.509ex;" alt="{\displaystyle \mathrm {pgcd} }"></span> <a href="/wiki/Plus_grand_commun_diviseur" title="Plus grand commun diviseur">PGCD</a><br /><span class="mwe-math-element"><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 {ppcm} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ppcm} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2414692d872334199729ef7ec1ee384b8a57f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.553ex; height:2.009ex;" alt="{\displaystyle \mathrm {ppcm} }"></span> <a href="/wiki/Plus_petit_commun_multiple" title="Plus petit commun multiple">PPCM</a> </p><p><b>Combinatoires</b><br /><span class="mwe-math-element"><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 stretchy="false">(</mo> <mo stretchy="false">)</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/d7bc8aa05e1302397bb3e7877e842784991351df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.843ex;" alt="{\displaystyle ()}"></span> <a href="/wiki/Coefficient_binomial" title="Coefficient binomial">Coefficient binomial</a><br /><span class="mwe-math-element"><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> <a href="/wiki/Arrangement" title="Arrangement">Arrangement</a> </p> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <p><b>Ensembles de parties</b><br /><span class="mwe-math-element"><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> <a href="/wiki/Union_(math%C3%A9matiques)" title="Union (mathématiques)">Union</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \backslash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant" mathvariant="normal">∖</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \backslash }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e865e5b244a7a5d4fdf7a95fe4cea6d25e581ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \backslash }"></span> <a href="/wiki/Alg%C3%A8bre_des_parties_d%27un_ensemble#Difference" title="Algèbre des parties d'un ensemble">Différence</a><br /><span class="mwe-math-element"><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> <a href="/wiki/Intersection_(math%C3%A9matiques)" title="Intersection (mathématiques)">Intersection</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> <a href="/wiki/Alg%C3%A8bre_des_parties_d%27un_ensemble#Difference_symetrique" title="Algèbre des parties d'un ensemble">Différence symétrique</a> </p><p><b>Ordre total</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695d28931288a686335c3969dfd15bb76ea873db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \min }"></span> <a href="/wiki/Extremum" title="Extremum">Minimum</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e49fca3e322708b32d21eaa8b095dc05f09538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }"></span> <a href="/wiki/Extremum" title="Extremum">Maximum</a> </p><p><b>Treillis</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> <a href="/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné)">Borne inférieure</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> <a href="/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné)">Borne supérieure</a> </p> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <div style="column-count:2;column-gap:1em;text-align:center" class="colonnes"> <p><b>Ensembles</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> <a href="/wiki/Produit_cart%C3%A9sien" title="Produit cartésien">Produit cartésien</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\cup }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∪</mo> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\cup }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca2f2880f761544ce268abf4057fcc1f50c0e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.509ex;" alt="{\displaystyle {\dot {\cup }}}"></span> <a href="/wiki/Op%C3%A9ration_ensembliste#Somme_disjointe" title="Opération ensembliste">Somme disjointe</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97daf05d8be90237eb4d2e4a46d3733ca829bd1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:2.009ex;" alt="{\displaystyle {\hat {}}}"></span> <a href="/wiki/Op%C3%A9ration_ensembliste#Exponentiation" title="Opération ensembliste">Puissance ensembliste</a> </p><p><b>Groupes</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> <a href="/wiki/Somme_directe" title="Somme directe">Somme directe</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"></span> <a href="/wiki/Produit_libre" title="Produit libre">Produit libre</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wr }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≀</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wr }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4cda60813dda84640dc58dc142774abfa5a07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \wr }"></span> <a href="/wiki/Produit_en_couronne" title="Produit en couronne">Produit en couronne</a> </p><p><b>Modules</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" 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 \otimes }"></span> <a href="/wiki/Produit_tensoriel" title="Produit tensoriel">Produit tensoriel</a><br /><span class="mwe-math-element"><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 {Hom} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea3aafc91ebbd147d45c3c69e88431c48cbe9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle \mathrm {Hom} }"></span> <a href="/wiki/Foncteur_Hom" title="Foncteur Hom">Homomorphisme</a><br /><span class="mwe-math-element"><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 {Tor} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Tor} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab99a129c5d7d0c5726c68a87fff94e73a0d57fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.752ex; height:2.176ex;" alt="{\displaystyle \mathrm {Tor} }"></span> <a href="/wiki/Foncteur_Tor" title="Foncteur Tor">Torsion</a><br /><span class="mwe-math-element"><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 {Ext} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ext} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d62c5fa7a8cff017d6febd4d585be1f0bcd2799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ext} }"></span> <a href="/wiki/Foncteur_Ext" title="Foncteur Ext">Extension</a> </p><p><b>Arbres</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> <a href="/w/index.php?title=Enracinement_(math%C3%A9matiques)&action=edit&redlink=1" class="new" title="Enracinement (mathématiques) (page inexistante)">Enracinement</a> </p><p><b>Variétés connexes</b><br /><span class="mwe-math-element"><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"> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72e8254fd59fa4060c66c9310acbaf6df2ce894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.936ex; height:2.509ex;" alt="{\displaystyle \#}"></span> <a href="/wiki/Somme_connexe" title="Somme connexe">Somme connexe</a> </p><p><b>Espaces pointés</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> <a href="/wiki/Bouquet_(math%C3%A9matiques)" title="Bouquet (mathématiques)">Bouquet</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> <a href="/wiki/Smash-produit" title="Smash-produit">Smash-produit</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"></span> <a href="/wiki/Joint_(math%C3%A9matiques)" title="Joint (mathématiques)">Joint</a> </p> </div> </div></td><td rowspan="" colspan="" style="padding:0px;;;width:10em;"><div> <div style="column-count:2;column-gap:1em;text-align:center" class="colonnes"> <p><b>Fonctionnelles</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> <a href="/wiki/Composition_de_fonctions" title="Composition de fonctions">Composition de fonctions</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"></span> <a href="/wiki/Produit_de_convolution" title="Produit de convolution">Produit de convolution</a> </p><p><b>Vectorielles</b><br /><span class="mwe-math-element"><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> <a href="/wiki/Produit_scalaire" title="Produit scalaire">Produit scalaire</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> <a href="/wiki/Produit_vectoriel" title="Produit vectoriel">Produit vectoriel</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59549550bdbbf3ee3c3e699ef776a2fb75d925b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.509ex;" alt="{\displaystyle \times \,}"></span> <a href="/wiki/Produit_vectoriel_en_dimension_7" title="Produit vectoriel en dimension 7">Produit vectoriel généralisé</a> </p><p><b>Algébriques</b><br /><span class="mwe-math-element"><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 stretchy="false">[</mo> <mo>,</mo> <mo stretchy="false">]</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/f04349be7aa458edb6f1ab423189a8455c6e21f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.328ex; height:2.843ex;" alt="{\displaystyle [,]}"></span> <a href="/wiki/Crochet_de_Lie" title="Crochet de Lie">Crochet de Lie</a><br /><span class="mwe-math-element"><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 fence="false" stretchy="false">{</mo> <mo>,</mo> <mo fence="false" stretchy="false">}</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/4e6e62f0d087cac84c136e76f9c5e8c9f6af5f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.359ex; height:2.843ex;" alt="{\displaystyle \{,\}}"></span> <a href="/wiki/Crochet_de_Poisson" title="Crochet de Poisson">Crochet de Poisson</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> <a href="/wiki/Produit_ext%C3%A9rieur" title="Produit extérieur">Produit extérieur</a> </p><p><b>Homologiques</b><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \smile }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⌣</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \smile }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3885eca63e224e40912ea2b44c5fe85f3d4f8be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.309ex; margin-bottom: -0.48ex; width:2.324ex; height:1.343ex;" alt="{\displaystyle \smile }"></span> <a href="/wiki/Cup-produit" title="Cup-produit">Cup-produit</a><br /><span class="mwe-math-element"><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> <a href="/w/index.php?title=Produit_d%27intersection&action=edit&redlink=1" class="new" title="Produit d'intersection (page inexistante)">Produit d'intersection</a> </p><p><b>Séquentielles</b><br /><span class="mwe-math-element"><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> <a href="/wiki/Concat%C3%A9nation" title="Concaténation">Concaténation</a> </p> </div> </div></td></tr></tbody></table></td> </tr> <tr> <td class="navbox-banner" style="background-color:transparent; color:inherit;" colspan="2"><div class="liste-horizontale"><b>Logique booléenne</b> : <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 \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> <a href="/wiki/Fonction_ET" title="Fonction ET">ET (conjonction)</a></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> <a href="/wiki/Fonction_OU" title="Fonction OU">OU (disjonction)</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> <a href="/wiki/Fonction_OU_exclusif" title="Fonction OU exclusif">OU exclusif</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span> <a href="/wiki/Implication_(logique)" title="Implication (logique)">IMP (implication)</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></span> <a href="/wiki/%C3%89quivalence_logique" title="Équivalence logique">EQV (équivalence)</a></li></ul> </div></td></tr></tbody></table> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="3" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Connecteurs_logiques" title="Modèle:Palette Connecteurs logiques"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Connecteurs_logiques&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Connecteur_logique" title="Connecteur logique">Connecteurs logiques</a></div></th> </tr> <tr> <td class="navbox-list" style="text-align:center;" colspan="2"><a href="/wiki/Tautologie_(logique)" title="Tautologie (logique)">Tautologie</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span></td> <td class="navbox-image" rowspan="5" style="vertical-align:middle;padding-left:7px"><span typeof="mw:File"><a href="/wiki/Fichier:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/80px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="80" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/120px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/160px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></td> </tr> <tr> <td class="navbox-list navbox-even" style="text-align:center;" colspan="2"><div class="liste-horizontale"> <ul><li><a href="/wiki/Barre_de_Sheffer" title="Barre de Sheffer">NON-ET</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></li> <li><a href="/wiki/Implication_r%C3%A9ciproque" title="Implication réciproque">Implication réciproque</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftarrow }"></span></li> <li><a href="/wiki/Implication_(logique)" title="Implication (logique)">Implication</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></span></li> <li><a href="/wiki/Disjonction_logique" title="Disjonction logique">OU</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></span></li></ul> </div></td> </tr> <tr> <td class="navbox-list" style="text-align:center;" colspan="2"><div class="liste-horizontale"> <ul><li><a href="/wiki/N%C3%A9gation_logique" title="Négation logique">Négation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></li> <li><a href="/wiki/Fonction_OU_exclusif" title="Fonction OU exclusif">XOR</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 \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span></li> <li><a href="/wiki/%C3%89quivalence_logique" title="Équivalence logique">Équivalence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }"></span></li></ul> </div></td> </tr> <tr> <td class="navbox-list navbox-even" style="text-align:center;" colspan="2"><div class="liste-horizontale"> <ul><li><a href="/wiki/Fonction_NON-OU" title="Fonction NON-OU">NON-OU</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></li> <li><a href="/wiki/Non-implication" title="Non-implication">Non-implication</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></span></li> <li><a href="/wiki/Non-implication_r%C3%A9ciproque" title="Non-implication réciproque">Non-implication réciproque</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nleftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7694c9fc8eebe8a57c8156dd3c2caf022a619439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nleftarrow }"></span></li> <li><a href="/wiki/Conjonction_logique" title="Conjonction logique">ET</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></span></li></ul> </div></td> </tr> <tr> <td class="navbox-list" style="text-align:center;" colspan="2"><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></span></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Informatique" title="Portail de l’informatique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Circle-icons-computer.svg/24px-Circle-icons-computer.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Circle-icons-computer.svg/36px-Circle-icons-computer.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Circle-icons-computer.svg/48px-Circle-icons-computer.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Informatique" title="Portail:Informatique">Portail de l’informatique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:%C3%89lectricit%C3%A9_et_%C3%A9lectronique" title="Portail de l’électricité et de l’électronique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/24px-Octicons-circuit-board.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/36px-Octicons-circuit-board.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/48px-Octicons-circuit-board.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:%C3%89lectricit%C3%A9_et_%C3%A9lectronique" title="Portail:Électricité et électronique">Portail de l’électricité et de l’électronique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Logique" title="Portail de la logique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/48px-Logic.svg.png" decoding="async" width="48" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/72px-Logic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/96px-Logic.svg.png 2x" data-file-width="85" data-file-height="28" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Logique" title="Portail:Logique">Portail de la logique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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