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id="toc-Union_disjointe_d'une_famille_finie_ou_dénombrable_d'ensembles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Union_disjointe_d'une_famille_finie_ou_dénombrable_d'ensembles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Union disjointe d'une famille finie ou dénombrable d'ensembles</span> </div> </a> <ul id="toc-Union_disjointe_d'une_famille_finie_ou_dénombrable_d'ensembles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Union_disjointe_d'une_famille_quelconque_d'ensembles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Union_disjointe_d'une_famille_quelconque_d'ensembles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Union disjointe d'une famille quelconque d'ensembles</span> </div> </a> <ul id="toc-Union_disjointe_d'une_famille_quelconque_d'ensembles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Réunion_disjointe_d'espaces_topologiques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Réunion_disjointe_d'espaces_topologiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Réunion disjointe d'espaces topologiques</span> </div> </a> <ul id="toc-Réunion_disjointe_d'espaces_topologiques-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-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="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">Réunion disjointe</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 19 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-19" 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">19 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Uni%C3%B3_disjunta" title="Unió disjunta – catalan" lang="ca" hreflang="ca" data-title="Unió disjunta" 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/Disjunktn%C3%AD_sjednocen%C3%AD" title="Disjunktní sjednocení – tchèque" lang="cs" hreflang="cs" data-title="Disjunktní sjednocení" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Disjunkte_Vereinigung" title="Disjunkte Vereinigung – allemand" lang="de" hreflang="de" data-title="Disjunkte Vereinigung" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Disjoint_union" title="Disjoint union – anglais" lang="en" hreflang="en" data-title="Disjoint union" 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/Disa_kuna%C4%B5o" title="Disa kunaĵo – espéranto" lang="eo" hreflang="eo" data-title="Disa kunaĵo" 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/Uni%C3%B3n_disjunta" title="Unión disjunta – espagnol" lang="es" hreflang="es" data-title="Unión disjunta" 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/L%C3%B5ikumatu_%C3%BChend" title="Lõikumatu ühend – estonien" lang="et" hreflang="et" data-title="Lõikumatu ühend" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AC%D8%AA%D9%85%D8%A7%D8%B9_%D9%85%D8%AC%D8%B2%D8%A7" title="اجتماع مجزا – persan" lang="fa" hreflang="fa" data-title="اجتماع مجزا" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diszjunkt_uni%C3%B3" title="Diszjunkt unió – hongrois" lang="hu" hreflang="hu" data-title="Diszjunkt unió" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Somma_disgiunta" title="Somma disgiunta – italien" lang="it" hreflang="it" data-title="Somma disgiunta" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9B%B4%E5%92%8C" title="直和 – japonais" lang="ja" hreflang="ja" data-title="直和" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EB%A6%AC%ED%95%A9%EC%A7%91%ED%95%A9" title="분리합집합 – coréen" lang="ko" hreflang="ko" data-title="분리합집합" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Disjuncte_vereniging" title="Disjuncte vereniging – néerlandais" lang="nl" hreflang="nl" data-title="Disjuncte vereniging" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Suma_roz%C5%82%C4%85czna" title="Suma rozłączna – polonais" lang="pl" hreflang="pl" data-title="Suma rozłączna" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Uni%C3%A3o_disjunta" title="União disjunta – portugais" lang="pt" hreflang="pt" data-title="União disjunta" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B7%D1%8A%D1%8E%D0%BD%D0%BA%D1%82%D0%BD%D0%BE%D0%B5_%D0%BE%D0%B1%D1%8A%D0%B5%D0%B4%D0%B8%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Дизъюнктное объединение – russe" lang="ru" hreflang="ru" data-title="Дизъюнктное объединение" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B7%27%D1%8E%D0%BD%D0%BA%D1%82%D0%BD%D0%B5_%D0%BE%D0%B1%27%D1%94%D0%B4%D0%BD%D0%B0%D0%BD%D0%BD%D1%8F" title="Диз'юнктне об'єднання – ukrainien" lang="uk" hreflang="uk" data-title="Диз'юнктне об'єднання" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%8D%E4%BA%A4%E5%B9%B6" title="不交并 – chinois" lang="zh" hreflang="zh" data-title="不交并" 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 href="https://zh-yue.wikipedia.org/wiki/%E4%B8%8D%E4%BA%A4%E4%BD%B5%E9%9B%86" title="不交併集 – cantonais" lang="yue" hreflang="yue" data-title="不交併集" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q842620#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu 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data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirigé depuis <a href="/w/index.php?title=Union_disjointe&redirect=no" class="mw-redirect" title="Union disjointe">Union disjointe</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, la <b>réunion disjointe</b> est une <a href="/wiki/Op%C3%A9ration_ensembliste" title="Opération ensembliste">opération ensembliste</a>. Contrairement à l'<a href="/wiki/Union_(math%C3%A9matiques)" title="Union (mathématiques)">union usuelle</a>, le <a href="/wiki/Cardinalit%C3%A9_(math%C3%A9matiques)" title="Cardinalité (mathématiques)">cardinal d'une</a> <b>union disjointe</b> d'<a href="/wiki/Ensemble" title="Ensemble">ensembles</a> est toujours égal à la somme de leurs <a href="/wiki/Nombre_cardinal" title="Nombre cardinal">cardinaux</a>. L'union disjointe d'une famille d'ensembles correspond à leur <a href="/wiki/Somme_(cat%C3%A9gorie)" title="Somme (catégorie)">somme</a> en <a href="/wiki/Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories">théorie des catégories</a>, c'est pourquoi on l'appelle aussi <b>somme disjointe</b>. C’est une opération fréquente en <a href="/wiki/Topologie" title="Topologie">topologie</a> et en <a href="/wiki/Informatique_th%C3%A9orique" title="Informatique théorique">informatique théorique</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Union_disjointe_de_deux_ensembles">Union disjointe de deux ensembles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A9union_disjointe&veaction=edit&section=1" title="Modifier la section : Union disjointe de deux ensembles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A9union_disjointe&action=edit&section=1" title="Modifier le code source de la section : Union disjointe de deux ensembles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans une réunion <i>A</i>∪<i>B</i> de deux ensembles, l'origine des éléments y figurant est perdue et les éléments de l'intersection ne sont comptés qu'une seule fois. Dans certaines situations, on désire conserver cette information et prendre en compte deux fois les éléments de l'intersection. Pour cela, on réunit non pas directement <i>A</i> et <i>B</i>, mais deux <a href="/wiki/Ensembles_disjoints" title="Ensembles disjoints">ensembles disjoints</a>, copies de <i>A</i> et <i>B</i> de la forme  { α } × <i>A</i>  et  { β } × <i>B</i> , où α et β sont deux symboles quelconques distincts servant à identifier les ensembles <i>A</i> et <i>B</i> (par exemple 0 et 1) et × désigne le <a href="/wiki/Produit_cart%C3%A9sien" title="Produit cartésien">produit cartésien</a>. </p><p>L'union disjointe, encore appelée « somme disjointe » ou « somme cartésienne », de deux ensembles <i>A</i> et <i>B</i> est ainsi définie par : </p> <center><span class="mwe-math-element"><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\sqcup B=A+B=A{\dot {\cup }}B:=(\{0\}\times A)\cup (\{1\}\times B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊔</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∪</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>B</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\sqcup B=A+B=A{\dot {\cup }}B:=(\{0\}\times A)\cup (\{1\}\times B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e26b123e0c2373c8fd1a17b46daef510c08301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.448ex; height:3.009ex;" alt="{\displaystyle A\sqcup B=A+B=A{\dot {\cup }}B:=(\{0\}\times A)\cup (\{1\}\times B).}"></span></center> <dl><dt>Exemples</dt> <dd> <ul><li>Soient <i>A</i> la <a href="/wiki/Paire" title="Paire">paire</a> {1, 2} et <i>B</i> l'ensemble à trois éléments {2, 3, 4}. Leur réunion (ordinaire) n'a que quatre éléments car ces deux ensembles ne sont pas disjoints. Pour construire leur union disjointe, on commence par les « numéroter » par deux « indices » distincts arbitraires <i>a</i> et <i>b</i> : on pose <i>I</i> = {<i>a</i>, <i>b</i>}, <i>E<sub>a</sub> = A</i> et <i>E<sub>b</sub> = B</i>. Puis on prend la réunion de deux « copies » de <i>A</i> et <i>B</i> qui, elles, sont disjointes : {<i>a</i>}×<i>A</i> et {<i>b</i>}×<i>B</i>. La réunion disjointe de (<i>E<sub>i</sub></i>)<sub><i>i</i>∈<i>I</i></sub> est la partie<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 (\{a\}\times \{1,2\})\cup (\{b\}\times \{2,3,4\})=\{(a,1),(a,2),(b,2),(b,3),(b,4)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\{a\}\times \{1,2\})\cup (\{b\}\times \{2,3,4\})=\{(a,1),(a,2),(b,2),(b,3),(b,4)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ba0a7ecef155fa00999bdc17063c5dc165da90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.363ex; height:2.843ex;" alt="{\displaystyle (\{a\}\times \{1,2\})\cup (\{b\}\times \{2,3,4\})=\{(a,1),(a,2),(b,2),(b,3),(b,4)\}}"></span></span>de {<i>a</i>, <i>b</i>}×{1, 2, 3, 4}. Elle a 2 + 3 = 5 éléments.</li> <li>De même, l'union disjointe de la paire {1, 2} avec elle-même est (en choisissant arbitrairement deux indices distincts, par exemple cette fois : 0 et 1) :<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 \{1,2\}\sqcup \{1,2\}=\{(0,1),(0,2),(1,1),(1,2)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2\}\sqcup \{1,2\}=\{(0,1),(0,2),(1,1),(1,2)\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696a3b9ae6726fb0f4bc0f2b666a176c97e00557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.795ex; height:2.843ex;" alt="{\displaystyle \{1,2\}\sqcup \{1,2\}=\{(0,1),(0,2),(1,1),(1,2)\}.}"></span></span></li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Union_disjointe_d'une_famille_finie_ou_dénombrable_d'ensembles"><span id="Union_disjointe_d.27une_famille_finie_ou_d.C3.A9nombrable_d.27ensembles"></span>Union disjointe d'une famille finie ou dénombrable d'ensembles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A9union_disjointe&veaction=edit&section=2" title="Modifier la section : Union disjointe d'une famille finie ou dénombrable d'ensembles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A9union_disjointe&action=edit&section=2" title="Modifier le code source de la section : Union disjointe d'une famille finie ou dénombrable d'ensembles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La somme disjointe peut se généraliser à plus de deux ensembles. Par exemple, pour trois ensembles quelconques <i>A</i>, <i>B</i> et <i>C</i>: </p> <center><span class="mwe-math-element"><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\sqcup B\sqcup C=(\{0\}\times A)\cup (\{1\}\times B)\cup (\{2\}\times 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> <mo>=</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\sqcup B\sqcup C=(\{0\}\times A)\cup (\{1\}\times B)\cup (\{2\}\times C).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5236f6f09b4bd34e78e57edfc0f30b42ebb84e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.034ex; height:2.843ex;" alt="{\displaystyle A\sqcup B\sqcup C=(\{0\}\times A)\cup (\{1\}\times B)\cup (\{2\}\times C).}"></span></center> <p>On peut définir plus généralement la somme disjointe de <i>n</i> ensembles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0},E_{1},\cdots E_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0},E_{1},\cdots E_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502f8144960c5a671089bf14721ef09a0e75451e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.751ex; height:2.509ex;" alt="{\displaystyle E_{0},E_{1},\cdots E_{n-1}}"></span> quelconques : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}\sqcup E_{1}\sqcup \cdots \sqcup E_{n-1}=\bigcup _{i=0}^{n-1}(\{i\}\times E_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊔</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊔</mo> <mo>⋯</mo> <mo>⊔</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}\sqcup E_{1}\sqcup \cdots \sqcup E_{n-1}=\bigcup _{i=0}^{n-1}(\{i\}\times E_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3101333464f9b6b333f28d1c4df687bde6cea47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.168ex; height:7.176ex;" alt="{\displaystyle E_{0}\sqcup E_{1}\sqcup \cdots \sqcup E_{n-1}=\bigcup _{i=0}^{n-1}(\{i\}\times E_{i}).}"></span></center> <p>On peut également généraliser cette notion à des ensembles quelconques (non nécessairement finis) d'indices, et former par exemple des unions disjointes <a href="/wiki/Ensemble_d%C3%A9nombrable" title="Ensemble dénombrable">dénombrables</a>. </p> <dl><dt>Exemple</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 \bigsqcup _{n\in \mathbb {N} }\left[n,+\infty \right[=\{(n,x)\in \mathbb {N} \times \mathbb {R} \mid x\geq n\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mi>n</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>[</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∣</mo> <mi>x</mi> <mo>≥</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigsqcup _{n\in \mathbb {N} }\left[n,+\infty \right[=\{(n,x)\in \mathbb {N} \times \mathbb {R} \mid x\geq n\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1edeafe335c866578f1891126bff10ebc4d98a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.945ex; height:5.676ex;" alt="{\displaystyle \bigsqcup _{n\in \mathbb {N} }\left[n,+\infty \right[=\{(n,x)\in \mathbb {N} \times \mathbb {R} \mid x\geq n\}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Union_disjointe_d'une_famille_quelconque_d'ensembles"><span id="Union_disjointe_d.27une_famille_quelconque_d.27ensembles"></span>Union disjointe d'une famille quelconque d'ensembles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A9union_disjointe&veaction=edit&section=3" title="Modifier la section : Union disjointe d'une famille quelconque d'ensembles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A9union_disjointe&action=edit&section=3" title="Modifier le code source de la section : Union disjointe d'une famille quelconque d'ensembles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour toute <a href="/wiki/Famille_(math%C3%A9matiques)" title="Famille (mathématiques)">famille (<i>E<sub>i</sub></i>)<sub><i>i</i>∈<i>I</i></sub> d'ensembles</a>, les ensembles produits {<i>i</i>}×<i>E<sub>i</sub></i> (<i>i</i> parcourant l'ensemble <i>I</i> des indices de la famille) sont disjoints deux à deux. La réunion disjointe ∐<sub><i>i</i>∈<i>I</i></sub> <i>E<sub>i</sub></i> des <i>E<sub>i</sub></i> est, par définition, la <a href="/wiki/Union_(math%C3%A9matiques)#Union_d'une_famille_d'ensembles" title="Union (mathématiques)">réunion (ordinaire)</a> de ces ensembles disjoints. Formellement : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigsqcup _{i\in I}E_{i}=\{(i,x)\mid i\in I,\ x\in E_{i}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigsqcup _{i\in I}E_{i}=\{(i,x)\mid i\in I,\ x\in E_{i}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606802a7538fdd34f7fb5b889c2b0f42bcc7e2be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.581ex; height:5.676ex;" alt="{\displaystyle \bigsqcup _{i\in I}E_{i}=\{(i,x)\mid i\in I,\ x\in E_{i}\}.}"></span></center> <p>Il s'agit bien d'un ensemble car, vue sa définition, ∐<sub><i>i</i>∈<i>I</i></sub> <i>E<sub>i</sub></i> peut se décrire <a href="/wiki/Sch%C3%A9ma_d%27axiomes_de_compr%C3%A9hension" title="Schéma d'axiomes de compréhension">en compréhension</a> comme une partie de <i>I</i>×<i>E</i>, le produit cartésien de <i>I </i> par la réunion (ordinaire) <i>E</i> des <i>E<sub>i</sub></i>. </p><p>La définition de la somme disjointe souffre d'un arbitraire inessentiel. On peut définir la somme disjointe comme étant la réunion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{i\in I}(E_{i}\times \{i\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{i\in I}(E_{i}\times \{i\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada5067994e443a43719f8c3f817b5a7e7f27499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.874ex; height:5.676ex;" alt="{\displaystyle \bigcup _{i\in I}(E_{i}\times \{i\})}"></span> ou bien <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{i\in I}(\{i\}\times E_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{i\in I}(\{i\}\times E_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cfa9d181b8962d58adb85060d114151ce8f57a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.874ex; height:5.676ex;" alt="{\displaystyle \bigcup _{i\in I}(\{i\}\times E_{i})}"></span><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>. Ces deux possibilités correspondent respectivement à un marquage « à droite » ou « à gauche » des éléments de la réunion ordinaire <i>E</i>, selon l'indice associé à l'ensemble dont ils proviennent. Dans les deux cas, il existe une <a href="/wiki/Surjection" title="Surjection">surjection</a> de la somme disjointe sur la réunion, qui est une <a href="/wiki/Bijection" title="Bijection">bijection</a> si les ensembles de la famille (<i>E<sub>i</sub></i>)<sub><i>i</i>∈<i>I</i></sub> sont disjoints deux à deux. </p><p>On peut remarquer que la somme disjointe de deux ensembles vérifie la propriété fondamentale des <a href="/wiki/Couple_(math%C3%A9matiques)" title="Couple (mathématiques)">couples</a>. De plus, contrairement aux couples de Kuratowski, cette notion, qui n'utilise que des opérations ensemblistes élémentaires, peut s'appliquer aux <a href="/wiki/Classe_(math%C3%A9matiques)" title="Classe (mathématiques)">classes propres</a>. C'est pourquoi les sommes disjointes sont parfois appelées <i>couples généralisés</i>, et utilisées ainsi en <a href="/wiki/Th%C3%A9orie_des_ensembles_de_von_Neumann-Bernays-G%C3%B6del" title="Théorie des ensembles de von Neumann-Bernays-Gödel">théorie des classes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Réunion_disjointe_d'espaces_topologiques"><span id="R.C3.A9union_disjointe_d.27espaces_topologiques"></span>Réunion disjointe d'espaces topologiques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A9union_disjointe&veaction=edit&section=4" title="Modifier la section : Réunion disjointe d'espaces topologiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A9union_disjointe&action=edit&section=4" title="Modifier le code source de la section : Réunion disjointe d'espaces topologiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans la définition ci-dessus, si chaque <i>E<sub>i</sub></i> est un <a href="/wiki/Espace_topologique" title="Espace topologique">espace topologique</a>, on dispose d'une topologie naturelle sur ∐<sub><i>i</i>∈<i>I</i></sub> <i>E<sub>i</sub></i>, dont les <a href="/wiki/Ouvert_(topologie)" title="Ouvert (topologie)">ouverts</a> sont les réunions disjointes ∐<sub><i>i</i>∈<i>I</i></sub> <i>U<sub>i</sub></i> où chaque <i>U<sub>i</sub></i> est un ouvert de <i>E<sub>i</sub></i>. </p><p>Cette construction, appelée <a href="/wiki/Somme_topologique" title="Somme topologique">somme topologique</a>, joue le rôle de somme dans la <a href="/wiki/Cat%C3%A9gorie_des_espaces_topologiques" title="Catégorie des espaces topologiques">catégorie des espaces topologiques</a>. Alliée avec l'<a href="/wiki/Topologie_quotient" title="Topologie quotient">espace quotient</a>, elle permet de construire de nombreux espaces, notamment les <a href="/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie)" title="Variété (géométrie)">variétés topologiques</a> et les <a href="/wiki/CW-complexe" title="CW-complexe">complexes cellulaires</a> ou <a href="/wiki/Complexe_simplicial" title="Complexe simplicial">simpliciaux</a>. </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=R%C3%A9union_disjointe&veaction=edit&section=5" 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=R%C3%A9union_disjointe&action=edit&section=5" 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> <ul><li>Cet article est partiellement ou en totalité issu de l'article intitulé « <span class="plainlinks"><a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Produit_cart%C3%A9sien&oldid=16104779">Produit cartésien</a> » <small>(voir <span class="plainlinks"><a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Produit_cart%C3%A9sien&oldid=16104779&action=history">la liste des auteurs</a></span>)</small></span>.</li></ul> <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">N. Bourbaki, <i>Théorie des ensembles</i>, 1970, p. II.30, donne la première de ces deux définition, mais il utilise la seconde dans <i>Algèbre, chapitres 1 à 3</i>, 1970, p. I.80.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A9union_disjointe&veaction=edit&section=6" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A9union_disjointe&action=edit&section=6" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Multiensemble" title="Multiensemble">Multiensemble</a> : généralisation de la notion d'ensemble, où l'on permet plusieurs occurrences (indiscernables) d'un même élément ; l'union de deux multiensembles ayant des éléments communs n’amène pas à les disjoindre comme ci-dessus mais à cumuler les nombres d'occurrences de chaque élément. </p> <ul id="bandeau-portail" class="bandeau-portail"><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&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ce document provient de « <a dir="ltr" href="https://fr.wikipedia.org/w/index.php?title=Réunion_disjointe&oldid=217995725">https://fr.wikipedia.org/w/index.php?title=Réunion_disjointe&oldid=217995725</a> ».</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Cat%C3%A9gorie:Accueil" title="Catégorie:Accueil">Catégories</a> : <ul><li><a href="/wiki/Cat%C3%A9gorie:Th%C3%A9orie_des_ensembles" title="Catégorie:Théorie des ensembles">Théorie des ensembles</a></li><li><a href="/wiki/Cat%C3%A9gorie:Th%C3%A9orie_des_cat%C3%A9gories" title="Catégorie:Théorie des catégories">Théorie des catégories</a></li><li><a href="/wiki/Cat%C3%A9gorie:Topologie_g%C3%A9n%C3%A9rale" title="Catégorie:Topologie générale">Topologie générale</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Catégories cachées : <ul><li><a href="/wiki/Cat%C3%A9gorie:Portail:Math%C3%A9matiques/Articles_li%C3%A9s" title="Catégorie:Portail:Mathématiques/Articles liés">Portail:Mathématiques/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Portail:Sciences/Articles_li%C3%A9s" title="Catégorie:Portail:Sciences/Articles liés">Portail:Sciences/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Projet:Math%C3%A9matiques/Articles" title="Catégorie:Projet:Mathématiques/Articles">Projet:Mathématiques/Articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La dernière modification de cette page a été faite le 25 août 2024 à 06:50.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A9dia:Citation_et_r%C3%A9utilisation_du_contenu_de_Wikip%C3%A9dia" title="Wikipédia:Citation et réutilisation du contenu de Wikipédia">Droit d'auteur</a> : les textes sont disponibles sous <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fr">licence Creative Commons attribution, partage dans les mêmes conditions</a> ; d’autres conditions peuvent s’appliquer. 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