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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Definições_básicas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definições_básicas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definições básicas</span> </div> </a> <button aria-controls="toc-Definições_básicas-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>Alternar a subsecção Definições básicas</span> </button> <ul id="toc-Definições_básicas-sublist" class="vector-toc-list"> <li id="toc-Definição_1:_Ordem_parcial_ampla_ou_não_estrita" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definição_1:_Ordem_parcial_ampla_ou_não_estrita"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definição 1: Ordem parcial ampla ou não estrita</span> </div> </a> <ul id="toc-Definição_1:_Ordem_parcial_ampla_ou_não_estrita-sublist" class="vector-toc-list"> <li id="toc-1.a_Reflexividade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#1.a_Reflexividade"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>1.a Reflexividade</span> </div> </a> <ul id="toc-1.a_Reflexividade-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-1.b_Antissimetria" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#1.b_Antissimetria"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>1.b Antissimetria</span> </div> </a> <ul id="toc-1.b_Antissimetria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-1.c_Transitividade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#1.c_Transitividade"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>1.c Transitividade</span> </div> </a> <ul id="toc-1.c_Transitividade-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definição_2:_Ordem_parcial_estrita" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definição_2:_Ordem_parcial_estrita"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definição 2: Ordem parcial estrita</span> </div> </a> <ul id="toc-Definição_2:_Ordem_parcial_estrita-sublist" class="vector-toc-list"> <li id="toc-2.a_Irreflexividade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2.a_Irreflexividade"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>2.a Irreflexividade</span> </div> </a> <ul id="toc-2.a_Irreflexividade-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2.b_Assimetria" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2.b_Assimetria"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>2.b Assimetria</span> </div> </a> <ul id="toc-2.b_Assimetria-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definição_3:_Correspondência_entre_ordens_estritas_e_amplas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definição_3:_Correspondência_entre_ordens_estritas_e_amplas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Definição 3: Correspondência entre ordens estritas e amplas</span> </div> </a> <ul id="toc-Definição_3:_Correspondência_entre_ordens_estritas_e_amplas-sublist" class="vector-toc-list"> <li id="toc-3.a_Correspondência" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#3.a_Correspondência"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>3.a Correspondência</span> </div> </a> <ul id="toc-3.a_Correspondência-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Relações_de_ordem_linear_ou_total" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relações_de_ordem_linear_ou_total"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relações de ordem linear ou total</span> </div> </a> <button aria-controls="toc-Relações_de_ordem_linear_ou_total-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>Alternar a subsecção Relações de ordem linear ou total</span> </button> <ul id="toc-Relações_de_ordem_linear_ou_total-sublist" class="vector-toc-list"> <li id="toc-Definição_4:_Totalidade_ou_linearidade" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definição_4:_Totalidade_ou_linearidade"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Definição 4: Totalidade ou linearidade</span> </div> </a> <ul id="toc-Definição_4:_Totalidade_ou_linearidade-sublist" class="vector-toc-list"> <li id="toc-4.a_Totalidade_ou_linearidade_(para_ordens_amplas)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#4.a_Totalidade_ou_linearidade_(para_ordens_amplas)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>4.a Totalidade ou linearidade (para ordens amplas)</span> </div> </a> <ul id="toc-4.a_Totalidade_ou_linearidade_(para_ordens_amplas)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4.b_Totalidade_ou_linearidade_(para_ordens_estritas)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#4.b_Totalidade_ou_linearidade_(para_ordens_estritas)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>4.b Totalidade ou linearidade (para ordens estritas)</span> </div> </a> <ul id="toc-4.b_Totalidade_ou_linearidade_(para_ordens_estritas)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Relações_de_ordem_densa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relações_de_ordem_densa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relações de ordem densa</span> </div> </a> <button aria-controls="toc-Relações_de_ordem_densa-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>Alternar a subsecção Relações de ordem densa</span> </button> <ul id="toc-Relações_de_ordem_densa-sublist" class="vector-toc-list"> <li id="toc-Definição_5:_Densidade" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definição_5:_Densidade"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Definição 5: Densidade</span> </div> </a> <ul id="toc-Definição_5:_Densidade-sublist" class="vector-toc-list"> <li id="toc-5_Densidade_(para_ordens_estritas)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#5_Densidade_(para_ordens_estritas)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>5 Densidade (para ordens estritas)</span> </div> </a> <ul id="toc-5_Densidade_(para_ordens_estritas)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Inversa_de_uma_ordem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inversa_de_uma_ordem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inversa de uma ordem</span> </div> </a> <ul id="toc-Inversa_de_uma_ordem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementos_distinguidos_numa_ordem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementos_distinguidos_numa_ordem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Elementos distinguidos numa ordem</span> </div> </a> <button aria-controls="toc-Elementos_distinguidos_numa_ordem-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>Alternar a subsecção Elementos distinguidos numa ordem</span> </button> <ul id="toc-Elementos_distinguidos_numa_ordem-sublist" class="vector-toc-list"> <li id="toc-Mínimo_e_máximo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mínimo_e_máximo"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Mínimo e máximo</span> </div> </a> <ul id="toc-Mínimo_e_máximo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimal_e_maximal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimal_e_maximal"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Minimal e maximal</span> </div> </a> <ul id="toc-Minimal_e_maximal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cotas_inferior_(minorante)_e_superior_(majorante)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cotas_inferior_(minorante)_e_superior_(majorante)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Cotas inferior (minorante) e superior (majorante)</span> </div> </a> <ul id="toc-Cotas_inferior_(minorante)_e_superior_(majorante)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Boa_ordem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Boa_ordem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Boa ordem</span> </div> </a> <ul id="toc-Boa_ordem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <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="Alternar o índice" > <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">Alternar o índice</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">Relação de ordem</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="Ir para um artigo noutra língua. Disponível em 20 línguas" > <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-20" 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">20 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D1%87%D1%8B%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BA%D1%83" title="Дачыненне парадку — bielorrusso" lang="be" hreflang="be" data-title="Дачыненне парадку" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%99%C4%95%D1%80%D0%BA%D0%B5%D0%B2_%D1%82%D0%B8%D0%B2%C4%95%D0%BC%C4%95" title="Йĕркев тивĕмĕ — chuvash" lang="cv" hreflang="cv" data-title="Йĕркев тивĕмĕ" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ordnungsrelation" title="Ordnungsrelation — alemão" lang="de" hreflang="de" data-title="Ordnungsrelation" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70894304 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Partial_order" title="Partial order — inglês" lang="en" hreflang="en" data-title="Partial order" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Osaline_j%C3%A4rjestus" title="Osaline järjestus — estónio" lang="et" hreflang="et" data-title="Osaline järjestus" data-language-autonym="Eesti" data-language-local-name="estónio" 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%AA%D8%B1%D8%AA%DB%8C%D8%A8_%D8%AC%D8%B2%D8%A6%DB%8C" title="ترتیب جزئی — persa" lang="fa" hreflang="fa" data-title="ترتیب جزئی" data-language-autonym="فارسی" data-language-local-name="persa" 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/J%C3%A4rjestysrelaatio" title="Järjestysrelaatio — finlandês" lang="fi" hreflang="fi" data-title="Järjestysrelaatio" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Relation_d%27ordre" title="Relation d'ordre — francês" lang="fr" hreflang="fr" data-title="Relation d'ordre" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%A1_%D7%A1%D7%93%D7%A8" title="יחס סדר — hebraico" lang="he" hreflang="he" data-title="יחס סדר" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rendezett_halmaz" title="Rendezett halmaz — húngaro" lang="hu" hreflang="hu" data-title="Rendezett halmaz" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Rangala_relato" title="Rangala relato — ido" lang="io" hreflang="io" data-title="Rangala relato" 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/Relazione_d%27ordine" title="Relazione d'ordine — italiano" lang="it" hreflang="it" data-title="Relazione d'ordine" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Uerdnung_(Mathematik)" title="Uerdnung (Mathematik) — luxemburguês" lang="lb" hreflang="lb" data-title="Uerdnung (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburguês" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Relazion_d%27orden" title="Relazion d'orden — lombardo" lang="lmo" hreflang="lmo" data-title="Relazion d'orden" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Relacion_d%27%C3%B2rdre" title="Relacion d'òrdre — occitano" lang="oc" hreflang="oc" data-title="Relacion d'òrdre" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Relassion_d%27%C3%B3rdin" title="Relassion d'órdin — Piedmontese" lang="pms" hreflang="pms" data-title="Relassion d'órdin" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Rela%C8%9Bie_de_ordine" title="Relație de ordine — romeno" lang="ro" hreflang="ro" data-title="Relație de ordine" data-language-autonym="Română" data-language-local-name="romeno" 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%9E%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BF%D0%BE%D1%80%D1%8F%D0%B4%D0%BA%D0%B0" title="Отношение порядка — russo" lang="ru" hreflang="ru" data-title="Отношение порядка" data-language-autonym="Русский" data-language-local-name="russo" 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href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem_total&redirect=no" class="mw-redirect" title="Relação de ordem total">Relação de ordem total</a></u>)</span></span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pt" dir="ltr"><p>Em <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a> e em <a href="/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática">lógica matemática</a>, especialmente em <a href="/wiki/Teoria_dos_conjuntos" title="Teoria dos conjuntos">teoria dos conjuntos</a> e em <a href="/wiki/Teoria_das_rela%C3%A7%C3%B5es" class="mw-redirect" title="Teoria das relações">teoria das relações</a>, uma <b>relação de ordem</b> é uma <a href="/wiki/Rela%C3%A7%C3%A3o_bin%C3%A1ria" title="Relação binária">relação binária</a> que pretende captar o sentido intuitivo de relações como o maior e o menor, o anterior e o posterior, etc. Foram <a href="/wiki/Defini%C3%A7%C3%A3o" title="Definição">definidos</a> muitos tipos de relações de ordem e diferentes obras usam os termos "ordem" e "relação de ordem" de maneiras diversas, pelo qual existe uma <a href="/wiki/Ambiguidade" title="Ambiguidade">ambiguidade</a> na literatura. Os tópicos "relações de ordens" estão fortemente vinculados ao <a href="/wiki/Conjunto_parcialmente_ordenado" title="Conjunto parcialmente ordenado">conjunto parcialmente ordenado</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definições_básicas"><span id="Defini.C3.A7.C3.B5es_b.C3.A1sicas"></span>Definições básicas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=1" title="Editar secção: Definições básicas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=1" title="Editar código-fonte da secção: Definições básicas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definição_1:_Ordem_parcial_ampla_ou_não_estrita"><span id="Defini.C3.A7.C3.A3o_1:_Ordem_parcial_ampla_ou_n.C3.A3o_estrita"></span>Definição 1: Ordem parcial ampla ou não estrita</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=2" title="Editar secção: Definição 1: Ordem parcial ampla ou não estrita" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=2" title="Editar código-fonte da secção: Definição 1: Ordem parcial ampla ou não estrita"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado um <a href="/wiki/Conjunto" title="Conjunto">conjunto</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}"> <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> e uma <a href="/wiki/Rela%C3%A7%C3%A3o_bin%C3%A1ria" title="Relação binária">relação binária</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> sobre <span class="mwe-math-element"><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> <mo>:</mo> </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/ef11a9011a94c416a5051ff441edd14571e6319d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.035ex; height:2.176ex;" alt="{\displaystyle A:}"></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 R\subseteq A\times A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq A\times A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519807a3a3b96cd5a6b9648f5a6082188ea14e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.836ex; height:2.509ex;" alt="{\displaystyle R\subseteq A\times A,}"></span> dizemos 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> é uma <i>relação de ordem (parcial) ampla (ou não estrita) sobre <span class="mwe-math-element"><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></i> se satisfaz as seguintes condições:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="1.a_Reflexividade">1.a Reflexividade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=3" title="Editar secção: 1.a Reflexividade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=3" title="Editar código-fonte da secção: 1.a Reflexividade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x\in A\;\;R(x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x\in A\;\;R(x,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8c5dfb8d7122c64b5ad6d4ed9c3a744b9225e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.344ex; height:2.843ex;" alt="{\displaystyle \forall \ x\in A\;\;R(x,x)}"></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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span> (ou seja, todo elemento está relacionado consigo mesmo); </p> <div class="mw-heading mw-heading4"><h4 id="1.b_Antissimetria">1.b Antissimetria</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=4" title="Editar secção: 1.b Antissimetria" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=4" title="Editar código-fonte da secção: 1.b Antissimetria"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y\in A\;\left(R(x,y)\wedge R(y,x)\Rightarrow x=y\right);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\;\left(R(x,y)\wedge R(y,x)\Rightarrow x=y\right);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14a2a8f8cc7b1e52cb4e5826742cf898b71b656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.817ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\;\left(R(x,y)\wedge R(y,x)\Rightarrow x=y\right);}"></span> e </p> <div class="mw-heading mw-heading4"><h4 id="1.c_Transitividade">1.c Transitividade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=5" title="Editar secção: 1.c Transitividade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=5" title="Editar código-fonte da secção: 1.c Transitividade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y,z\in A\;\left(R(x,y)\wedge R(y,z)\Rightarrow R(x,z)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y,z\in A\;\left(R(x,y)\wedge R(y,z)\Rightarrow R(x,z)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8bfce5edd7a136d71916575e90c1c69d3b6735b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.105ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y,z\in A\;\left(R(x,y)\wedge R(y,z)\Rightarrow R(x,z)\right)}"></span> </p><p>Quando uma relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> satisfaz as condições acima, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45f0bb548f5fe8470b7257809f044c967628eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.092ex; height:2.843ex;" alt="{\displaystyle R(x,y)}"></span> é escrito como <span class="mwe-math-element"><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\leq y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6616694036cf80c75a6bf94273b2e902cfc41884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.343ex;" alt="{\displaystyle x\leq y.}"></span> A relação habitual de menor ou igual em conjuntos numéricos, <a href="/wiki/N%C3%BAmero_natural" title="Número natural"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></a>, <a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span></a>, <a href="/wiki/N%C3%BAmero_racional" title="Número racional"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span></a>, <a href="/wiki/N%C3%BAmero_real" title="Número real"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></a>, cumpre com essas condições explicando essa notação. </p><p>Um exemplo típico é a relação de inclusão (ampla) entre conjuntos: <span class="mwe-math-element"><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\subseteq B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7517fdf97968868c52de7a3de77b995cca2d284b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.252ex; height:2.343ex;" alt="{\displaystyle A\subseteq B.}"></span> geralmente definida sobre o conjunto das partes 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 A:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> </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/ef11a9011a94c416a5051ff441edd14571e6319d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.035ex; height:2.176ex;" alt="{\displaystyle A:}"></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 {\mathcal {P}}\left(A\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(A\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c1588c402efa77190eea04e971d0dede790bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.677ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(A\right).}"></span> Um outro exemplo é a relação "<span class="mwe-math-element"><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> divide <span class="mwe-math-element"><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>": seja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a339dbd4a64016f4d222b5cd5d840d77041924a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ^{+}}"></span> o conjunto dos números naturais maiores que zero. Para <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {N} ^{+},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {N} ^{+},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bceb16ff08e806fb048a60a6171d2467802b1f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.195ex; height:2.843ex;" alt="{\displaystyle x,y\in \mathbb {N} ^{+},}"></span> dizemos que <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> divide <span class="mwe-math-element"><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></i>, em símbolos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x|y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x|y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2491cf336f81df9d2e0a2d98b44adbe6297a9605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.132ex; height:2.843ex;" alt="{\displaystyle x|y}"></span> se e somente se existe um <span class="mwe-math-element"><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\in \mathbb {N} ^{+},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {N} ^{+},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982d7682c59426ccb86b05fa6c224823978b65ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.765ex; height:2.843ex;" alt="{\displaystyle z\in \mathbb {N} ^{+},}"></span> tal 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 z.x=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>.</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z.x=y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27b0d44e15c0c2b2f70d9e293d9b73922d74234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.353ex; height:2.009ex;" alt="{\displaystyle z.x=y.}"></span> Pode ser demonstrado que a relação "divide" assim definida satisfaz as condições da <a href="#Definição_1:_Ordem_parcial_ampla_ou_não_estrita">Definição 1.</a> </p> <div class="mw-heading mw-heading3"><h3 id="Definição_2:_Ordem_parcial_estrita"><span id="Defini.C3.A7.C3.A3o_2:_Ordem_parcial_estrita"></span>Definição 2: Ordem parcial estrita</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=6" title="Editar secção: Definição 2: Ordem parcial estrita" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=6" title="Editar código-fonte da secção: Definição 2: Ordem parcial estrita"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado um <a href="/wiki/Conjunto" title="Conjunto">conjunto</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}"> <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> e uma <a href="/wiki/Rela%C3%A7%C3%A3o_bin%C3%A1ria" title="Relação binária">relação binária</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> sobre <span class="mwe-math-element"><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> <mo>:</mo> </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/ef11a9011a94c416a5051ff441edd14571e6319d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.035ex; height:2.176ex;" alt="{\displaystyle A:}"></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 R\subseteq A\times A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq A\times A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519807a3a3b96cd5a6b9648f5a6082188ea14e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.836ex; height:2.509ex;" alt="{\displaystyle R\subseteq A\times A,}"></span> dizemos 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> é uma <i>relação de ordem (parcial) estrita sobre <span class="mwe-math-element"><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></i> se satisfaz <a href="#1.c_Transitividade:">transitividade</a> e: </p> <div class="mw-heading mw-heading4"><h4 id="2.a_Irreflexividade">2.a Irreflexividade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=7" title="Editar secção: 2.a Irreflexividade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=7" title="Editar código-fonte da secção: 2.a Irreflexividade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x\in A\;\;\neg R(x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x\in A\;\;\neg R(x,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716a73f2d23886454dfd3152f1b72e46ea48500b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.894ex; height:2.843ex;" alt="{\displaystyle \forall \ x\in A\;\;\neg R(x,x)}"></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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span> (ou seja, nenhum elemento está relacionado consigo mesmo) </p><p>Se uma relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> satisfaz <a href="#1.c_Transitividade:">transitividade</a> e <a href="#2.a_Antirreflexividade:">irreflexividade</a>, pode ser demonstrado 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> também satisfaz: </p> <div class="mw-heading mw-heading4"><h4 id="2.b_Assimetria">2.b Assimetria</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=8" title="Editar secção: 2.b Assimetria" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=8" title="Editar código-fonte da secção: 2.b Assimetria"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y\in A\;\left(R(x,y)\Rightarrow \neg R(y,x)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\;\left(R(x,y)\Rightarrow \neg R(y,x)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260b87c0a28cd93ab2ffd3d78a26aa86c284a557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.201ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\;\left(R(x,y)\Rightarrow \neg R(y,x)\right).}"></span> </p><p>Analogamente, pode ser demonstrado que se uma relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> satisfaz <a href="#1.c_Transitividade:">transitividade</a> e <a href="#2.b_Assimetria:">assimetria</a>, então também satisfaz <a href="#2.a_Irreflexividade:">irreflexividade</a>, fornecendo uma definição alternativa de ordem parcial estrita, preferida por alguns autores. </p><p>Quando uma relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> é uma relação de ordem parcial estrita, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45f0bb548f5fe8470b7257809f044c967628eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.092ex; height:2.843ex;" alt="{\displaystyle R(x,y)}"></span> é escrito como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b187075203efe00ad723001c9944c7202c367c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.176ex;" alt="{\displaystyle x<y.}"></span> </p><p>Um conjunto que possui uma relação de ordem é chamado de <i>conjunto parcialmente ordenado</i>. </p><p>Em contextos não matemáticos é mais comum utilizar as ordens em sentido estrito. Por exemplo, dizemos que João é mais alto que Pedro no sentido que a altura de João é estritamente maior que a de Pedro. Também pode ser verificado que a relação "<span class="mwe-math-element"><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> é antepassado 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 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>" também é uma ordem estrita. </p> <div class="mw-heading mw-heading3"><h3 id="Definição_3:_Correspondência_entre_ordens_estritas_e_amplas"><span id="Defini.C3.A7.C3.A3o_3:_Correspond.C3.AAncia_entre_ordens_estritas_e_amplas"></span>Definição 3: Correspondência entre ordens estritas e amplas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=9" title="Editar secção: Definição 3: Correspondência entre ordens estritas e amplas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=9" title="Editar código-fonte da secção: Definição 3: Correspondência entre ordens estritas e amplas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dada uma ordem estrita ou uma ordem ampla, pode ser definida a outra ordem correspondente, segundo:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="3.a_Correspondência"><span id="3.a_Correspond.C3.AAncia"></span>3.a Correspondência</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=10" title="Editar secção: 3.a Correspondência" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=10" title="Editar código-fonte da secção: 3.a Correspondência"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 x\leqslant y\Leftrightarrow \left(x<y\lor x=y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⩽</mo> <mi>y</mi> <mo stretchy="false">⇔</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∨</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leqslant y\Leftrightarrow \left(x<y\lor x=y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22497af07d2c2cf7a067677e37f845f7ee7e1f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.757ex; height:2.843ex;" alt="{\displaystyle x\leqslant y\Leftrightarrow \left(x<y\lor x=y\right)}"></span> </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 x<y\Leftrightarrow \left(x\leqslant y\land x\neq y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">⇔</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>⩽</mo> <mi>y</mi> <mo>∧</mo> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y\Leftrightarrow \left(x\leqslant y\land x\neq y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2454b97d1b990b90c713ad8d160502eaeb62df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.757ex; height:2.843ex;" alt="{\displaystyle x<y\Leftrightarrow \left(x\leqslant y\land x\neq y\right)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Relações_de_ordem_linear_ou_total"><span id="Rela.C3.A7.C3.B5es_de_ordem_linear_ou_total"></span>Relações de ordem linear ou total</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=11" title="Editar secção: Relações de ordem linear ou total" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=11" title="Editar código-fonte da secção: Relações de ordem linear ou total"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dada um relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f035e033d7d2c784a07e01448f7605945dfd435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle R,}"></span> dizemos 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 x,y\in A,\;\;x\neq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in A,\;\;x\neq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5f83e2dd5eb2d55fe91b37ff5273436bdc842d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.011ex; height:2.676ex;" alt="{\displaystyle x,y\in A,\;\;x\neq y}"></span> são <i>incomparáveis</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\parallel y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∥</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\parallel y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4954c55407a080588c4d5f5e976330cbd18daa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.938ex; height:2.843ex;" alt="{\displaystyle x\parallel y}"></span> se e somente se <span class="mwe-math-element"><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 R(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg R(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f926de08d8e3bb1405bc001bb2b4fa4df27f9f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.643ex; height:2.843ex;" alt="{\displaystyle \neg R(x,y)}"></span> nem <span class="mwe-math-element"><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 R(y,x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg R(y,x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/609766e7b37a0d5d76948dc9484e8e9caa949b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.29ex; height:2.843ex;" alt="{\displaystyle \neg R(y,x).}"></span> Uma <i>relação de ordem linear ou total</i> não têm elementos incomparáveis. </p> <div class="mw-heading mw-heading3"><h3 id="Definição_4:_Totalidade_ou_linearidade"><span id="Defini.C3.A7.C3.A3o_4:_Totalidade_ou_linearidade"></span>Definição 4: Totalidade ou linearidade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=12" title="Editar secção: Definição 4: Totalidade ou linearidade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=12" title="Editar código-fonte da secção: Definição 4: Totalidade ou linearidade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> uma relação sobre <span class="mwe-math-element"><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> <mo>,</mo> </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/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> no caso de uma ordem ampla, a totalidade (linearidade) está dada por: </p> <div class="mw-heading mw-heading4"><h4 id="4.a_Totalidade_ou_linearidade_(para_ordens_amplas)"><span id="4.a_Totalidade_ou_linearidade_.28para_ordens_amplas.29"></span>4.a Totalidade ou linearidade (para ordens amplas)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=13" title="Editar secção: 4.a Totalidade ou linearidade (para ordens amplas)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=13" title="Editar código-fonte da secção: 4.a Totalidade ou linearidade (para ordens amplas)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y\in A\left(x\leqslant y\lor y\leqslant x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>⩽</mo> <mi>y</mi> <mo>∨</mo> <mi>y</mi> <mo>⩽</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\left(x\leqslant y\lor y\leqslant x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e10b911264f4d8728be904501f64b5f37c338f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.922ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\left(x\leqslant y\lor y\leqslant x\right)}"></span> Também denominado "dicotomia". </p><p>No caso das ordens estritas: </p> <div class="mw-heading mw-heading4"><h4 id="4.b_Totalidade_ou_linearidade_(para_ordens_estritas)"><span id="4.b_Totalidade_ou_linearidade_.28para_ordens_estritas.29"></span>4.b Totalidade ou linearidade (para ordens estritas)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=14" title="Editar secção: 4.b Totalidade ou linearidade (para ordens estritas)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=14" title="Editar código-fonte da secção: 4.b Totalidade ou linearidade (para ordens estritas)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y\in A\left(x\neq y\Rightarrow x<y\lor y<x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∨</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\left(x\neq y\Rightarrow x<y\lor y<x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d914878777b9ae566ef8cca064249c0eb753a665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.12ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\left(x\neq y\Rightarrow x<y\lor y<x\right)}"></span> </p><p>Também denominado "tricotomia", pois pode ser escrito equivalentemente: </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 \forall \ x,y\in A\left(x=y\lor x<y\lor y<x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>∨</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∨</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\left(x=y\lor x<y\lor y<x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe31e41bc7ff51936fa006771e6fe95bca7421c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.089ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\left(x=y\lor x<y\lor y<x\right)}"></span> </p><p>As ordens dos conjuntos numéricos, <a href="/wiki/N%C3%BAmero_natural" title="Número natural"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></a>, <a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span></a>, <a href="/wiki/N%C3%BAmero_racional" title="Número racional"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span></a>, <a href="/wiki/N%C3%BAmero_real" title="Número real"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></a> são lineares. Dado um conjunto <span class="mwe-math-element"><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> com dois ou mais elementos, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\wp }\left(A\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">℘</mi> </mrow> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\wp }\left(A\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd67e77a5396f3b179f9f3c0e9dfc98eb6b9d92f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.452ex; height:2.843ex;" alt="{\displaystyle {\wp }\left(A\right),}"></span> o <a href="/wiki/Conjunto_das_partes" class="mw-redirect" title="Conjunto das partes">conjunto das partes</a> 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 A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </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/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> não está linearmente ordenado por <a href="/wiki/Inclus%C3%A3o" class="mw-redirect" title="Inclusão">inclusão</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 \left(\subseteq \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mo>⊆</mo> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\subseteq \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca368d5e39fe18bcde38d83c9f35549a7f30cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.617ex; height:2.843ex;" alt="{\displaystyle \left(\subseteq \right)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Relações_de_ordem_densa"><span id="Rela.C3.A7.C3.B5es_de_ordem_densa"></span>Relações de ordem densa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=15" title="Editar secção: Relações de ordem densa" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=15" title="Editar código-fonte da secção: Relações de ordem densa"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A ideia intuitiva de densidade de uma ordem corresponde a conceber que entre dois elementos comparáveis existe uma quantidade infinita de elementos. </p> <div class="mw-heading mw-heading3"><h3 id="Definição_5:_Densidade"><span id="Defini.C3.A7.C3.A3o_5:_Densidade"></span>Definição 5: Densidade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=16" title="Editar secção: Definição 5: Densidade" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=16" title="Editar código-fonte da secção: Definição 5: Densidade"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uma relação de ordem estrita, parcial ou total, é denominada <i>densa</i> se entre dois elementos sempre existe um outro: </p> <div class="mw-heading mw-heading4"><h4 id="5_Densidade_(para_ordens_estritas)"><span id="5_Densidade_.28para_ordens_estritas.29"></span>5 Densidade (para ordens estritas)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=17" title="Editar secção: 5 Densidade (para ordens estritas)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=17" title="Editar código-fonte da secção: 5 Densidade (para ordens estritas)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \forall \ x,y\in A\ \left(x<y\Rightarrow \exists \ z\in S\left(x<z<y\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">∃</mi> <mtext> </mtext> <mi>z</mi> <mo>∈</mo> <mi>S</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo><</mo> <mi>z</mi> <mo><</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ x,y\in A\ \left(x<y\Rightarrow \exists \ z\in S\left(x<z<y\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c43206c59bfd6014136ab215ae456e30ed32529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.219ex; height:2.843ex;" alt="{\displaystyle \forall \ x,y\in A\ \left(x<y\Rightarrow \exists \ z\in S\left(x<z<y\right)\right)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Inversa_de_uma_ordem">Inversa de uma ordem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=18" title="Editar secção: Inversa de uma ordem" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=18" title="Editar código-fonte da secção: Inversa de uma ordem"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se uma relação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> é uma ordem estrita, então a <a href="/wiki/Rela%C3%A7%C3%A3o_(matem%C3%A1tica)#Relação_inversa" title="Relação (matemática)">relação inversa</a> 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 R:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470e63632cae3e604afbf95bd86a9d950ed7e4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.056ex; height:2.176ex;" alt="{\displaystyle R:}"></span> </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 R^{-1}=\left\{\left\langle y,x\right\rangle \mid \left\langle x,y\right\rangle \in R\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>⟨</mo> <mrow> <mi>y</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>⟩</mo> </mrow> <mo>∣</mo> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>⟩</mo> </mrow> <mo>∈</mo> <mi>R</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}=\left\{\left\langle y,x\right\rangle \mid \left\langle x,y\right\rangle \in R\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecdccd32b094b930cd584de1fea976e14ecdc0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.719ex; height:3.176ex;" alt="{\displaystyle R^{-1}=\left\{\left\langle y,x\right\rangle \mid \left\langle x,y\right\rangle \in R\right\}}"></span> </p><p>também é uma relação de ordem estrita. A inversa 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 <}"> <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/33737c89a17785dacc8638b4d66db3d5c8670de1" 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 <}"></span>" é geralmente escrita "<span class="mwe-math-element"><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/1b27b77ab4e3293ea9ce65cef60fea655c398423" 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 >}"></span>". De maneira análoga, para uma relação de ordem ampla "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leqslant }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⩽</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leqslant }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641be8b31b3bebbb9122010d73df358f4cf7203a" 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 \leqslant }"></span>" pode ser definida a sua inversa "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geqslant }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⩾</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \geqslant }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e7d18b196b7b9060634a9c96a324be85704341" 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 \geqslant }"></span>", que também é uma relação de ordem ampla. </p><p>Apesar dessa propriedade ser denominada às vezes de "dualidade", não é uma dualidade em sentido estrito, como a que possuem as <a href="/wiki/%C3%81lgebra_de_Boole" class="mw-redirect" title="Álgebra de Boole">álgebras de Boole</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Elementos_distinguidos_numa_ordem">Elementos distinguidos numa ordem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=19" title="Editar secção: Elementos distinguidos numa ordem" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=19" title="Editar código-fonte da secção: Elementos distinguidos numa ordem"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alguns elementos de um conjunto ordenado podem ser caraterizados usando a relação de ordem. Apesar das definições abaixo serem expressadas somente para ordens amplas, "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" 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 \leq }"></span>", ou estritas, "<span class="mwe-math-element"><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/33737c89a17785dacc8638b4d66db3d5c8670de1" 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 <}"></span>", definições correspondentes podem ser estabelecidas usando <a href="#Definição_3:_Correspondência_entre_ordens_estritas_e_amplas">Definição 3</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mínimo_e_máximo"><span id="M.C3.ADnimo_e_m.C3.A1ximo"></span>Mínimo e máximo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=20" title="Editar secção: Mínimo e máximo" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=20" title="Editar código-fonte da secção: Mínimo e máximo"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dada uma relação de ordem ampla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leqslant }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⩽</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leqslant }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641be8b31b3bebbb9122010d73df358f4cf7203a" 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 \leqslant }"></span> sobre um conjunto <span class="mwe-math-element"><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> <mo>,</mo> </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/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> um elemento <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> é denominado <i>mínimo</i> ou <i>primeiro elemento</i> se e somente se: </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 \forall \ b\in A\left(a\leqslant b\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>⩽</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ b\in A\left(a\leqslant b\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c7d36f733de7a9a52b2e5a386f891f41a3e922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.977ex; height:2.843ex;" alt="{\displaystyle \forall \ b\in A\left(a\leqslant b\right)}"></span> </p><p>De maneira simétrica, <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> é denominado <i>máximo</i> ou <i>último elemento</i> se e somente se: </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 \forall \ b\in A\left(a\geqslant b\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>⩾</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ b\in A\left(a\geqslant b\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776670b043f33ede2bfe48f4602fdade78d8bffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.977ex; height:2.843ex;" alt="{\displaystyle \forall \ b\in A\left(a\geqslant b\right)}"></span> </p><p>O conjunto <a href="/wiki/N%C3%BAmero_natural" title="Número natural"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></a> tem mínimo, mas não tem máximo. Os conjuntos <a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span></a>, <a href="/wiki/N%C3%BAmero_racional" title="Número racional"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span></a> e <a href="/wiki/N%C3%BAmero_real" title="Número real"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></a> não têm nem máximo, nem mínimo. O intervalo </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 \left[0,1\right]=\left\{x\in \mathbb {R} :\;\;0\leqslant x\leqslant 1\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[0,1\right]=\left\{x\in \mathbb {R} :\;\;0\leqslant x\leqslant 1\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997954359ab663fcc08aef0c84e0d32015d64772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.003ex; height:2.843ex;" alt="{\displaystyle \left[0,1\right]=\left\{x\in \mathbb {R} :\;\;0\leqslant x\leqslant 1\right\}}"></span> </p><p>tem mínimo 0 e máximo 1. Dado um conjunto <span class="mwe-math-element"><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> e considerando a ordem <a href="/wiki/Inclus%C3%A3o" class="mw-redirect" title="Inclusão">inclusão</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 \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" 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 \subseteq }"></span>, o conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp \left(A\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp \left(A\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/177757784594ff1c1a4f79704487fa5ebefe9599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.418ex; height:2.843ex;" alt="{\displaystyle \wp \left(A\right)}"></span> , <a href="/wiki/Conjunto_das_partes" class="mw-redirect" title="Conjunto das partes">das partes</a> 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 A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </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/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> tem mínimo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> é máximo <span class="mwe-math-element"><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> <mo>.</mo> </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/8a71bf21ad35b8fe05555041d54d1e17eeb0f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}"></span> Se um conjunto tem mínimo, então tem um único mínimo. O mesmo vale para o máximo. </p> <div class="mw-heading mw-heading3"><h3 id="Minimal_e_maximal">Minimal e maximal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=21" title="Editar secção: Minimal e maximal" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=21" title="Editar código-fonte da secção: Minimal e maximal"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dada uma relação de ordem estrita <span class="mwe-math-element"><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/33737c89a17785dacc8638b4d66db3d5c8670de1" 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 <}"></span> sobre um conjunto <span class="mwe-math-element"><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> <mo>,</mo> </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/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> um elemento <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> é denominado <i>minimal</i> quando não existe outro elemento que seja menor que ele: </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 \nexists \ x\in A,\;\;x<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∄</mi> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nexists \ x\in A,\;\;x<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb6c82f03e85a5a619a42f77b518933fdb291ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.19ex; width:15.96ex; height:2.676ex;" alt="{\displaystyle \nexists \ x\in A,\;\;x<a}"></span> </p><p>Analogamente, um elemento de um conjunto parcialmente ordenado é <i>maximal</i> quando não existe outro elemento que seja maior que ele: </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 \nexists \ x\in A,\;\;x>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∄</mi> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nexists \ x\in A,\;\;x>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35152e372adaa65e8e8ebff503b493473120a4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.19ex; width:15.96ex; height:2.676ex;" alt="{\displaystyle \nexists \ x\in A,\;\;x>a}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Cotas_inferior_(minorante)_e_superior_(majorante)"><span id="Cotas_inferior_.28minorante.29_e_superior_.28majorante.29"></span>Cotas inferior (minorante) e superior (majorante)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=22" title="Editar secção: Cotas inferior (minorante) e superior (majorante)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=22" title="Editar código-fonte da secção: Cotas inferior (minorante) e superior (majorante)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um elemento <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> é uma <i>cota inferior</i> ou <i><a href="/wiki/Majorante_(matem%C3%A1tica)#Definição" class="mw-redirect" title="Majorante (matemática)">minorante</a></i> de um subconjunto <span class="mwe-math-element"><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\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}"></span> se e somente se: </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 \forall \ b\in B\left(a\leqslant b\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>⩽</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ b\in B\left(a\leqslant b\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed0fda668b78047054eb07545860134f8ed499d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.998ex; height:2.843ex;" alt="{\displaystyle \forall \ b\in B\left(a\leqslant b\right)}"></span> </p><p>Um elemento <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span> é uma <i>cota superior</i> ou <i><a href="/wiki/Majorante_(matem%C3%A1tica)" class="mw-redirect" title="Majorante (matemática)">majorante</a></i> de um subconjunto <span class="mwe-math-element"><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\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}"></span> se e somente se: </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 \forall \ b\in B\left(a\geqslant b\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>⩾</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ b\in B\left(a\geqslant b\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee66bf3c18605269f690a6bcd6a70e1767c337b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.998ex; height:2.843ex;" alt="{\displaystyle \forall \ b\in B\left(a\geqslant b\right)}"></span> </p><p>Às vezes os elementos acima são denominados de <i>limite inferior</i> e <i>limite superior</i>, mas este conceito não deve ser confundido com o de <a href="/wiki/Limite_de_uma_sequ%C3%AAncia" title="Limite de uma sequência">limite de uma sequência</a>. </p><p>Se consideramos o intervalo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[0,1\right]\subseteq \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[0,1\right]\subseteq \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d220b1bf3b0a1adca73dfb2e46d330036018d7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.076ex; height:2.843ex;" alt="{\displaystyle \left[0,1\right]\subseteq \mathbb {R} ,}"></span> então qualquer <span class="mwe-math-element"><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\leqslant 0,x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⩽</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leqslant 0,x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456d3236bcaf221865603615d1cdb8f115ca1fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.473ex; height:2.509ex;" alt="{\displaystyle x\leqslant 0,x\in \mathbb {R} }"></span> é cota inferior do intervalo e qualquer <span class="mwe-math-element"><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\geqslant 1,x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⩾</mo> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geqslant 1,x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d401050ea3a1ce62a20234cc1886b0c7bdcf11ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.473ex; height:2.509ex;" alt="{\displaystyle x\geqslant 1,x\in \mathbb {R} }"></span> é cota superior. </p> <div class="mw-heading mw-heading2"><h2 id="Boa_ordem">Boa ordem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=23" title="Editar secção: Boa ordem" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=23" title="Editar código-fonte da secção: Boa ordem"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uma relação de ordem estrita <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> sobre um conjunto <span class="mwe-math-element"><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> é denominada uma <i>boa ordem</i> se e somente se todo subconjunto não vazio 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 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> tem primeiro elemento segundo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle R.}"></span> Em símbolos, uma relação "<span class="mwe-math-element"><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/33737c89a17785dacc8638b4d66db3d5c8670de1" 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 <}"></span>" sobre <span class="mwe-math-element"><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> é uma <i>boa ordem</i> se e somente se: </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/33737c89a17785dacc8638b4d66db3d5c8670de1" 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 <}"></span> é <a href="#2.a_Irreflexividade:">irreflexiva</a>, <a href="#1.c_Transitividade:">transitiva</a> e</li></ul> <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 \forall \ B\subseteq A\left(B\neq \varnothing \Rightarrow \left(\exists \ a\in B\;\;\forall \ b\in B\left(a\neq b\Rightarrow a<b\right)\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">⇒</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∃</mi> <mtext> </mtext> <mi>a</mi> <mo>∈</mo> <mi>B</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>≠</mo> <mi>b</mi> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo><</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \ B\subseteq A\left(B\neq \varnothing \Rightarrow \left(\exists \ a\in B\;\;\forall \ b\in B\left(a\neq b\Rightarrow a<b\right)\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9742a9ac278b533b43d871198507d4b5a4ce5a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.705ex; height:2.843ex;" alt="{\displaystyle \forall \ B\subseteq A\left(B\neq \varnothing \Rightarrow \left(\exists \ a\in B\;\;\forall \ b\in B\left(a\neq b\Rightarrow a<b\right)\right)\right)}"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup></li></ul> <p>Um conjunto com uma relação de boa ordem é denominado <i>bem ordenado</i>. Por exemplo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> é bem ordenado pela relação natural desse conjunto (ver <a href="/wiki/Princ%C3%ADpio_da_boa-ordena%C3%A7%C3%A3o" class="mw-redirect" title="Princípio da boa-ordenação">Princípio da boa-ordenação</a>), mas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"></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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> não são, segundo as suas ordens naturais. O conceito de boa ordem é importante para definir matematicamente os <a href="/wiki/N%C3%BAmero_ordinal" title="Número ordinal">números ordinais</a> em teoria dos conjuntos. </p><p>Uma boa ordem é sempre uma ordem linear, pois se para <span class="mwe-math-element"><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\in A,a\neq 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>a</mi> <mo>≠</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in A,a\neq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6125fd76410d10e7631295ad1721aab5023422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.205ex; height:2.676ex;" alt="{\displaystyle a,b\in A,a\neq b}"></span> consideramos o conjunto <span class="mwe-math-element"><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"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </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/630f3e36a24bd17ada8cad79ac18c91b859e21bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.233ex; height:2.843ex;" alt="{\displaystyle \{a,b\},}"></span> ele tem primeiro elemento, de modo que 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 a<b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> </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/46f9e3247a4373970cad2b3b37920af5d23ec7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a<b,}"></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 b<a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa0fb542f7b226476199bf09efc79193a37a32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.973ex; height:2.176ex;" alt="{\displaystyle b<a.}"></span> </p> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="#Birkhoff1948LatticeTheory">BIRKHOFF (1948), p. 1.</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="#DaveyPriestley2002IntroductiontoLatticesAndOrder">DAVEY PRIESTLEY (2002)</a>, p. 2.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation web"><a class="external text" href="https://pt.m.wikipedia.org/wiki/Rela%C3%A7%C3%A3o_bem-fundada">«Relação Bem-fundada»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.btitle=Rela%C3%A7%C3%A3o+Bem-fundada&rft.genre=unknown&rft_id=https%3A%2F%2Fpt.m.wikipedia.org%2Fwiki%2FRela%25C3%25A7%25C3%25A3o_bem-fundada&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=24" title="Editar secção: Bibliografia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=24" title="Editar código-fonte da secção: Bibliografia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="Birkhoff1948LatticeTheory" class="citation book">BIRKHOFF, Garrett (1948). <i>Lattice Theory</i> (em inglês). New York: American Mathematical Society</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.au=BIRKHOFF%2C+Garrett&rft.btitle=Lattice+Theory&rft.date=1948&rft.genre=book&rft.place=New+York&rft.pub=American+Mathematical+Society&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="DaveyPriestley2002IntroductiontoLatticesAndOrder" class="citation book">DAVEY, B.A.; PRIESTLEY, H.A (2002). <i>Introduction to Lattices and Order</i> (em inglês) 2nd. ed. Cambridge: Cambridge University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-0-521-78451-1" title="Especial:Fontes de livros/978-0-521-78451-1">978-0-521-78451-1</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.au=DAVEY%2C+B.A.&rft.btitle=Introduction+to+Lattices+and+Order&rft.date=2002&rft.edition=2nd.&rft.genre=book&rft.isbn=978-0-521-78451-1&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">A referência emprega parâmetros obsoletos <code style="color:inherit; border:inherit; padding:inherit;">|coautor=</code> (<a href="/wiki/Ajuda:Erros_nas_refer%C3%AAncias#deprecated_params" title="Ajuda:Erros nas referências">ajuda</a>)</span></li></ul> <ul><li><cite id="Fraisse2000theoryrelations" class="citation book">FRAÏSSÉ, Roland (2000). <i>Theory of Relations</i> (em inglês) 1rst. (revised) ed. Amsterdam: Elsevier. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/0-444-50542-3" title="Especial:Fontes de livros/0-444-50542-3">0-444-50542-3</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.au=FRA%C3%8FSS%C3%89%2C+Roland&rft.btitle=Theory+of+Relations&rft.date=2000&rft.edition=1rst.+%28revised%29&rft.genre=book&rft.isbn=0-444-50542-3&rft.place=Amsterdam&rft.pub=Elsevier&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="Roman2008LatticesOrderedSets" class="citation book">ROMAN, Steven (2008). <i>Lattices and Ordered Sets</i> (em inglês). New York: Springer. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/978-0-387-78900-2" title="Especial:Fontes de livros/978-0-387-78900-2">978-0-387-78900-2</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.au=ROMAN%2C+Steven&rft.btitle=Lattices+and+Ordered+Sets&rft.date=2008&rft.genre=book&rft.isbn=978-0-387-78900-2&rft.place=New+York&rft.pub=Springer&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="Rosenstein1982LinearOrderings" class="citation book">ROSENSTEIN, Joseph G (1982). <i>Linear Orderings</i> (em inglês) 2nd. ed. New York: Academic Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/0-12-597680-1" title="Especial:Fontes de livros/0-12-597680-1">0-12-597680-1</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARela%C3%A7%C3%A3o+de+ordem&rft.au=ROSENSTEIN%2C+Joseph+G&rft.btitle=Linear+Orderings&rft.date=1982&rft.edition=2nd.&rft.genre=book&rft.isbn=0-12-597680-1&rft.place=New+York&rft.pub=Academic+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&veaction=edit&section=25" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rela%C3%A7%C3%A3o_de_ordem&action=edit&section=25" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Rela%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Relação (matemática)">Relação (matemática)</a></li> <li><a href="/wiki/Topologia_da_ordem" title="Topologia da ordem">Topologia da ordem</a>: uma relação de ordem parcial gera uma <a href="/wiki/Espa%C3%A7o_topol%C3%B3gico" title="Espaço topológico">topologia</a>, que tem como <a href="/wiki/Base_(topologia)" title="Base (topologia)">base</a> os conjuntos do tipo {x | x < b}, {x | x > a} e {x | a < x < b}.</li> <li><a href="/wiki/Corpo_ordenado" title="Corpo ordenado">Corpo ordenado</a>: quando o conjunto ordenado tem uma estrutura algébrica de <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpo</a>, e a ordem e as operações algébricas são compatíveis.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Relação_de_ordem&oldid=64590320">https://pt.wikipedia.org/w/index.php?title=Relação_de_ordem&oldid=64590320</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:Teoria_da_ordem" title="Categoria:Teoria da ordem">Teoria da ordem</a></li><li><a href="/wiki/Categoria:%C3%81lgebra" title="Categoria:Álgebra">Álgebra</a></li><li><a href="/wiki/Categoria:Teoria_dos_conjuntos" title="Categoria:Teoria dos conjuntos">Teoria dos conjuntos</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!CS1_ingl%C3%AAs-fontes_em_l%C3%ADngua_(en)" title="Categoria:!CS1 inglês-fontes em língua (en)">!CS1 inglês-fontes em língua (en)</a></li><li><a href="/wiki/Categoria:!P%C3%A1ginas_que_usam_refer%C3%AAncias_com_par%C3%A2metros_obsoletas" title="Categoria:!Páginas que usam referências com parâmetros obsoletas">!Páginas que usam referências com parâmetros obsoletas</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"> Esta página foi editada pela última vez às 05h23min de 19 de outubro de 2022.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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