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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ezkutatu</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Nabigazioa </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Azala" title="Azala bisitatu [z]" accesskey="z"><span>Azala</span></a></li><li id="n-Txikipedia" class="mw-list-item"><a href="/wiki/Txikipedia:Azala"><span>Txikipedia</span></a></li><li id="n-Ikusgela" class="mw-list-item"><a href="/wiki/Atari:Hezkuntza/Ikusgela"><span>Ikusgela</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Txokoa" title="Proiektuaren inguruan, zer egin dezakezu, non aurkitu nahi duzuna"><span>Txokoa</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Berezi:AzkenAldaketak" title="Wikiko azken aldaketen zerrenda. [r]" accesskey="r"><span>Aldaketa berriak</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Berezi:Ausazkoa" title="Ausazko orrialde bat kargatu [x]" accesskey="x"><span>Ausazko orria</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Laguntza:Sarrera" title="Aurkitzeko lekua."><span>Laguntza</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Azala" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Entziklopedia askea" src="/static/images/mobile/copyright/wikipedia-tagline-eu.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Berezi:Bilatu" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Wikipedia(e)n bilatu [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Bilatu</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Wikipedia wikian bilatu" aria-label="Wikipedia wikian bilatu" autocapitalize="sentences" title="Wikipedia(e)n bilatu [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Berezi:Bilatu"> </div> <button class="cdx-button cdx-search-input__end-button">Bilatu</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Tresna pertsonalak"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Itxura"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Itxura" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Itxura</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_eu.wikipedia.org&uselang=eu" class=""><span>Dohaintza egin</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Berezi:KontuaSortu&returnto=Azpimultzo" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea." class=""><span>Sortu kontua</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Berezi:SaioaHasi&returnto=Azpimultzo" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o" class=""><span>Hasi saioa</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Aukera gehiago" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Tresna pertsonalak" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Tresna pertsonalak</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_eu.wikipedia.org&uselang=eu"><span>Dohaintza egin</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:KontuaSortu&returnto=Azpimultzo" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Sortu kontua</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:SaioaHasi&returnto=Azpimultzo" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Hasi saioa</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Izena eman gabeko erabiltzaileentzako orrialdeak <a href="/wiki/Laguntza:Sarrera" aria-label="Artikuluak aldatzeari buruz gehiago ikasi"><span>gehiago ikasi</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Berezi:NireEkarpenak" title="IP helbide honetatik egindako aldaketen zerrenda [y]" accesskey="y"><span>Ekarpenak</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Berezi:NireEztabaida" title="Zure IParen eztabaida [n]" accesskey="n"><span>Eztabaida</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Gunea"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Edukiak" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Edukiak</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ezkutatu</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">⇑ Gora</div> </a> </li> <li id="toc-Definizioa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definizioa</span> </div> </a> <button aria-controls="toc-Definizioa-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>Erakutsi/ezkutatu Definizioa azpiatal</span> </button> <ul id="toc-Definizioa-sublist" class="vector-toc-list"> <li id="toc-Azpimultzo_propioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Azpimultzo_propioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Azpimultzo propioa</span> </div> </a> <ul id="toc-Azpimultzo_propioa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ezaugarriak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ezaugarriak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ezaugarriak</span> </div> </a> <ul id="toc-Ezaugarriak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propietateak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propietateak"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Propietateak</span> </div> </a> <ul id="toc-Propietateak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adibideak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Adibideak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Adibideak</span> </div> </a> <ul id="toc-Adibideak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Kanpo estekak</span> </div> </a> <ul id="toc-Kanpo_estekak-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="Edukiak" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Eduki taularen ikusgarritasuna aldatu" > <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">Eduki taularen ikusgarritasuna aldatu</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">Azpimultzo</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="Joan beste hizkuntza batean idatzitako artikulu batera. 67 hizkuntzatan eskuragarri." > <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-67" 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">67 hizkuntza</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B3%E1%88%85%E1%89%B3%E1%8B%AD_%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ታህታይ ስብስብ – amharera" lang="am" hreflang="am" data-title="ታህታይ ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="amharera" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9" title="مجموعة جزئية – arabiera" lang="ar" hreflang="ar" data-title="مجموعة جزئية" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B4%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Падмноства – bielorrusiera" lang="be" hreflang="be" data-title="Падмноства" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B4%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Падмноства – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Падмноства" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – bulgariera" lang="bg" hreflang="bg" data-title="Подмножество" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%B8%E0%A7%87%E0%A6%9F" title="উপসেট – bengalera" lang="bn" hreflang="bn" data-title="উপসেট" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Subconjunt" title="Subconjunt – katalana" lang="ca" hreflang="ca" data-title="Subconjunt" data-language-autonym="Català" data-language-local-name="katalana" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%DB%8E%D8%B1%DA%A9%DB%86%D9%85%DB%95%DA%B5" title="ژێرکۆمەڵ – erdialdeko kurduera" lang="ckb" hreflang="ckb" data-title="ژێرکۆمەڵ" data-language-autonym="کوردی" data-language-local-name="erdialdeko kurduera" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Podmno%C5%BEina" title="Podmnožina – txekiera" lang="cs" hreflang="cs" data-title="Podmnožina" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%B9%D0%B9%D1%8B%D1%88" title="Аййыш – txuvaxera" lang="cv" hreflang="cv" data-title="Аййыш" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Is-set" title="Is-set – galesa" lang="cy" hreflang="cy" data-title="Is-set" data-language-autonym="Cymraeg" data-language-local-name="galesa" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Delm%C3%A6ngde" title="Delmængde – daniera" lang="da" hreflang="da" data-title="Delmængde" data-language-autonym="Dansk" data-language-local-name="daniera" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Teilmenge" title="Teilmenge – alemana" lang="de" hreflang="de" data-title="Teilmenge" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%BF%CF%83%CF%8D%CE%BD%CE%BF%CE%BB%CE%BF" title="Υποσύνολο – greziera" lang="el" hreflang="el" data-title="Υποσύνολο" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Subset" title="Subset – ingelesa" lang="en" hreflang="en" data-title="Subset" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Subaro" title="Subaro – esperantoa" lang="eo" hreflang="eo" data-title="Subaro" data-language-autonym="Esperanto" data-language-local-name="esperantoa" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Subconjunto" title="Subconjunto – gaztelania" lang="es" hreflang="es" data-title="Subconjunto" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Alamhulk" title="Alamhulk – estoniera" lang="et" hreflang="et" data-title="Alamhulk" data-language-autonym="Eesti" data-language-local-name="estoniera" 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%B2%DB%8C%D8%B1%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87" title="زیرمجموعه – persiera" lang="fa" hreflang="fa" data-title="زیرمجموعه" data-language-autonym="فارسی" data-language-local-name="persiera" 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/Osajoukko" title="Osajoukko – finlandiera" lang="fi" hreflang="fi" data-title="Osajoukko" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Alambhulk" title="Alambhulk – Võro" lang="vro" hreflang="vro" data-title="Alambhulk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Subset" title="Subset – fijiera" lang="fj" hreflang="fj" data-title="Subset" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijiera" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Inclusion_(math%C3%A9matiques)" title="Inclusion (mathématiques) – frantsesa" lang="fr" hreflang="fr" data-title="Inclusion (mathématiques)" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Subconxunto" title="Subconxunto – galiziera" lang="gl" hreflang="gl" data-title="Subconxunto" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%AA-%D7%A7%D7%91%D7%95%D7%A6%D7%94" title="תת-קבוצה – hebreera" lang="he" hreflang="he" data-title="תת-קבוצה" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%89%E0%A4%AA%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF" title="उपसमुच्चय – hindia" lang="hi" hreflang="hi" data-title="उपसमुच्चय" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Podskup" title="Podskup – kroaziera" lang="hr" hreflang="hr" data-title="Podskup" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%A9szhalmaz" title="Részhalmaz – hungariera" lang="hu" hreflang="hu" data-title="Részhalmaz" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%B6%D5%A9%D5%A1%D5%A2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ենթաբազմություն – armeniera" lang="hy" hreflang="hy" data-title="Ենթաբազմություն" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Subinsimul" title="Subinsimul – interlingua" lang="ia" hreflang="ia" data-title="Subinsimul" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Himpunan_bagian" title="Himpunan bagian – indonesiera" lang="id" hreflang="id" data-title="Himpunan bagian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Subensemblo" title="Subensemblo – idoa" lang="io" hreflang="io" data-title="Subensemblo" data-language-autonym="Ido" data-language-local-name="idoa" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutmengi" title="Hlutmengi – islandiera" lang="is" hreflang="is" data-title="Hlutmengi" data-language-autonym="Íslenska" data-language-local-name="islandiera" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Inclusione_(matematica)" title="Inclusione (matematica) – italiera" lang="it" hreflang="it" data-title="Inclusione (matematica)" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E9%9B%86%E5%90%88" title="部分集合 – japoniera" lang="ja" hreflang="ja" data-title="部分集合" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%EB%B6%84%EC%A7%91%ED%95%A9" title="부분집합 – koreera" lang="ko" hreflang="ko" data-title="부분집합" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Binkom" title="Binkom – kurduera" lang="ku" hreflang="ku" data-title="Binkom" data-language-autonym="Kurdî" data-language-local-name="kurduera" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Inclusion" title="Inclusion – Lombard" lang="lmo" hreflang="lmo" data-title="Inclusion" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Poaibis" title="Poaibis – lituaniera" lang="lt" hreflang="lt" data-title="Poaibis" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – mazedoniera" lang="mk" hreflang="mk" data-title="Подмножество" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D1%8D%D0%B4_%D0%BE%D0%BB%D0%BE%D0%BD%D0%BB%D0%BE%D0%B3" title="Дэд олонлог – mongoliera" lang="mn" hreflang="mn" data-title="Дэд олонлог" data-language-autonym="Монгол" data-language-local-name="mongoliera" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Subset" title="Subset – malaysiera" lang="ms" hreflang="ms" data-title="Subset" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Deelverzameling" title="Deelverzameling – nederlandera" lang="nl" hreflang="nl" data-title="Deelverzameling" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Delmengd" title="Delmengd – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Delmengd" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Delmengde" title="Delmengde – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Delmengde" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podzbi%C3%B3r" title="Podzbiór – poloniera" lang="pl" hreflang="pl" data-title="Podzbiór" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Subconjunto" title="Subconjunto – portugesa" lang="pt" hreflang="pt" data-title="Subconjunto" data-language-autonym="Português" data-language-local-name="portugesa" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Submul%C8%9Bime" title="Submulțime – errumaniera" lang="ro" hreflang="ro" data-title="Submulțime" data-language-autonym="Română" data-language-local-name="errumaniera" 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%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – errusiera" lang="ru" hreflang="ru" data-title="Подмножество" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Suttanzemi" title="Suttanzemi – siziliera" lang="scn" hreflang="scn" data-title="Suttanzemi" data-language-autonym="Sicilianu" data-language-local-name="siziliera" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Podskup" title="Podskup – serbokroaziera" lang="sh" hreflang="sh" data-title="Podskup" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Subset" title="Subset – Simple English" lang="en-simple" hreflang="en-simple" data-title="Subset" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Podmno%C5%BEina" title="Podmnožina – eslovakiera" lang="sk" hreflang="sk" data-title="Podmnožina" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Podmno%C5%BEica" title="Podmnožica – esloveniera" lang="sl" hreflang="sl" data-title="Podmnožica" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D1%81%D0%BA%D1%83%D0%BF" title="Подскуп – serbiera" lang="sr" hreflang="sr" data-title="Подскуп" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Delm%C3%A4ngd" title="Delmängd – suediera" lang="sv" hreflang="sv" data-title="Delmängd" data-language-autonym="Svenska" data-language-local-name="suediera" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%9F%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="உட்கணம் – tamilera" lang="ta" hreflang="ta" data-title="உட்கணம்" data-language-autonym="தமிழ்" data-language-local-name="tamilera" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%8B%E0%B8%95%E0%B8%A2%E0%B9%88%E0%B8%AD%E0%B8%A2" title="เซตย่อย – thailandiera" lang="th" hreflang="th" data-title="เซตย่อย" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Subpangkat" title="Subpangkat – tagaloa" lang="tl" hreflang="tl" data-title="Subpangkat" data-language-autonym="Tagalog" data-language-local-name="tagaloa" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alt_k%C3%BCme" title="Alt küme – turkiera" lang="tr" hreflang="tr" data-title="Alt küme" data-language-autonym="Türkçe" data-language-local-name="turkiera" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Підмножина – ukrainera" lang="uk" hreflang="uk" data-title="Підмножина" data-language-autonym="Українська" data-language-local-name="ukrainera" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BA%ADp_h%E1%BB%A3p_con" title="Tập hợp con – vietnamera" lang="vi" hreflang="vi" data-title="Tập hợp con" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/D%C3%AAelverzoamelienge" title="Dêelverzoamelienge – West Flemish" lang="vls" hreflang="vls" data-title="Dêelverzoamelienge" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – wu txinera" lang="wuu" hreflang="wuu" data-title="子集" data-language-autonym="吴语" data-language-local-name="wu txinera" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – txinera" lang="zh" hreflang="zh" data-title="子集" data-language-autonym="中文" data-language-local-name="txinera" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="子集" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – kantonera" lang="yue" hreflang="yue" data-title="子集" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177646#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Izen-tarteak"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Azpimultzo" title="Eduki orrialdea ikusi [c]" accesskey="c"><span>Artikulua</span></a></li><li id="ca-talk" class="new vector-tab-noicon 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bertsioak. 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vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tresnak</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ezkutatu</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Ekintzak </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected 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href="/wiki/Berezi:ZerkLotzenDuHona/Azpimultzo" title="Orri honetaranzko esteka duten wiki orri guztien zerrenda [j]" accesskey="j"><span>Honanzko esteka duten orriak</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Berezi:RecentChangesLinked/Azpimultzo" rel="nofollow" title="Orri honetatik esteka duten orrietako azken aldaketak [k]" accesskey="k"><span>Lotutako orrietako aldaketak</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=eu" title="Irudiak edo media fitxategiak igo [u]" accesskey="u"><span>Fitxategia igo</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Berezi:OrrialdeBereziak" title="Orri berezi guztien zerrenda [q]" accesskey="q"><span>Orri bereziak</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Azpimultzo&oldid=9319881" title="Orriaren bertsio honetaranzko esteka iraunkorra"><span>Lotura iraunkorra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Azpimultzo&action=info" title="Orrialde honi buruzko informazio gehiago"><span>Orri honen datuak</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Berezi:CiteThisPage&page=Azpimultzo&id=9319881&wpFormIdentifier=titleform" title="Orri honen aipua egiteko moduari buruzko informazioa"><span>Artikulu hau aipatu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Berezi:UrlShortener&url=https%3A%2F%2Feu.wikipedia.org%2Fwiki%2FAzpimultzo"><span>URL laburra lortu</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Berezi:QrCode&url=https%3A%2F%2Feu.wikipedia.org%2Fwiki%2FAzpimultzo"><span>QR kodea jaitsi</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Inprimatu/esportatu </div> <div 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href="https://commons.wikimedia.org/wiki/Category:Subsets" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177646" title="Datuen biltegi elementu batera lotuta [g]" accesskey="g"><span>Wikidata itema</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Itxura"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Itxura</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ezkutatu</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Wikipedia, Entziklopedia askea</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="eu" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Venn_A_subset_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/150px-Venn_A_subset_B.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/225px-Venn_A_subset_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/300px-Venn_A_subset_B.svg.png 2x" data-file-width="155" data-file-height="155" /></a><figcaption><a href="/wiki/Eulerren_diagrama" class="mw-redirect" title="Eulerren diagrama">Eulerren diagrama</a> honen bidez<br /> erakusten da <i>A</i> <i>B</i>-ren azpimultzo propio bat dela eta alderantziz <i>B</i> <i>A</i>-ren gainmultzo propio bat</figcaption></figure> <p><a href="/wiki/Matematika" title="Matematika">Matematikan</a>, bereziki <a href="/wiki/Multzo-teoria" title="Multzo-teoria">multzo-teorian</a>, <b>azpimultzoa</b> <a href="/wiki/Multzo" title="Multzo">multzo</a> bateko zenbait elementuz osatutako edozein multzoa da. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizioa">Definizioa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=1" title="Aldatu atal hau: «Definizioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=1" title="Edit section's source code: Definizioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: #FFFFFF; color: #000000"> <p>Izan bitez <i>A</i> eta <i>B</i> bi <a href="/wiki/Multzo" title="Multzo">multzo</a> non <i>A</i>-ren elementu bakoitza <i>B</i>-ren elementua ere den. Orduan: </p> <ul><li><i>A</i> <i>B</i>-ren <b>azpimultzoa</b> da, eta <i>A</i> ⊆ <i>B</i> adierazten da</li> <li><i>B</i> <i>A</i>-ren <b>gainmultzoa</b> da, eta <i>B</i> ⊇ <i>A</i> adierazten da</li></ul> </blockquote> <p>Azpimultzoen arteko diferentzia, multzo baten barnean eta bestean ez dauden elementuek osatzen dute. </p> <div class="mw-heading mw-heading3"><h3 id="Azpimultzo_propioa">Azpimultzo propioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=2" title="Aldatu atal hau: «Azpimultzo propioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=2" title="Edit section's source code: Azpimultzo propioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jatorrizko multzoaren osaera bera ez duen azpimultzoa. </p> <blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: #FFFFFF; color: #000000"> <p>Izan bedi <i>A</i> <i>B</i>-ren azpimultzo bat ezen <i>A</i> ≠ <i>B</i> baita. Orduan esaten da <i>A</i> <i>B</i>-ren <b>azpimultzo propio</b> bat dela, eta <i>A</i> ⊊ <i>B</i> adierazten da. <br /> (Era berean, esaten da <i>B</i> <i>A</i>-ren <b>gainmultzo propio</b> bat dela, <i>B</i> ⊋ <i>A</i>) </p> </blockquote> <p><i>A</i> ⊂ <i>B</i> eta <i>B</i> ⊃ <i>A</i> notazioak ere erabiltzen dira, baina haiek azpimultzoa adieraz dezakete, <i>A</i> ⊆ <i>B</i> eta <i>B</i> ⊇ <i>A</i>; edo azpimultzo propio, <i>A</i> ⊊ <i>B</i> eta <i>B</i> ⊋ <i>A</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Ezaugarriak">Ezaugarriak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=3" title="Aldatu atal hau: «Ezaugarriak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=3" title="Edit section's source code: Ezaugarriak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A multzoa B multzoaren <b>azpimultzoa</b> da baldin eta soilik baldin haien ebakidura A bada.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\iff A\cap B=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\iff A\cap B=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc7b1e82c35d16ca5270282324a3e7a44ab90c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.434ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\iff A\cap B=A}"></span></dd></dl> <ul><li>A multzoa B multzoaren <b>azpimultzoa</b> da baldin eta soilik baldin haien bildura B bada.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\iff A\cup B=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\iff A\cup B=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43129c5c866a6f8962450cf85b8ee06718427b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.455ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\iff A\cup B=B}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Propietateak">Propietateak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=4" title="Aldatu atal hau: «Propietateak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=4" title="Edit section's source code: Propietateak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Azpimultzo hutsa ∅ bezala adierazten da, multzo guztien azpimultzoa da. Izan ere, « ∅-k ez du elementurik An dagoena» eta «∅-ko elementu guztiak Akoak dira» esanahi berdina dute. Hau edozein An gertatzen da, ∅k ez baitu elementurik.</li> <li>A multzoko elementuak B multzoan badaude, eta Bko elementu guztiak C multzoan badaude, Ako elementuak C multzoaren barnean daude. Hau da, A Bren azpimultzoa bada, eta B Cren azpimultzoa bada, orduan A Cren azpimultzoa da.</li> <li>Bi multzo elkarren azpimultzo badira, hau da, Ako elementu guztiak Bn badaude eta Bko elementu guztiak An badaude, bi multzoek elementu berdinak dituzte. Beste modu batera esanda, A Bren azpimultzoa bada, eta B Aren azpimultzoa bada, orduan A=B</li></ul> <div class="mw-heading mw-heading2"><h2 id="Adibideak">Adibideak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=5" title="Aldatu atal hau: «Adibideak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=5" title="Edit section's source code: Adibideak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>{1,2,3} multzoa {-563,1,2,3,68} multzoaren azpimultzo propio bat da.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}\subset \{-563,1,2,3,68\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>563</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>68</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}\subset \{-563,1,2,3,68\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877089ccf779ac67669128bc97dfe1bf65c62a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.547ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}\subset \{-563,1,2,3,68\}}"></span></dd></dl> <ul><li>[0;1] <a href="/wiki/Tarte_(argipena)" class="mw-disambig" title="Tarte (argipena)">tartea</a> <a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">zenbaki errealen</a> multzoko azpimultzo propio bat da.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0;1]\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>;</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0;1]\subset R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7673bcd7c828519df8b4f791b00f52449a3846d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.515ex; height:2.843ex;" alt="{\displaystyle [0;1]\subset R}"></span></dd></dl> <ul><li>{46,189,1264} multzoa beraren azpimultzo bat da, eta <a href="/wiki/Zenbaki_arrunt" title="Zenbaki arrunt">zenbaki arrunten</a> multzoko azpimultzo propio bat da.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{46,189,1264\}\subseteq \{46,189,1264\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>46</mn> <mo>,</mo> <mn>189</mn> <mo>,</mo> <mn>1264</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>46</mn> <mo>,</mo> <mn>189</mn> <mo>,</mo> <mn>1264</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{46,189,1264\}\subseteq \{46,189,1264\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a694874e019820bfb433d604b6536b076b95c584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.808ex; height:2.843ex;" alt="{\displaystyle \{46,189,1264\}\subseteq \{46,189,1264\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{46,189,1264\}\subset N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>46</mn> <mo>,</mo> <mn>189</mn> <mo>,</mo> <mn>1264</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{46,189,1264\}\subset N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a06853f4de66504e126c1c3daf0bc876660150f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.017ex; height:2.843ex;" alt="{\displaystyle \{46,189,1264\}\subset N}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azpimultzo&veaction=edit&section=6" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Azpimultzo&action=edit&section=6" title="Edit section's source code: Kanpo estekak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="clear:both;"></div><style data-mw-deduplicate="TemplateStyles:r7786466">.mw-parser-output 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