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class="vector-dropdown-label-text">Pangunahing pagpipilian</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Pangunahing pagpipilian</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">ilipat sa gilid</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">itago</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Maglibot </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Unang_Pahina" title="Puntahan ang unang pahina [z]" accesskey="z"><span>Unang pahina</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Tingnan_ang_buod" title="Mga gabay sa pagbabasa ng laktaw-laktaw sa Wikipedia"><span>Mga nilalaman</span></a></li><li id="n-Napiling-nilalaman" class="mw-list-item"><a href="/wiki/Wikipedia:Mga_napiling_artikulo"><span>Napiling nilalaman</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Natatangi:Alin_man" title="Magpalabas ng kahit-anong artikulo [x]" accesskey="x"><span>Alinmang artikulo</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Patungkol"><span>Patungkol sa Wikipedia</span></a></li><li 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[r]" accesskey="r"><span>Mga huling binago</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Wikipedia:Makipag-ugnayan_sa_amin" title="Paano makokontak ang Wikipedia"><span>Makipag-ugnayan</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Wikipedia:Tulong" title="Ang lugar para maghanap ng tulong sa Wikipedia."><span>Tulong</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Unang_Pahina" 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="Ang Malayang Ensiklopedya" src="/static/images/mobile/copyright/wikipedia-tagline-tl.svg" width="119" height="13" style="width: 7.4375em; 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/Natatangi:Maghanap" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Maghanap sa Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Hanapin</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="Maghanap sa Wikipedia" aria-label="Maghanap sa Wikipedia" autocapitalize="sentences" title="Maghanap sa Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Natatangi:Maghanap"> </div> <button class="cdx-button cdx-search-input__end-button">Hanapin</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Mga pansariling kagamitan"> <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="Itsura"> <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="Itsura" > <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">Itsura</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_tl.wikipedia.org&uselang=tl" class=""><span>Donasyon</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=Natatangi:Likhain_ang_kuwenta&returnto=Pangkat+%28matematika%29" title="Mas maganda kung gagawa ka ng account at magla-login; gayunpaman, hindi ka pinipilit gawin ito" class=""><span>Gumawa ng account</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=Natatangi:Paglagda_ng_tagagamit&returnto=Pangkat+%28matematika%29" title="Mas maganda kung magla-login ka; gayunpaman, hindi ka pinipilit gawin ito [o]" accesskey="o" class=""><span>Mag-login</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="Marami pang pagpipilian" > <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="Mga pansariling kagamitan" > <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">Mga pansariling kagamitan</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal" 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_tl.wikipedia.org&uselang=tl"><span>Donasyon</span></a></li><li id="pt-contribute" class="mw-list-item"><a href="/wiki/Natatangi:Contribute"><span class="vector-icon mw-ui-icon-edit mw-ui-icon-wikimedia-edit"></span> <span>Mag-ambag</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Natatangi:Likhain_ang_kuwenta&returnto=Pangkat+%28matematika%29" title="Mas maganda kung gagawa ka ng account at magla-login; gayunpaman, hindi ka pinipilit gawin ito"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Gumawa ng account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Natatangi:Paglagda_ng_tagagamit&returnto=Pangkat+%28matematika%29" title="Mas maganda kung magla-login ka; gayunpaman, hindi ka pinipilit gawin ito [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Mag-login</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"> Mga pahina para sa naka-logout na mga patnugot o editor <a href="/wiki/Wikipedia:Maligayang_pagdating!" aria-label="Alamin pa ang tungkol sa pagpapatnugot"><span>alamin pa</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Natatangi:Usapan_ko" title="Usapang tungkol sa mga pagbabagong ginawa sa ip address na ito [n]" accesskey="n"><span>Usapan</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="Sayt"> <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="Mga nilalaman" 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">Mga nilalaman</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ilipat sa gilid</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">itago</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">Simula</div> </a> </li> <li id="toc-Kasaysayan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kasaysayan"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Kasaysayan</span> </div> </a> <button aria-controls="toc-Kasaysayan-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>Ipakita/Itago ang subseksyon na Kasaysayan</span> </button> <ul id="toc-Kasaysayan-sublist" class="vector-toc-list"> <li id="toc-Intuitibong_teorya_ng_pangkat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intuitibong_teorya_ng_pangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Intuitibong teorya ng pangkat</span> </div> </a> <ul id="toc-Intuitibong_teorya_ng_pangkat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aksiyomatikong_teorya_ng_pangkat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aksiyomatikong_teorya_ng_pangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Aksiyomatikong teorya ng pangkat</span> </div> </a> <ul id="toc-Aksiyomatikong_teorya_ng_pangkat-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kahulugan_at_notasyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kahulugan_at_notasyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Kahulugan at notasyon</span> </div> </a> <button aria-controls="toc-Kahulugan_at_notasyon-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>Ipakita/Itago ang subseksyon na Kahulugan at notasyon</span> </button> <ul id="toc-Kahulugan_at_notasyon-sublist" class="vector-toc-list"> <li id="toc-Notasyong_pantala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notasyong_pantala"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Notasyong pantala</span> </div> </a> <ul id="toc-Notasyong_pantala-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semantikong_kahulugan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semantikong_kahulugan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Semantikong kahulugan</span> </div> </a> <ul id="toc-Semantikong_kahulugan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notasyong_pampangkat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notasyong_pampangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Notasyong pampangkat</span> </div> </a> <ul id="toc-Notasyong_pampangkat-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pagkamiyembro" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pagkamiyembro"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Pagkamiyembro</span> </div> </a> <ul id="toc-Pagkamiyembro-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basyong_pangkat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basyong_pangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Basyong pangkat</span> </div> </a> <ul id="toc-Basyong_pangkat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_subpangkat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_subpangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mga subpangkat</span> </div> </a> <ul id="toc-Mga_subpangkat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_natatanging_pangkat_ng_mga_bilang_sa_matematika" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_natatanging_pangkat_ng_mga_bilang_sa_matematika"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Mga natatanging pangkat ng mga bilang sa matematika</span> </div> </a> <ul id="toc-Mga_natatanging_pangkat_ng_mga_bilang_sa_matematika-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_punsiyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_punsiyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Mga punsiyon</span> </div> </a> <ul id="toc-Mga_punsiyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kardinalidad" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kardinalidad"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Kardinalidad</span> </div> </a> <ul id="toc-Kardinalidad-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_kapangyarihang_pangkat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_kapangyarihang_pangkat"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Mga kapangyarihang pangkat</span> </div> </a> <ul id="toc-Mga_kapangyarihang_pangkat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_partisyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_partisyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Mga partisyon</span> </div> </a> <ul id="toc-Mga_partisyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_basikong_operasyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_basikong_operasyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Mga basikong operasyon</span> </div> </a> <ul id="toc-Mga_basikong_operasyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_aplikasyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_aplikasyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Mga aplikasyon</span> </div> </a> <ul id="toc-Mga_aplikasyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prinsipyo_ng_ingklusyon_at_eksklusyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Prinsipyo_ng_ingklusyon_at_eksklusyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Prinsipyo ng ingklusyon at eksklusyon</span> </div> </a> <ul id="toc-Prinsipyo_ng_ingklusyon_at_eksklusyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_sanggunian" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_sanggunian"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Mga sanggunian</span> </div> </a> <ul id="toc-Mga_sanggunian-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="Mga nilalaman" 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="Itago/Ipakita ang talaan ng mga nilalaman" > <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">Itago/Ipakita ang talaan ng mga nilalaman</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">Pangkat (matematika)</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="Pumunta sa isang artikulo na nakasulat sa ibang wika. Mayroon ito sa 102 (na) wika" > <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-102" 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">102 wika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Menge_(Mathematik)" title="Menge (Mathematik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Menge (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ስብስብ – Amharic" lang="am" hreflang="am" data-title="ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" 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%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مجموعة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="مجموعة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Conxuntu" title="Conxuntu – Asturian" lang="ast" hreflang="ast" data-title="Conxuntu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxluqlar" title="Çoxluqlar – Azerbaijani" lang="az" hreflang="az" data-title="Çoxluqlar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA" title="Күмәклек – Bashkir" lang="ba" hreflang="ba" data-title="Күмәклек" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Мноства – Belarusian" lang="be" hreflang="be" data-title="Мноства" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%9C%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Мноства – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мноства" 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%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Bulgarian" lang="bg" hreflang="bg" data-title="Множество" data-language-autonym="Български" data-language-local-name="Bulgarian" 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%B8%E0%A7%87%E0%A6%9F" title="সেট – Bangla" lang="bn" hreflang="bn" data-title="সেট" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Skup_(matematika)" title="Skup (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Skup (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjunt" title="Conjunt – Catalan" lang="ca" hreflang="ca" data-title="Conjunt" data-language-autonym="Català" data-language-local-name="Catalan" 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%A9%DB%86%D9%85%DB%95%DA%B5%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="کۆمەڵە (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کۆمەڵە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" 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/Mno%C5%BEina" title="Množina – Czech" lang="cs" hreflang="cs" data-title="Množina" data-language-autonym="Čeština" data-language-local-name="Czech" 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%99%D1%8B%D1%88" title="Йыш – Chuvash" lang="cv" hreflang="cv" data-title="Йыш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" 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/Set_(mathemateg)" title="Set (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Set (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/M%C3%A6ngde" title="Mængde – Danish" lang="da" hreflang="da" data-title="Mængde" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Menge_(Mathematik)" title="Menge (Mathematik) – German" lang="de" hreflang="de" data-title="Menge (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" 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%A3%CF%8D%CE%BD%CE%BF%CE%BB%CE%BF" title="Σύνολο – Greek" lang="el" hreflang="el" data-title="Σύνολο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" 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/Set_(mathematics)" title="Set (mathematics) – English" lang="en" hreflang="en" data-title="Set (mathematics)" data-language-autonym="English" data-language-local-name="English" 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/Aro_(matematiko)" title="Aro (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Aro (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" 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/Conjunto" title="Conjunto – Spanish" lang="es" hreflang="es" data-title="Conjunto" data-language-autonym="Español" data-language-local-name="Spanish" 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/Hulk" title="Hulk – Estonian" lang="et" hreflang="et" data-title="Hulk" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Multzo" title="Multzo – Basque" lang="eu" hreflang="eu" data-title="Multzo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="مجموعه (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="مجموعه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Joukko" title="Joukko – Finnish" lang="fi" hreflang="fi" data-title="Joukko" data-language-autonym="Suomi" data-language-local-name="Finnish" 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/Hulk" title="Hulk – Võro" lang="vro" hreflang="vro" data-title="Hulk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ensemble" title="Ensemble – French" lang="fr" hreflang="fr" data-title="Ensemble" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Insiemi" title="Insiemi – Friulian" lang="fur" hreflang="fur" data-title="Insiemi" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tacar" title="Tacar – Irish" lang="ga" hreflang="ga" data-title="Tacar" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%9B%86%E5%90%88" title="集合 – Gan" lang="gan" hreflang="gan" data-title="集合" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Ansanm" title="Ansanm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Ansanm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Seata" title="Seata – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Seata" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Conxunto" title="Conxunto – Galician" lang="gl" hreflang="gl" data-title="Conxunto" data-language-autonym="Galego" data-language-local-name="Galician" 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%A7%D7%91%D7%95%D7%A6%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קבוצה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="קבוצה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" 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%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="समुच्चय (गणित) – Hindi" lang="hi" hreflang="hi" data-title="समुच्चय (गणित)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Skup" title="Skup – Croatian" lang="hr" hreflang="hr" data-title="Skup" data-language-autonym="Hrvatski" data-language-local-name="Croatian" 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/Halmaz_(matematika)" title="Halmaz (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Halmaz (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բազմություն – Armenian" lang="hy" hreflang="hy" data-title="Բազմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" 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/Insimul" title="Insimul – Interlingua" lang="ia" hreflang="ia" data-title="Insimul" 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_(matematika)" title="Himpunan (matematika) – Indonesian" lang="id" hreflang="id" data-title="Himpunan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ensemblo" title="Ensemblo – Ido" lang="io" hreflang="io" data-title="Ensemblo" data-language-autonym="Ido" data-language-local-name="Ido" 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/Mengi" title="Mengi – Icelandic" lang="is" hreflang="is" data-title="Mengi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Insieme" title="Insieme – Italian" lang="it" hreflang="it" data-title="Insieme" data-language-autonym="Italiano" data-language-local-name="Italian" 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%9B%86%E5%90%88" title="集合 – Japanese" lang="ja" hreflang="ja" data-title="集合" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Set_(matimatix)" title="Set (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Set (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%9A%E1%83%94" title="სიმრავლე – Georgian" lang="ka" hreflang="ka" data-title="სიმრავლე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D1%96%D0%BA" title="Бөлік – Kazakh" lang="kk" hreflang="kk" data-title="Бөлік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3_(%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4)" title="ಗಣ (ಗಣಿತ) – Kannada" lang="kn" hreflang="kn" data-title="ಗಣ (ಗಣಿತ)" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%91%ED%95%A9" title="집합 – Korean" lang="ko" hreflang="ko" data-title="집합" data-language-autonym="한국어" data-language-local-name="Korean" 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/Kom" title="Kom – Kurdish" lang="ku" hreflang="ku" data-title="Kom" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Copia" title="Copia – Latin" lang="la" hreflang="la" data-title="Copia" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Insemma" title="Insemma – Lombard" lang="lmo" hreflang="lmo" data-title="Insemma" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/El%C9%94ng%C9%94%CC%81t%C9%9B%CC%82_lisang%C3%A1" title="Elɔngɔ́tɛ̂ lisangá – Lingala" lang="ln" hreflang="ln" data-title="Elɔngɔ́tɛ̂ lisangá" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Aib%C4%97" title="Aibė – Lithuanian" lang="lt" hreflang="lt" data-title="Aibė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kopa" title="Kopa – Latvian" lang="lv" hreflang="lv" data-title="Kopa" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Macedonian" lang="mk" hreflang="mk" data-title="Множество" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%82_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="ഗണം (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="ഗണം (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9E%D0%BB%D0%BE%D0%BD%D0%BB%D0%BE%D0%B3" title="Олонлог – Mongolian" lang="mn" hreflang="mn" data-title="Олонлог" data-language-autonym="Монгол" data-language-local-name="Mongolian" 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/Set" title="Set – Malay" lang="ms" hreflang="ms" data-title="Set" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%85%E1%80%AF" title="အစု – Burmese" lang="my" hreflang="my" data-title="အစု" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D1%81%D1%81%D0%B0%D0%B5%D0%B2%D0%BA%D1%81" title="Вейссаевкс – Erzya" lang="myv" hreflang="myv" data-title="Вейссаевкс" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Koppel_(Mathematik)" title="Koppel (Mathematik) – Low German" lang="nds" hreflang="nds" data-title="Koppel (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verzameling_(wiskunde)" title="Verzameling (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Verzameling (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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/Mengd" title="Mengd – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Mengd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mengde" title="Mengde – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Mengde" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Ensemble" title="Ensemble – Novial" lang="nov" hreflang="nov" data-title="Ensemble" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Ensemble" title="Ensemble – Occitan" lang="oc" hreflang="oc" data-title="Ensemble" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%88%E0%A9%B1%E0%A8%9F_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਸੈੱਟ (ਗਣਿਤ) – Punjabi" lang="pa" hreflang="pa" data-title="ਸੈੱਟ (ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zbi%C3%B3r" title="Zbiór – Polish" lang="pl" hreflang="pl" data-title="Zbiór" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Ansem" title="Ansem – Piedmontese" lang="pms" hreflang="pms" data-title="Ansem" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Conjunto" title="Conjunto – Portuguese" lang="pt" hreflang="pt" data-title="Conjunto" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Tantachisqa" title="Tantachisqa – Quechua" lang="qu" hreflang="qu" data-title="Tantachisqa" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mul%C8%9Bime" title="Mulțime – Romanian" lang="ro" hreflang="ro" data-title="Mulțime" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Russian" lang="ru" hreflang="ru" data-title="Множество" data-language-autonym="Русский" data-language-local-name="Russian" 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/Nzemi" title="Nzemi – Sicilian" lang="scn" hreflang="scn" data-title="Nzemi" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" 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/Skup" title="Skup – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Skup" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Set" title="Set – Simple English" lang="en-simple" hreflang="en-simple" data-title="Set" 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/Mno%C5%BEina" title="Množina – Slovak" lang="sk" hreflang="sk" data-title="Množina" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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/Mno%C5%BEica" title="Množica – Slovenian" lang="sl" hreflang="sl" data-title="Množica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Qaybta_(xisaab)" title="Qaybta (xisaab) – Somali" lang="so" hreflang="so" data-title="Qaybta (xisaab)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Bashk%C3%ABsit%C3%AB" title="Bashkësitë – Albanian" lang="sq" hreflang="sq" data-title="Bashkësitë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BA%D1%83%D0%BF" title="Скуп – Serbian" lang="sr" hreflang="sr" data-title="Скуп" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A4ngd" title="Mängd – Swedish" lang="sv" hreflang="sv" data-title="Mängd" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Mynga_(matymatyka)" title="Mynga (matymatyka) – Silesian" lang="szl" hreflang="szl" data-title="Mynga (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="கணம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="கணம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%AE%E0%B0%BF%E0%B0%A4%E0%B1%81%E0%B0%B2%E0%B1%81" title="సమితులు – Telugu" lang="te" hreflang="te" data-title="సమితులు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</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%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="เซต (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="เซต (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCme" title="Küme – Turkish" lang="tr" hreflang="tr" data-title="Küme" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Множина – Ukrainian" lang="uk" hreflang="uk" data-title="Множина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D9%85_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="طاقم (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="طاقم (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/To%CA%BBplam_(matematika)" title="Toʻplam (matematika) – Uzbek" lang="uz" hreflang="uz" data-title="Toʻplam (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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_(to%C3%A1n_h%E1%BB%8Dc)" title="Tập hợp (toán học) – Vietnamese" lang="vi" hreflang="vi" data-title="Tập hợp (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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/Verzoamelienge" title="Verzoamelienge – West Flemish" lang="vls" hreflang="vls" data-title="Verzoamelienge" 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/%E9%9B%86%E5%90%88%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="集合（数学） – Wu" lang="wuu" hreflang="wuu" data-title="集合（数学）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9E%D0%BB%D0%BD" title="Олн – Kalmyk" lang="xal" hreflang="xal" data-title="Олн" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%98%E1%83%90%E1%83%A0%E1%83%94" title="მიარე – Mingrelian" lang="xmf" hreflang="xmf" data-title="მიარე" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A2%D7%96%D7%A2%D7%9E%D7%9C_(%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7)" title="געזעמל (מאטעמאטיק) – Yiddish" lang="yi" hreflang="yi" data-title="געזעמל (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学) – Chinese" lang="zh" hreflang="zh" data-title="集合 (数学)" data-language-autonym="中文" data-language-local-name="Chinese" 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/%E9%9B%86" title="集 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="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-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chi%CC%8Dp-ha%CC%8Dp" title="Chi̍p-ha̍p – Minnan" lang="nan" hreflang="nan" data-title="Chi̍p-ha̍p" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%86%E5%90%88" title="集合 – Cantonese" lang="yue" hreflang="yue" data-title="集合" data-language-autonym="粵語" data-language-local-name="Cantonese" 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/Q36161#sitelinks-wikipedia" title="Baguhin ang link sa ibang wika" class="wbc-editpage">baguhin ang mga link</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="Namespace"> <div id="p-associated-pages" class="vector-menu 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[j]" accesskey="j"><span>Mga nakaturo rito</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Natatangi:Nakakawing_ng_kamakailang_pagbabago/Pangkat_(matematika)" rel="nofollow" title="Mga huling binago sa mga kaugnay na pahina mula sa pahinang ito [k]" accesskey="k"><span>Kaugnay na pagbabago</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=tl" title="Magkarga ng mga larawan o iba pang midya [u]" accesskey="u"><span>Mag-upload ng file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Natatangi:Natatanging_mga_pahina" title="Tala ng mga natatanging pahina [q]" accesskey="q"><span>Natatanging pahina</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Pangkat_(matematika)&oldid=2023090" title="Ibigay ang palagiang kawing para sa kasalukuyang bersyon ng pahinang ito"><span>Permanenteng link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Pangkat_(matematika)&action=info" title="More information about this page"><span>Impormasyon ng pahina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Natatangi:Sipiin&page=Pangkat_%28matematika%29&id=2023090&wpFormIdentifier=titleform" title="Impormasyon sa pagsangguni sa pahinang ito"><span>Sumangguni</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Natatangi:UrlShortener&url=https%3A%2F%2Ftl.wikipedia.org%2Fwiki%2FPangkat_%28matematika%29"><span>Kumuha ng pinaikling URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Natatangi:QrCode&url=https%3A%2F%2Ftl.wikipedia.org%2Fwiki%2FPangkat_%28matematika%29"><span>I-download ang QR code</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"> I-print/I-export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Natatangi:Aklat&bookcmd=book_creator&referer=Pangkat+%28matematika%29"><span>Lumikha ng isang aklat</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Natatangi:DownloadAsPdf&page=Pangkat_%28matematika%29&action=show-download-screen"><span>I-download bilang PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Pangkat_(matematika)&printable=yes" title="Ipakita ang bersyon ng pahinang itong madaling maililimbag [p]" accesskey="p"><span>Bersyong mapi-print</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Sa iba pang proyekto </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Sets_(mathematics)" 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/Q36161" title="Link sa nakakonektang item sa repositoryo ng datos [g]" accesskey="g"><span>Item na Wikidata</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="Itsura"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div 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</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="tl" dir="ltr"><div role="note" class="hatnote navigation-not-searchable">Huwag ikalito sa <a href="/wiki/Grupo_(matematika)" title="Grupo (matematika)">Grupo (matematika)</a>.</div> <div role="note" class="hatnote navigation-not-searchable">Tungkol sa ang tinatawag ng mga matematiko na "intuitibong" teorya ng pangkat ang artikulo na ito. Para sa mas mahigpit na modernong <a href="/wiki/Aksiyoma" class="mw-disambig" title="Aksiyoma">aksiyomatikong</a> talakayan ng mga pangkat, tingnan ang <a href="/wiki/Teorya_ng_pangkat" title="Teorya ng pangkat">Teorya ng pangkat</a>.</div> <div role="note" class="hatnote navigation-not-searchable">Tingnan din: <a href="/wiki/Pangkat" class="mw-disambig" title="Pangkat">Pangkat</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Talaksan:Example_of_a_set.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/220px-Example_of_a_set.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/330px-Example_of_a_set.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/440px-Example_of_a_set.svg.png 2x" data-file-width="450" data-file-height="400" /></a><figcaption>Isang pangkat ng mga poligono sa isang <a href="/w/index.php?title=Diyagrama_ni_Euler&action=edit&redlink=1" class="new" title="Diyagrama ni Euler (hindi pa naisusulat)">diyagrama ni Euler</a>.</figcaption></figure> <p>Sa <a href="/wiki/Matematika" title="Matematika">matematika</a>, ang isang <b>pangkat</b> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">set</i>) ay isang koleksyon ng mga natatanging <a href="/w/index.php?title=Elemento_(matematika)&action=edit&redlink=1" class="new" title="Elemento (matematika) (hindi pa naisusulat)">elemento</a>.<sup id="cite_ref-JainAhmad1995_1-0" class="reference"><a href="#cite_note-JainAhmad1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Goldberg1986_2-0" class="reference"><a href="#cite_note-Goldberg1986-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CormenCormen2001_3-0" class="reference"><a href="#cite_note-CormenCormen2001-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Maaring kahit anumang bagay ang mga elementong binubuo ng isang pangkat: mga tao, mga titik ng <a href="/wiki/Alpabeto" title="Alpabeto">alpabeto</a>, mga <a href="/wiki/Bilang" title="Bilang">bilang</a>, mga punto sa espasyo, mga <a href="/wiki/Guhit" class="mw-disambig" title="Guhit">guhit</a>, mga hugis pang-<a href="/wiki/Heometriya" title="Heometriya">heometriya</a>, mga <a href="/wiki/Baryable" title="Baryable">baryable</a>, o kahit ibang mga pangkat.<sup id="cite_ref-FOOTNOTEHalmos19601_4-0" class="reference"><a href="#cite_note-FOOTNOTEHalmos19601-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Magkatumbas ang dalawang pangkat kung at kung lamang sila ay mayroong tumpak na mga parehong elemento.<sup id="cite_ref-Stoll_5-0" class="reference"><a href="#cite_note-Stoll-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Nasa lahat ng dako ng makabagong matematika ang mga pangkat. Sa katunayan, ang <a href="/wiki/Teorya_ng_pangkat" title="Teorya ng pangkat">teorya ng pangkat</a>, mas partikular ang teorya ng pangkat ng Zermelo–Fraenkel, ay naging pamantayang paraan upang magbigay ng mahigpit na mga pundasyon para sa lahat ng sangay ng matematika simula pa noong unang kalahati ng <a href="/wiki/Ika-20_dantaon" title="Ika-20 dantaon">ika-20 dantaon</a>.<sup id="cite_ref-FOOTNOTEHalmos19601_4-1" class="reference"><a href="#cite_note-FOOTNOTEHalmos19601-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Kasaysayan">Kasaysayan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=1" title="Baguhin seksiyon: Kasaysayan" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=1" title="Edit section's source code: Kasaysayan"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/wiki/Teorya_ng_pangkat#Kasaysayan" title="Teorya ng pangkat">Teorya ng pangkat § Kasaysayan</a></div> <p>Lumitaw ang konsepto ng isang pangkat sa matematika sa pagtatapos ng ika-19 na siglo. Ang Alemang salita para sa isang pangkat, <i>Menge</i>, ay nilalang ni <a href="/w/index.php?title=Bernard_Bolzano&action=edit&redlink=1" class="new" title="Bernard Bolzano (hindi pa naisusulat)">Bernard Bolzano</a> sa kaniyang aklat ng <i>Paradoxien des Unendlichen</i> (<a href="/wiki/Wikang_Tagalog" title="Wikang Tagalog">Tagalog</a>: <i lang="tl">Mga Kabalintunaan ng Walang Hangganan</i>). </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Talaksan:Passage_with_the_set_definition_of_Georg_Cantor.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/220px-Passage_with_the_set_definition_of_Georg_Cantor.png" decoding="async" width="220" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/330px-Passage_with_the_set_definition_of_Georg_Cantor.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/440px-Passage_with_the_set_definition_of_Georg_Cantor.png 2x" data-file-width="1401" data-file-height="594" /></a><figcaption>Teksto na may pagsasalin sa <a href="/wiki/Ingles" class="mw-redirect" title="Ingles">Ingles</a> ng orihinal na kahulugan ni Georg Cantor para sa isang pangkat. Dito isinasalin ng <a href="/wiki/Wikang_Aleman" title="Wikang Aleman">Alemang</a> salitang <i>Menge</i> (<a href="/wiki/Wikang_Tagalog" title="Wikang Tagalog">Tagalog</a>: <i lang="tl">pangkat</i>) bilang <i>aggregate</i> (<a href="/wiki/Wikang_Tagalog" title="Wikang Tagalog">Tagalog</a>: <i lang="tl">kapulungan</i>).</figcaption></figure> <p>Ibinigay ni <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, isa sa mga tagapagtatag ng teorya ng pangkat, ang kasunod na kahulugan sa simula ng kaniyang <i>Beiträge zur Begründung der transfiniten Mengenlehre</i> (<a href="/wiki/Wikang_Tagalog" title="Wikang Tagalog">Tagalog</a>: <i lang="tl">Mga Ambag para sa Pundasyon ng Teorya ng Pangkat</i>):<sup id="cite_ref-Cantor_6-0" class="reference"><a href="#cite_note-Cantor-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r2106284">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}</style><blockquote class="templatequote"><p>Ang pangkat ay isang kapulungan sa kabuuan ng mga natatanging bagay ng nating persepsyon o nating pag-iisip, na tinatawag na mga elemento ng pangkat.</p></blockquote> <p>Nagpasok ni <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> ang katangian sa pagitan ng isang pangkat at isang <a href="/w/index.php?title=Klase_(teorya_ng_pangkat)&action=edit&redlink=1" class="new" title="Klase (teorya ng pangkat) (hindi pa naisusulat)">klase</a> (ang pangkat ay isang klase, pero ilang mga klase, tulad ng klase ng lahat ng mga pangkat, hindi ay mga pangkat, halimbawa <a href="/w/index.php?title=Kabalintunaan_ni_Russell&action=edit&redlink=1" class="new" title="Kabalintunaan ni Russell (hindi pa naisusulat)">kabalintunaan ni Russell</a>):<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2106284"><blockquote class="templatequote"><p>Kapag inatupag ng mga <a href="/wiki/Matematiko" title="Matematiko">matematiko</a> ang kapulungan, <i lang="de">Menge</i>, <i lang="fr">ensemble</i>, o katumbas na pangalan, karaniwan, lalo na kung saan ang bilang ng mga termino na kinabibilangan ay may hangganan, may palagay ng mismong bagay (sa katunayan isang klase) bilang tala ng niyang mga termino, at palagay na posible na may iisang termino, na nasa kasong iyon <i>ay</i> isang klase.</p></blockquote> <div class="mw-heading mw-heading3"><h3 id="Intuitibong_teorya_ng_pangkat">Intuitibong teorya ng pangkat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=2" title="Baguhin seksiyon: Intuitibong teorya ng pangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=2" title="Edit section's source code: Intuitibong teorya ng pangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Intuitibong_teorya_ng_pangkat&action=edit&redlink=1" class="new" title="Intuitibong teorya ng pangkat (hindi pa naisusulat)">Intuitibong teorya ng pangkat</a></div> <p>Ang pangunahing propyedad ng isang pangkat ay na maaaring sumaklaw ng mga elemento, na tinatawag din na <i>miyembro</i>. Katumbas ang dalawang pangkat kung may parehong mga elemento. Mas eksakto na, katumbas ang mga pangkat ng <i>A</i> at <i>B</i> kung bawa't isang elemento ng <i>A</i> ay isang elemento ng <i>B</i>, at bawa't isang elemento ng <i>B</i> ay isang elemento ng <i>A</i>. Ang itong propyedad ay tinatawag na <i><a href="/w/index.php?title=Ekstensiyonalidad&action=edit&redlink=1" class="new" title="Ekstensiyonalidad (hindi pa naisusulat)">ekstensiyonalidad</a></i> (<a href="/wiki/Wikang_Kastila" title="Wikang Kastila">Kastila</a>: <i lang="es">extensionalidad</i>, <a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">extensionality</i>) <i>ng mga pangkat</i>. </p><p>Ang simpleng konsepto ng isang pangkat ay naging napaka-kapaki-pakinabang sa matematika, pero lumilitaw ang mga <a href="/w/index.php?title=Kabalintunaan&action=edit&redlink=1" class="new" title="Kabalintunaan (hindi pa naisusulat)">kabalintunaan</a> kung walang takda nasa pagpapagawa ng mga pangkat: </p> <ul><li>Ang <a href="/w/index.php?title=Kabalintunaan_ni_Russell&action=edit&redlink=1" class="new" title="Kabalintunaan ni Russell (hindi pa naisusulat)">kabalintunaan ni Russell</a> ay ipinapakita na ang "pangkat ng lahat ng mga pangkat na hindi sumasaklaw ng mga mismo", kumbaga <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a>, ay imposible.</li> <li>Ang <a href="/w/index.php?title=Kabalintunaan_ni_Cantor&action=edit&redlink=1" class="new" title="Kabalintunaan ni Cantor (hindi pa naisusulat)">kabalintunaan ni Cantor</a> ay ipinapakita na ang "pangkat ng lahat ng mga pangkat" ay imposible.</li></ul> <p>Ipinapaliwanag ng <a href="/w/index.php?title=Intuitibong_teorya_ng_pangkat&action=edit&redlink=1" class="new" title="Intuitibong teorya ng pangkat (hindi pa naisusulat)">intuitibong teorya ng pangkat</a> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">naive set theory</i>) ang pangkat bilang anumang koleksiyon na <a href="/w/index.php?title=Ekspresyon_na_naayos_na_ipinapaliwanag&action=edit&redlink=1" class="new" title="Ekspresyon na naayos na ipinapaliwanag (hindi pa naisusulat)">maayos na ipinapaliwanag</a> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">well-defined</i>) na may mga natatanging elemento, pero lumilitaw ang mga problema dahil sa labo ng terminong <i>maayos na ipinapaliwanag</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Aksiyomatikong_teorya_ng_pangkat">Aksiyomatikong teorya ng pangkat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=3" title="Baguhin seksiyon: Aksiyomatikong teorya ng pangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=3" title="Edit section's source code: Aksiyomatikong teorya ng pangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sa kasunod na mga pagsisikap para lutasin ang itong mga kabalintunaan sapul nang orihinal na pormulasyon ng intuitibong teorya ng pangkat, ang mga propyedad ng mga pangkat ay ipinapaliwanag ng mga <a href="/wiki/Aksiyoma" class="mw-disambig" title="Aksiyoma">aksiyoma</a>. Naiintindihan ng aksiyomatikong teorya ng pangkat ang konsepto ng isang pangkat bilang isang <a href="/w/index.php?title=Primitibong_nosyon&action=edit&redlink=1" class="new" title="Primitibong nosyon (hindi pa naisusulat)">primitibong nosyon</a>. Ang layon ng mga aksiyoma ay idulot ang basikong balangkas para hinuhain ang <a href="/wiki/Katotohanan" title="Katotohanan">katotohanan</a> o <a href="/wiki/Kasinungalingan" title="Kasinungalingan">kasinungalingan</a> ng partikular na matematikal na mga proposisyon (mga sabi) tungkol sa mga pangkat, sa pamamagitan ng <a href="/wiki/Lohika_ng_unang_orden" title="Lohika ng unang orden">lohika ng unang orden</a>. Gayunman, ayon sa <a href="/w/index.php?title=Mga_teorema_ng_di-pagkakumpleto_ni_G%C3%B6del&action=edit&redlink=1" class="new" title="Mga teorema ng di-pagkakumpleto ni Gödel (hindi pa naisusulat)">mga teorema ng di-pagkakumpleto ni Gödel</a> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">Gödel's incompleteness theorems</i>), imposible ang paggamit ng lohika ng unang orden para patunayan na anumang aksiyomatikong teorya ng pangkat ay walang kabalintunaan. </p> <div class="mw-heading mw-heading2"><h2 id="Kahulugan_at_notasyon">Kahulugan at notasyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=4" title="Baguhin seksiyon: Kahulugan at notasyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=4" title="Edit section's source code: Kahulugan at notasyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Karaniwang ipinapaliwanag ng mga tekstong matematikal ang mga pangkat sa <a href="/wiki/Malaking_titik" class="mw-redirect" title="Malaking titik">malalaking titik</a> sa <i>italiko</i>, tulag ng <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, <span class="texhtml mvar" style="font-style:italic;">C</span>.<sup id="cite_ref-:1_8-0" class="reference"><a href="#cite_note-:1-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Ang isang pangkat ay natatawag din na <i>koleksiyon</i> o <i>pamilya</i>, lalo na kung ang kaniyang mga elementong mismo ay mga pangkat </p> <div class="mw-heading mw-heading3"><h3 id="Notasyong_pantala">Notasyong pantala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=5" title="Baguhin seksiyon: Notasyong pantala" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=5" title="Edit section's source code: Notasyong pantala"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sa notasyong pantala (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">roster notation, enumeration notation</i>) ipinapaliwanag ang isang pangkat ng tala ng niyang mga elemento sa pagitan ng mga <a href="/wiki/Kulot_na_panaklong" class="mw-redirect" title="Kulot na panaklong">kulot na panaklong</a>, na hinihiwalayan ng mga <a href="/w/index.php?title=Kuwit&action=edit&redlink=1" class="new" title="Kuwit (hindi pa naisusulat)">kuwit</a>: </p> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;"><span class="texhtml"><i>A</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span></div> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;"><span class="texhtml"><i>B</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>.</div> <p>Sa isang pangkat, mahalaga lang ang pagkakasama ng bawa't isang elemento, kaya hindi makabuluhan ang pagsasaayos sa notasyong pantala (sa kabilang banda, sa isang <a href="/w/index.php?title=Sekwensiya&action=edit&redlink=1" class="new" title="Sekwensiya (hindi pa naisusulat)">sekwensiya</a>, <a href="/w/index.php?title=Tupla&action=edit&redlink=1" class="new" title="Tupla (hindi pa naisusulat)">tupla</a>, o <a href="/wiki/Permutasyon" title="Permutasyon">permutasyon</a> ng isang pangkat, makahuluhan ang pagsasaayos). Halimbawa, ang <span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span> at <span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span> ay kinakatawan ang parehong pangkat. </p><p>Para sa mga pangkat na may maraming elemento, lalo na kung may implisitong padron, ang tala ng mga miyembro ay maaaring daglatin ng <a href="/w/index.php?title=Elipsis&action=edit&redlink=1" class="new" title="Elipsis (hindi pa naisusulat)">elipsis</a> "<span class="texhtml">...</span>" Halimbawa, ang pangkat ng unang libong positibong <a href="/wiki/Buumbilang" title="Buumbilang">buumbilang</a> ay maaaring tukuyin sa notasyong pantala ganito: </p> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;"><span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>.</div> <div class="mw-heading mw-heading3"><h3 id="Semantikong_kahulugan">Semantikong kahulugan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=6" title="Baguhin seksiyon: Semantikong kahulugan" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=6" title="Edit section's source code: Semantikong kahulugan"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ang ibang paraan para ipaliwanag ang isang pangkat ay sa pamamagitan ng isang panuntunan para itakda ang mga elemento: </p> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;">Ipaliwanag <span class="texhtml mvar" style="font-style:italic;">A</span> bilang pangkat na ang kaninong mga miyembro ay unang apat na positibong <a href="/wiki/Buumbilang" title="Buumbilang">buumbilang</a>.</div> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;">Ipaliwanag <span class="texhtml mvar" style="font-style:italic;">B</span> bilang pangkat ng mga <a href="/wiki/Kulay" title="Kulay">kulay</a> ng <a href="/wiki/Watawat_ng_Pilipinas" title="Watawat ng Pilipinas">watawat ng Pilipinas</a>.</div> <p>Ang ganyang kahulugan ay tinatawag na isang <i>semantikong paglalarawan</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Notasyong_pampangkat">Notasyong pampangkat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=7" title="Baguhin seksiyon: Notasyong pampangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=7" title="Edit section's source code: Notasyong pampangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Notasyong_pampangkat&action=edit&redlink=1" class="new" title="Notasyong pampangkat (hindi pa naisusulat)">Notasyong pampangkat</a></div> <p>Tinutukoy ng notasyong pampangkat ang isang pangkat bilang isang pagpili mula sa isang mas malaking pangkat. Ang itong pagpili ay itinatakda ng isang kondisyon sa mga elemento. Halimbawa, isang pangkat ng <span class="texhtml mvar" style="font-style:italic;">F</span> ay naipapaliwanag ganito: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\{n\mid n{\text{ ay buumbilang, at }}0\leq n\leq 19\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∣</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> ay buumbilang, at </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>19</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\{n\mid n{\text{ ay buumbilang, at }}0\leq n\leq 19\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72001fd8ed8dd3c742dab708995873c8583df118" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.06ex; height:2.843ex;" alt="{\displaystyle F=\{n\mid n{\text{ ay buumbilang, at }}0\leq n\leq 19\}.}"></span> </p><p>Sa itong notasyon, ang <a href="/w/index.php?title=Bertikal_na_harang&action=edit&redlink=1" class="new" title="Bertikal na harang (hindi pa naisusulat)">bertikal na harang</a> "|" ay nangangahulugan ng "kung saan", at nababasa ang paglalarawan bilang "Ang <span class="texhtml mvar" style="font-style:italic;">F</span> ay pangkat ng lahat ng bilang <span class="texhtml mvar" style="font-style:italic;">n</span> kung saan <span class="texhtml mvar" style="font-style:italic;">n</span> ay buumbilang nasa interbalo mula sa 0 hanggang sa 19, inklusibo". Gumagamit ang ilang mga awtor ng isang <a href="/w/index.php?title=Tutuldok&action=edit&redlink=1" class="new" title="Tutuldok (hindi pa naisusulat)">tutuldok</a> imbes na bertikal na harang. </p> <div class="mw-heading mw-heading2"><h2 id="Pagkamiyembro">Pagkamiyembro</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=8" title="Baguhin seksiyon: Pagkamiyembro" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=8" title="Edit section's source code: Pagkamiyembro"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Elemento_(matematika)&action=edit&redlink=1" class="new" title="Elemento (matematika) (hindi pa naisusulat)">Elemento (matematika)</a></div> <p>Kung ang <span class="texhtml mvar" style="font-style:italic;">B</span> ay isang pangkat at ang <span class="texhtml mvar" style="font-style:italic;">x</span> ay isang elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>, ito ay sinusulat sa maikling anyo bilang <span class="texhtml"><i>x</i> ∈ <i>B</i></span>, na nababasa bilang "kinabilangan ng <i>x</i> ang <i>B</i>", o "ang <i>x</i> ay nasa <i>B</i>". Ang sabing "ang <i>y</i> hindi ay elemento sa <i>B</i>" ay sinusulat bilang <span class="texhtml"><i>y</i> ∉ <i>B</i></span>, na nababasa bilang "<i>y</i> hindi ay nasa <i>B</i>". </p><p>Halimbawa, tungkol sa mga pangkat ng <span class="texhtml"><i>A</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>, <span class="texhtml"><i>B</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>, and <span class="texhtml"><i>F</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>, </p> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;"><span class="texhtml">4 ∈ <i>A</i></span> at <span class="texhtml">12 ∈ <i>F</i></span>; at</div> <div style="padding-left: 3em; padding-right:0em; overflow: hidden;"><span class="texhtml">20 ∉ <i>F</i></span> and <span class="texhtml">berde ∉ <i>B</i></span>.</div> <div class="mw-heading mw-heading2"><h2 id="Basyong_pangkat">Basyong pangkat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=9" title="Baguhin seksiyon: Basyong pangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=9" title="Edit section's source code: Basyong pangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Basyong_pangkat&action=edit&redlink=1" class="new" title="Basyong pangkat (hindi pa naisusulat)">Basyong pangkat</a></div> <p>Ang <i>basyong pangkat</i> ay nag-iisang pangkat na wala mga miyembro. Sinusulat <span class="texhtml">∅</span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></span> o <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a> o <span class="texhtml">ϕ</span> o (or <span class="texhtml mvar" style="font-style:italic;">ϕ</span>). </p> <div class="mw-heading mw-heading2"><h2 id="Mga_subpangkat">Mga subpangkat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=10" title="Baguhin seksiyon: Mga subpangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=10" title="Edit section's source code: Mga subpangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/wiki/Subpangkat" title="Subpangkat">Subpangkat</a></div> <p>Kung ang bawa't isang elemento ng pangkat ng <i>A</i> ay din nasa <i>B</i>, inilalarawan na <i>A</i> bilang isang <i>subpangkat ng B</i>, o <i>sinasaklaw sa B</i>, na sinusulat <i>A</i> ⊆ <i>B</i>, o <i>B</i> ⊇ <i>A</i>. Nababasa ang huli na <i>Sumasaklaw ang B ng A,</i> <i>Isinasama ng A ang B</i>, o <i>Ang B ay superpangkat ng A</i>. Ang <a href="/w/index.php?title=Kaugnayan_(matematika)&action=edit&redlink=1" class="new" title="Kaugnayan (matematika) (hindi pa naisusulat)">kaugnayan</a> ng mga pangkat ay tinatawag na <i>pagkakasama</i> o <i>saklaw</i>. Katumbas ang dalawang pangkat kung sumasaklaw ng isa't isa: kung <i>A</i> ⊆ <i>B</i> at <i>B</i> ⊆ <i>A</i>, ay katumbas sa <i>A</i> = <i>B</i>. </p><p>Kung ang <i>A</i> ay subpangkat ng <i>B</i>, pero ang <i>A</i> hindi ay katumbas sa <i>B</i>, ang <i>A</i> ay tinatawag isang <i>angkop na subpangkat ng B</i>. Naisusulat ito na <i>A</i> ⊊ <i>B</i>. Gayon din ang <i>B</i> ⊋ <i>A</i> ay nangangahulugan na <i>Ang B ay angkop na superpangkat ng A</i>. Kumbaga, sumasaklaw ang <i>B</i> ang <i>A</i>, at hindi ay katumbas sa <i>A</i>. </p><p>Ang pangatlong pares ng mga operador ⊂ at ⊃ ay kakaibang ginagawit ng kakaibang mga awtor: gumamit ang ilang mga awtor ng <i>A</i> ⊂ <i>B</i> at <i>B</i> ⊃ <i>A</i> para ibig sabihin na ang <i>A</i> ay anumang subpangkat ng <i>B</i> (at hindi naman kailangan isang angkop na superpangkat), habang inilalaan ng ibang mga awtor para mga kaso kung saan <i>A</i> ay angkop na subpangkat ng <i>B</i>. </p><p>Mga halimbawa: </p> <ul><li>Ang pangkat ng lahat ng mga <a href="/wiki/Tao" title="Tao">tao</a> ay isang angkop na subpangkat ng pangkat ng lahat ng <a href="/wiki/Mamalya" title="Mamalya">mamalya</a>.</li> <li><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a> ⊂ <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a>.</li> <li><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a> ⊆ <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a>.</li></ul> <p>Ang basyong pangkat ay subpangkat ng bawa't isang pangkat, at ang bawa't isang pangkat ay subpangkat ng mismo: </p> <ul><li>∅ ⊆ <i>A</i>.</li> <li><i>A</i> ⊆ <i>A</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Mga_natatanging_pangkat_ng_mga_bilang_sa_matematika">Mga natatanging pangkat ng mga bilang sa matematika</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=11" title="Baguhin seksiyon: Mga natatanging pangkat ng mga bilang sa matematika" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=11" title="Edit section's source code: Mga natatanging pangkat ng mga bilang sa matematika"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Talaksan:NumberSetinC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/220px-NumberSetinC.svg.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/330px-NumberSetinC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/440px-NumberSetinC.svg.png 2x" data-file-width="600" data-file-height="470" /></a><figcaption>Ang mga <a href="/wiki/Likas_na_bilang" title="Likas na bilang">likas na bilang</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 \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> ay sinasaklaw sa mga <a href="/wiki/Buumbilang" title="Buumbilang">buumbilang</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 \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>, na sinasaklaw sa mga <a href="/wiki/Makatwirang_bilang" title="Makatwirang bilang">makatwirang bilang</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 \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>, na sinasaklaw sa mga <a href="/wiki/Tunay_na_bilang" title="Tunay na bilang">tunay na bilang</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 \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>, na sinasaklaw sa mga <a href="/wiki/Komplikadong_bilang" title="Komplikadong bilang">komplikadong bilang</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>.</figcaption></figure> <p>Napakahahalagang ang ilang mga pangkat sa matematika, kaya ang mga <a href="/wiki/Matematiko" title="Matematiko">matematiko</a> madalas na tinutukoy ang mga ito, kaya nagtamo ang ganyang mga pangkat ng mga natatanging pangalan at kaugalian para kumilala ng mga ito. </p><p>Ang mga itong mahahalagang pangkat ay kinakatawan sa mga tekstong matematikal sa <b>bold</b> (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }"></span>) o <b>bold pampisara</b> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i lang="en">blackboard bold</i>, e.g. <span class="mwe-math-element"><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>).<sup id="cite_ref-Tourlakis2003_9-0" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Sumasaklaw: </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 \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }"></span> o <span class="mwe-math-element"><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>, ang pangkat ng lahat ng mga <a href="/wiki/Likas_na_bilang" title="Likas na bilang">likas na bilang</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 \mathbf {N} =\{0,1,2,3,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e56e443c36d1169793ec83704be5b091096fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.402ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}"></span> (madalas na pinapalayas ng mga awtor ang <span class="texhtml">0</span>);<sup id="cite_ref-Tourlakis2003_9-1" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }"></span> o <span class="mwe-math-element"><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>, ang pangkat ng lahat ng mga <a href="/wiki/Buumbilang" title="Buumbilang">buumbilang</a> (kung positibo, negatibo o sero): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc04b5e7bcf1d9a5b99026491b203989de97b6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.09ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}"></span>;<sup id="cite_ref-Tourlakis2003_9-2" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"></span> o <span class="mwe-math-element"><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>, ang pangkat ng lahat ng mga <a href="/wiki/Makatwirang_bilang" title="Makatwirang bilang">makatwirang bilang</a> (kumbaga, ang pangkat ng lahat ng <a href="/wiki/Hatimbilang" title="Hatimbilang">hatimbilang</a>, parehong angkop at hindi angkop): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>,</mo> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd8d6f8bfd25dc94c7f5819bc780c74715d211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.238ex; height:5.009ex;" alt="{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}"></span>. Halimbawa, <span class="texhtml">−<span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">7</span><span class="visualhide">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">4</span></span> ∈ <b>Q</b></span> at <span class="texhtml">5 = <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">5</span><span class="visualhide">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">1</span></span> ∈ <b>Q</b></span>;<sup id="cite_ref-Tourlakis2003_9-3" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></span> o <span class="mwe-math-element"><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>, ang pangkat ng lahat ng mga <a href="/wiki/Tunay_na_bilang" title="Tunay na bilang">tunay na bilang</a>, na sumusuklaw ng lahat ng makatwirang bilang at lahat ng <a href="/w/index.php?title=Hindi_makatwirang_bilang&action=edit&redlink=1" class="new" title="Hindi makatwirang bilang (hindi pa naisusulat)">hindi makatwirang bilang</a> (na sumusuklaw ng mga <a href="/w/index.php?title=Alhebraikong_bilang&action=edit&redlink=1" class="new" title="Alhebraikong bilang (hindi pa naisusulat)">alhebraikong bilang</a> tulad ng <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> na hindi naisusulat bilang mga hatimbilang, at saka mga <a href="/w/index.php?title=Transendental_na_bilang&action=edit&redlink=1" class="new" title="Transendental na bilang (hindi pa naisusulat)">transendental na bilang</a> tulad ng <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> and <a href="/w/index.php?title=E_(konstanteng_matematikal)&action=edit&redlink=1" class="new" title="E (konstanteng matematikal) (hindi pa naisusulat)"><span class="texhtml"><i>e</i></span></a>);<sup id="cite_ref-Tourlakis2003_9-4" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"></span> o <span class="mwe-math-element"><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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>, ang pangkat ng lahat ng mga <a href="/wiki/Komplikadong_bilang" title="Komplikadong bilang">komplikadong bilang</a>: <span class="texhtml"><b>C</b> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>, halimbawa, <span class="texhtml">1 + 2<i>i</i> ∈ <b>C</b></span>.<sup id="cite_ref-Tourlakis2003_9-5" class="reference"><a href="#cite_note-Tourlakis2003-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <p>Ang bawa't isang pangkat sa itaas ay may mga elemento na walang hangganan. Ang bawa't isa ay subpangkat ng mga pangkat na itinatala sa ilalim. </p><p>Ang mga pangkat ng mga positibo o negatibong bilang ay minsan na ipinapahiwatig ng <sup>+</sup> at <sup>-</sup>. Halimbawa, kinakatawan ng <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fafdbdb9efd367bcb05e6e310a9c85326281bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.843ex;" alt="{\displaystyle \mathbf {Q} ^{+}}"></span> ang pangkat ng mga positibong makatwirang bilang. </p> <div class="mw-heading mw-heading2"><h2 id="Mga_punsiyon">Mga punsiyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=12" title="Baguhin seksiyon: Mga punsiyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=12" title="Edit section's source code: Mga punsiyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ang isang <i><a href="/wiki/Punsiyon_(matematika)" title="Punsiyon (matematika)">punsiyon</a></i> (o <i><a href="/w/index.php?title=Mapa_(matematika)&action=edit&redlink=1" class="new" title="Mapa (matematika) (hindi pa naisusulat)">pagmamapa</a></i>) mula sa isang pangkat ng <span class="texhtml mvar" style="font-style:italic;">A</span> hanggang sa isang pangkat ng <span class="texhtml mvar" style="font-style:italic;">B</span> ay panuntunan na nagtatalaga sa bawa't isang elementong "input" ng <span class="texhtml mvar" style="font-style:italic;">A</span> ng isang "output" na ay isang elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>. Mas pormal na, isang punsiyon ay natatanging uri ng <a href="/w/index.php?title=Kaugnayan_(matematika)&action=edit&redlink=1" class="new" title="Kaugnayan (matematika) (hindi pa naisusulat)">kaugnayan</a>, isa na nag-uugnay ng bawa't isang elemento ng <span class="texhtml mvar" style="font-style:italic;">A</span> sa <i>eksakto na isang</i> elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>. Ang isang punsiyon ay tinatawag na: </p> <ul><li><a href="/w/index.php?title=Inyektibo&action=edit&redlink=1" class="new" title="Inyektibo (hindi pa naisusulat)">inyektibo</a> kung nagmamapa anumang dalawang kakaibang elemento ng <span class="texhtml mvar" style="font-style:italic;">A</span> sa <i>kakaibang</i> mga elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>,</li> <li><a href="/w/index.php?title=Suryektibo&action=edit&redlink=1" class="new" title="Suryektibo (hindi pa naisusulat)">suryektibo</a> kung para bawa't isang elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>, may kahit isa lang elemento ng <span class="texhtml mvar" style="font-style:italic;">A</span> na nagmamapa sa ito, at</li> <li><a href="/w/index.php?title=Biyektibo&action=edit&redlink=1" class="new" title="Biyektibo (hindi pa naisusulat)">biyektibo</a> kung ang punsiyon ay parehong inyektibo at suryektibo — sa itong kaso, bawa't isang elemento ng <span class="texhtml mvar" style="font-style:italic;">A</span> ay inirereto sa nag-iisang elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span>, at bawa't isang elemento ng <span class="texhtml mvar" style="font-style:italic;">B</span> ay inirereto sa nag-iisang elemento ng <span class="texhtml mvar" style="font-style:italic;">A</span>, kaya walang elemento na walang kapares.</li></ul> <p>Ang isang inyektibong punsiyon ay tinatawag na isang <i>inyeksiyon</i>, ang isang suryektibong punsiyon ay tinatawag na isang <i>suryeksiyon</i>, at ang isang biyektibong punsiyon ay tinatawag na isang <i>biyeksiyon</i> o <i>isa-sa-isang tugma</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Kardinalidad">Kardinalidad</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=13" title="Baguhin seksiyon: Kardinalidad" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=13" title="Edit section's source code: Kardinalidad"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Kardinalidad&action=edit&redlink=1" class="new" title="Kardinalidad (hindi pa naisusulat)">Kardinalidad</a></div> <p>Ang kardinalidad ng isang pangkat ng <span class="texhtml"><i>S</i></span>, na sinusulat na <span class="texhtml"><a href="/w/index.php?title=Padron:Mabs&action=edit&redlink=1" class="new" title="Padron:Mabs (hindi pa naisusulat)">Padron:Mabs</a></span>, ay dami ng mga miyembro ng <span class="texhtml"><i>S</i></span>. Halimbawa, kung <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a>}}, ang kardinalidad ay 3: <span class="texhtml"><a href="/w/index.php?title=Padron:Mabs&action=edit&redlink=1" class="new" title="Padron:Mabs (hindi pa naisusulat)">Padron:Mabs</a> = 3</span>. Sa notasyong pantala, ang mga miyembro na inuulit hindi ay ibinibilang, kaya <span class="texhtml"><a href="/w/index.php?title=Padron:Mabs&action=edit&redlink=1" class="new" title="Padron:Mabs (hindi pa naisusulat)">Padron:Mabs</a> = 3</span> din. </p><p>Mas pormal na, nagkakaroon ang dalawang pangkat ng parehong kardinalidad kung may isa-sa-isang tugma sa pagitan ng nila. </p><p>Ang kardinalidad ng basyong pangkat ay sero. </p> <div class="mw-heading mw-heading2"><h2 id="Mga_kapangyarihang_pangkat">Mga kapangyarihang pangkat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=14" title="Baguhin seksiyon: Mga kapangyarihang pangkat" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=14" title="Edit section's source code: Mga kapangyarihang pangkat"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/wiki/Kapangyarihang_pangkat" title="Kapangyarihang pangkat">Kapangyarihang pangkat</a></div> <p>Ang kapangyarihang pangkat ng isang pangkat ng <span class="texhtml"><i>S</i></span> ay pangkat ng lahat ng mga <a href="/wiki/Subpangkat" title="Subpangkat">subpangkat</a> ng <span class="texhtml"><i>S</i></span>. Ang <a href="/w/index.php?title=Basyong_pangkat&action=edit&redlink=1" class="new" title="Basyong pangkat (hindi pa naisusulat)">basyong pangkat</a> at <span class="texhtml"><i>S</i></span> mismo ay mga elemento ng kapangyarihang pangkat ng <span class="texhtml"><i>S</i></span>, dahil sila ay parehong mga subpangkat ng <span class="texhtml"><i>S</i></span>. Halimbawa, ang kapangyarihang pangkat ng <span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span> ay <span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>. Ang kapangyarihang pangkat ng isang pangkat ng <span class="texhtml"><i>S</i></span> ay madalas na isinusulat na <span class="texhtml"><i>P</i>(<i>S</i>)</span> o <span class="texhtml">2<sup><i>S</i></sup></span>. </p><p>Kung nagkakaroon ang <span class="texhtml"><i>S</i></span> ng <span class="texhtml"><i>n</i></span> elemento, nagkakaroon ang <span class="texhtml"><i>P</i>(<i>S</i>)</span> ng <span class="texhtml">2<sup><i>n</i></sup></span> elemento. Halimbawa, nagkakaroon ang <span class="texhtml"><a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span> ng tatlong elemento, at nagkakaroon ang kaniyang kapangyarihang pangkat ng <span class="texhtml">2<sup>3</sup> = 8</span> elemento, bilang ipinakita sa itaas. </p><p>Kung ang <span class="texhtml"><i>S</i></span> ay impinito (kung naililista o hindi), ang <span class="texhtml"><i>P</i>(<i>S</i>)</span> hindi ay naililista. Saka, laging "mas malaki" ang kapangyarihang pangkat kaysa sa orihinal na pangkat, sa muwang na ang anumang tangka para parisan ang mga elemento ng <span class="texhtml"><i>S</i></span> sa mga elemento ng <span class="texhtml"><i>P</i>(<i>S</i>)</span> ay mag-iiwan ng ilang mga elemento ng <span class="texhtml"><i>P</i>(<i>S</i>)</span> na walang pares. (Hindi kailanman may isang <a href="/w/index.php?title=Biyeksiyon&action=edit&redlink=1" class="new" title="Biyeksiyon (hindi pa naisusulat)">biyeksiyon</a> mula sa <span class="texhtml"><i>S</i></span> hanggang sa <span class="texhtml"><i>P</i>(<i>S</i>)</span>.) </p> <div class="mw-heading mw-heading2"><h2 id="Mga_partisyon">Mga partisyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=15" title="Baguhin seksiyon: Mga partisyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=15" title="Edit section's source code: Mga partisyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Partisyon_ng_isang_pangkat&action=edit&redlink=1" class="new" title="Partisyon ng isang pangkat (hindi pa naisusulat)">Partisyon ng isang pangkat</a></div> <p>Ang isang partisyon ng isang pangkat ng <span class="texhtml"><i>S</i></span> ay isang pangkat ng mga di-basyong pangkat ng <span class="texhtml"><i>S</i></span>, kung saan bawa't isang <span class="texhtml"><i>x</i></span> nasa <span class="texhtml"><i>S</i></span> ay nasa eksakto isa sa mga ito subpangkat. </p> <div class="mw-heading mw-heading2"><h2 id="Mga_basikong_operasyon">Mga basikong operasyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=16" title="Baguhin seksiyon: Mga basikong operasyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=16" title="Edit section's source code: Mga basikong operasyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Alhebra_ng_mga_pangkat&action=edit&redlink=1" class="new" title="Alhebra ng mga pangkat (hindi pa naisusulat)">Alhebra ng mga pangkat</a></div> <div role="note" class="hatnote navigation-not-searchable">Tingnan din: <a href="/wiki/Teorya_ng_pangkat" title="Teorya ng pangkat">Teorya ng pangkat</a></div> <div class="mw-heading mw-heading2"><h2 id="Mga_aplikasyon">Mga aplikasyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=17" title="Baguhin seksiyon: Mga aplikasyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=17" title="Edit section's source code: Mga aplikasyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nasa lahat ng dako ng makabagong matematika ang mga pangkat. Halimbawa, ang mga <a href="/w/index.php?title=Alhebraikong_istruktura&action=edit&redlink=1" class="new" title="Alhebraikong istruktura (hindi pa naisusulat)">alhebraikong istruktura</a> sa <a href="/wiki/Alhebrang_basal" title="Alhebrang basal">alhebrang basal</a>, tulad ng mga <a href="/wiki/Grupo_(matematika)" title="Grupo (matematika)">grupo</a>, <a href="/w/index.php?title=Kampo_(matematika)&action=edit&redlink=1" class="new" title="Kampo (matematika) (hindi pa naisusulat)">kampo</a>, at <a href="/w/index.php?title=Singsing_(matematika)&action=edit&redlink=1" class="new" title="Singsing (matematika) (hindi pa naisusulat)">singsing</a>, ay <a href="/w/index.php?title=Pagkakasara_(matematika)&action=edit&redlink=1" class="new" title="Pagkakasara (matematika) (hindi pa naisusulat)">nakasarang</a> pangkat sa ilalim ng isa o higit pang mga <a href="/wiki/Operasyon_(matematika)" title="Operasyon (matematika)">operasyon</a>. </p><p>Ang isa sa pangunahing mga aplikasyon sa intuitibong teorya ng pangkat ay konstruksiyon ng mga <a href="/w/index.php?title=Kaugnayan_(matematika)&action=edit&redlink=1" class="new" title="Kaugnayan (matematika) (hindi pa naisusulat)">kaugnayan</a>. Ang isang kaugnayan mula sa isang <a href="/wiki/Sakop_(matematika)" title="Sakop (matematika)">sakop</a> ng <span class="texhtml"><i>A</i></span> hanggang sa <a href="/wiki/Kasakop" title="Kasakop">kasakop</a> ng <span class="texhtml"><i>B</i></span> ay subpangkat ng <a href="/wiki/Produktong_Cartesian" title="Produktong Cartesian">produktong Cartesiang</a> <span class="texhtml"><i>A</i> × <i>B</i></span>. Halimbawa, pagdating sa pangkat ng <span class="texhtml"><i>S</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span> ng mga <a href="/wiki/Hugis" title="Hugis">hugis</a> sa <a href="/w/index.php?title=Papel,_gunting,_bato&action=edit&redlink=1" class="new" title="Papel, gunting, bato (hindi pa naisusulat)">laro</a> ng mismong pangalan, ang kaugnayang "tumatalo" mula sa <span class="texhtml"><i>S</i></span> hanggang sa <span class="texhtml"><i>S</i></span> ay pangkat ng <span class="texhtml"><i>B</i> = <a href="/w/index.php?title=Padron:Mset&action=edit&redlink=1" class="new" title="Padron:Mset (hindi pa naisusulat)">Padron:Mset</a></span>. Kaya tumatalo ang <span class="texhtml"><i>x</i></span> ng <span class="texhtml"><i>y</i></span> sa laro kung ang pares ng <span class="texhtml">(<i>x</i>,<i>y</i>)</span> ay miyembro ng <span class="texhtml"><i>B</i></span>. Ang ibang halimbawa ay pangkat ng <span class="texhtml"><i>F</i></span> ng lahat ng mga pares ng <span class="texhtml">(<i>x</i>, <i>x</i><sup>2</sup>)</span>, where <span class="texhtml"><i>x</i></span> ay tunay. Ang itong kaugnayan ay isang subpangkat ng <span class="texhtml"><b>R</b> × <b>R</b></span>, kasi ang pangkat ng lahat ng mga <a href="/wiki/Parirami" title="Parirami">parirami</a> ay isang subpangkat ng pangkat ng lahat ng <a href="/wiki/Tunay_na_bilang" title="Tunay na bilang">tunay na bilang</a>. Dahil para bawa't isang <span class="texhtml"><i>x</i></span> sa <span class="texhtml"><b>R</b></span>, may isa at lang isang pares <span class="texhtml">(<i>x</i>,...)</span> sa <span class="texhtml"><i>F</i></span>, ang itong kaugnayan ay tinatawag na isang <a href="/wiki/Punsiyon_(matematika)" title="Punsiyon (matematika)">punsiyon</a>. Sa notasyong pampunsiyon, ang itong kaugnayan ay nasusulat na <span class="texhtml"><i>F</i>(<i>x</i>) = <i>x</i><sup>2</sup></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Prinsipyo_ng_ingklusyon_at_eksklusyon">Prinsipyo ng ingklusyon at eksklusyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=18" title="Baguhin seksiyon: Prinsipyo ng ingklusyon at eksklusyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=18" title="Edit section's source code: Prinsipyo ng ingklusyon at eksklusyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Pangunahing artikulo: <a href="/w/index.php?title=Prinsipyo_ng_ingklusyon_at_eksklusyon&action=edit&redlink=1" class="new" title="Prinsipyo ng ingklusyon at eksklusyon (hindi pa naisusulat)">Prinsipyo ng ingklusyon at eksklusyon</a></div> <div class="mw-heading mw-heading2"><h2 id="Mga_sanggunian">Mga sanggunian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pangkat_(matematika)&veaction=edit&section=19" title="Baguhin seksiyon: Mga sanggunian" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Pangkat_(matematika)&action=edit&section=19" title="Edit section's source code: Mga sanggunian"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2121818">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-JainAhmad1995-1"><span class="mw-cite-backlink"><a href="#cite_ref-JainAhmad1995_1-0">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r2132731">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFP._K._JainKhalil_AhmadOm_P._Ahuja1995" class="citation book cs1 cs1-prop-foreign-lang-source">P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yZ68h97pnAkC&pg=PA1"><i>Functional Analysis</i></a> (sa wikang Ingles). New Age International. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/978-81-224-0801-0" title="Natatangi:Mga pinagmulang aklat/978-81-224-0801-0"><bdi>978-81-224-0801-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.pages=1&rft.pub=New+Age+International&rft.date=1995&rft.isbn=978-81-224-0801-0&rft.au=P.+K.+Jain&rft.au=Khalil+Ahmad&rft.au=Om+P.+Ahuja&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyZ68h97pnAkC%26pg%3DPA1&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-Goldberg1986-2"><span class="mw-cite-backlink"><a href="#cite_ref-Goldberg1986_2-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFSamuel_Goldberg1986" class="citation book cs1 cs1-prop-foreign-lang-source">Samuel Goldberg (1 Enero 1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CmzFx9rB_FcC&pg=PA2"><i>Probability: An Introduction</i></a> (sa wikang Ingles). Courier Corporation. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/978-0-486-65252-8" title="Natatangi:Mga pinagmulang aklat/978-0-486-65252-8"><bdi>978-0-486-65252-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%3A+An+Introduction&rft.pages=2&rft.pub=Courier+Corporation&rft.date=1986-01-01&rft.isbn=978-0-486-65252-8&rft.au=Samuel+Goldberg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCmzFx9rB_FcC%26pg%3DPA2&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-CormenCormen2001-3"><span class="mw-cite-backlink"><a href="#cite_ref-CormenCormen2001_3-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFThomas_H.._CormenThomas_H_CormenCharles_E_LeisersonRonald_L_Rivest2001" class="citation book cs1 cs1-prop-foreign-lang-source">Thomas H.. Cormen; Thomas H Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NLngYyWFl_YC&pg=PA1070"><i>Introduction To Algorithms</i></a> (sa wikang Ingles). MIT Press. p. 1070. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/978-0-262-03293-3" title="Natatangi:Mga pinagmulang aklat/978-0-262-03293-3"><bdi>978-0-262-03293-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Algorithms&rft.pages=1070&rft.pub=MIT+Press&rft.date=2001&rft.isbn=978-0-262-03293-3&rft.au=Thomas+H..+Cormen&rft.au=Thomas+H+Cormen&rft.au=Charles+E+Leiserson&rft.au=Ronald+L+Rivest&rft.au=Clifford+Stein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNLngYyWFl_YC%26pg%3DPA1070&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-FOOTNOTEHalmos19601-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-FOOTNOTEHalmos19601_4-0">4.0</a></sup> <sup><a href="#cite_ref-FOOTNOTEHalmos19601_4-1">4.1</a></sup></span> <span class="reference-text"><a href="#CITEREFHalmos1960">Halmos 1960</a>, p. 1.<span class="error harv-error" style="display: none; font-size:100%"> harv error: no target: CITEREFHalmos1960 (<a href="/wiki/Kategorya:Harv_and_Sfn_template_errors" title="Kategorya:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-Stoll-5"><span class="mw-cite-backlink"><a href="#cite_ref-Stoll_5-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFStoll1974" class="citation book cs1 cs1-prop-foreign-lang-source">Stoll, Robert (1974). <span class="cs1-lock-registration" title="Kinakailangan ng malayang pagrehistro"><a rel="nofollow" class="external text" href="https://archive.org/details/setslogicaxiomat0000stol"><i>Sets, Logic and Axiomatic Theories</i></a></span> (sa wikang Ingles). W. H. Freeman and Company. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/setslogicaxiomat0000stol/page/5">5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+Logic+and+Axiomatic+Theories&rft.pages=5&rft.pub=W.+H.+Freeman+and+Company&rft.date=1974&rft.aulast=Stoll&rft.aufirst=Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsetslogicaxiomat0000stol&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-Cantor-6"><span class="mw-cite-backlink"><a href="#cite_ref-Cantor_6-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFCantorJourdain1895" class="citation journal cs1 cs1-prop-long-vol cs1-prop-foreign-lang-source">Cantor, Georg; Jourdain, Philip E.B. (Translator) (1895). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110610133240/http://brinkmann-du.de/mathe/fos/fos01_03.htm">"beiträge zur begründung der transfiniten Mengenlehre"</a> [contributions to the founding of the theory of transfinite numbers]. <i>Mathematische Annalen</i> (sa wikang Aleman). New York Dover Publications (1954 English translation). xlvi, xlix: 481–512, 207–246. Inarkibo mula sa <a rel="nofollow" class="external text" href="http://brinkmann-du.de/mathe/fos/fos01_03.htm">orihinal</a> noong 2011-06-10. <q>By an aggregate (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and <b>separate</b> objects m (p.85)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=beitr%C3%A4ge+zur+begr%C3%BCndung+der+transfiniten+Mengenlehre&rft.volume=xlvi%3Bxlix&rft.pages=481-512%2C+207-246&rft.date=1895&rft.aulast=Cantor&rft.aufirst=Georg&rft.au=Jourdain%2C+Philip+E.B.+%28Translator%29&rft_id=http%3A%2F%2Fbrinkmann-du.de%2Fmathe%2Ffos%2Ffos01_03.htm&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_journal" title="Padron:Cite journal">cite journal</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> (1903) <i><a href="/w/index.php?title=The_Principles_of_Mathematics&action=edit&redlink=1" class="new" title="The Principles of Mathematics (hindi pa naisusulat)">The Principles of Mathematics</a></i>, chapter VI: Classes (sa Ingles).</span> </li> <li id="cite_note-:1-8"><span class="mw-cite-backlink"><a href="#cite_ref-:1_8-0">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/sets-introduction.html">"Introduction to Sets"</a>. <i>www.mathsisfun.com</i> (sa wikang Ingles)<span class="reference-accessdate">. Nakuha noong <span class="nowrap">2020-08-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Introduction+to+Sets&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fsets%2Fsets-introduction.html&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_web" title="Padron:Cite web">cite web</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-Tourlakis2003-9"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Tourlakis2003_9-0">9.0</a></sup> <sup><a href="#cite_ref-Tourlakis2003_9-1">9.1</a></sup> <sup><a href="#cite_ref-Tourlakis2003_9-2">9.2</a></sup> <sup><a href="#cite_ref-Tourlakis2003_9-3">9.3</a></sup> <sup><a href="#cite_ref-Tourlakis2003_9-4">9.4</a></sup> <sup><a href="#cite_ref-Tourlakis2003_9-5">9.5</a></sup></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFGeorge_Tourlakis2003" class="citation book cs1 cs1-prop-foreign-lang-source">George Tourlakis (13 Pebrero 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nparMXao59QC&pg=PA137"><i>Lectures in Logic and Set Theory: Volume 2, Set Theory</i></a> (sa wikang Ingles). Cambridge University Press. p. 137. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/978-1-139-43943-5" title="Natatangi:Mga pinagmulang aklat/978-1-139-43943-5"><bdi>978-1-139-43943-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+in+Logic+and+Set+Theory%3A+Volume+2%2C+Set+Theory&rft.pages=137&rft.pub=Cambridge+University+Press&rft.date=2003-02-13&rft.isbn=978-1-139-43943-5&rft.au=George+Tourlakis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnparMXao59QC%26pg%3DPA137&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APangkat+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> </ol></div></div> <p><span typeof="mw:File"><a href="/wiki/Talaksan:Arithmetic_symbols.svg" class="mw-file-description" title="Matematika"><img alt="Matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/35px-Arithmetic_symbols.svg.png" decoding="async" width="35" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/53px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/70px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></a></span> <i>Ang lathalaing ito na tungkol sa <b>Matematika</b> ay isang <a href="/wiki/Wikipedia:Usbong" title="Wikipedia:Usbong">usbong</a>. 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