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href="/wiki/Speciaal:MijnBijdragen" title="Bijdragen IP-adres [y]" accesskey="y"><span>Bijdragen</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speciaal:MijnOverleg" title="Overlegpagina van de anonieme gebruiker van dit IP-adres [n]" accesskey="n"><span>Overleg</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="Site"> <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="Inhoud" 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">Inhoud</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">verbergen</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">Top</div> </a> </li> <li id="toc-Definitie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definitie</span> </div> </a> <ul id="toc-Definitie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beschrijving" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beschrijving"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Beschrijving</span> </div> </a> <ul id="toc-Beschrijving-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deelverzamelingen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Deelverzamelingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Deelverzamelingen</span> </div> </a> <ul id="toc-Deelverzamelingen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bewerkingen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bewerkingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bewerkingen</span> </div> </a> <ul id="toc-Bewerkingen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wetten_van_De_Morgan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wetten_van_De_Morgan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Wetten van De Morgan</span> </div> </a> <ul id="toc-Wetten_van_De_Morgan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kardinaliteit" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kardinaliteit"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Kardinaliteit</span> </div> </a> <button aria-controls="toc-Kardinaliteit-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>Kardinaliteit-subkopje inklappen</span> </button> <ul id="toc-Kardinaliteit-sublist" class="vector-toc-list"> <li id="toc-Machtsverzamelingen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Machtsverzamelingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Machtsverzamelingen</span> </div> </a> <ul id="toc-Machtsverzamelingen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Getallenverzamelingen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Getallenverzamelingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Getallenverzamelingen</span> </div> </a> <ul id="toc-Getallenverzamelingen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toepassingen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Toepassingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Toepassingen</span> </div> </a> <ul id="toc-Toepassingen-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="Inhoud" 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="Inhoudsopgave omschakelen" > <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">Inhoudsopgave omschakelen</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">Verzameling (wiskunde): verschil tussen versies</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="Ga naar een artikel in een andere taal. Beschikbaar in 102 talen" > <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 talen</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) – Zwitserduits" lang="gsw" hreflang="gsw" data-title="Menge (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Zwitserduits" 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 – Amhaars" lang="am" hreflang="am" data-title="ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="Amhaars" 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) – Arabisch" lang="ar" hreflang="ar" data-title="مجموعة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabisch" 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 – Asturisch" lang="ast" hreflang="ast" data-title="Conxuntu" data-language-autonym="Asturianu" data-language-local-name="Asturisch" 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 – Azerbeidzjaans" lang="az" hreflang="az" data-title="Çoxluqlar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" 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 – Basjkiers" lang="ba" hreflang="ba" data-title="Күмәклек" data-language-autonym="Башҡортса" data-language-local-name="Basjkiers" 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 – Belarussisch" lang="be" hreflang="be" data-title="Мноства" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" 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 – Bulgaars" lang="bg" hreflang="bg" data-title="Множество" data-language-autonym="Български" data-language-local-name="Bulgaars" 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 – Bengaals" lang="bn" hreflang="bn" data-title="সেট" data-language-autonym="বাংলা" data-language-local-name="Bengaals" 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) – Bosnisch" lang="bs" hreflang="bs" data-title="Skup (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" 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 – Catalaans" lang="ca" hreflang="ca" data-title="Conjunt" data-language-autonym="Català" data-language-local-name="Catalaans" 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) – Soranî" lang="ckb" hreflang="ckb" data-title="کۆمەڵە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Soranî" 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 – Tsjechisch" lang="cs" hreflang="cs" data-title="Množina" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" 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 – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Йыш" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" 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 – Deens" lang="da" hreflang="da" data-title="Mængde" data-language-autonym="Dansk" data-language-local-name="Deens" 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) – Duits" lang="de" hreflang="de" data-title="Menge (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="Duits" 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 – Grieks" lang="el" hreflang="el" data-title="Σύνολο" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" 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) – Engels" lang="en" hreflang="en" data-title="Set (mathematics)" data-language-autonym="English" data-language-local-name="Engels" 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 – Spaans" lang="es" hreflang="es" data-title="Conjunto" data-language-autonym="Español" data-language-local-name="Spaans" 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 – Estisch" lang="et" hreflang="et" data-title="Hulk" data-language-autonym="Eesti" data-language-local-name="Estisch" 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 – Baskisch" lang="eu" hreflang="eu" data-title="Multzo" data-language-autonym="Euskara" data-language-local-name="Baskisch" 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) – Perzisch" lang="fa" hreflang="fa" data-title="مجموعه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Perzisch" 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 – Fins" lang="fi" hreflang="fi" data-title="Joukko" data-language-autonym="Suomi" data-language-local-name="Fins" 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 – Frans" lang="fr" hreflang="fr" data-title="Ensemble" data-language-autonym="Français" data-language-local-name="Frans" 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 – Friulisch" lang="fur" hreflang="fur" data-title="Insiemi" data-language-autonym="Furlan" data-language-local-name="Friulisch" 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 – Iers" lang="ga" hreflang="ga" data-title="Tacar" data-language-autonym="Gaeilge" data-language-local-name="Iers" 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 – Ganyu" lang="gan" hreflang="gan" data-title="集合" data-language-autonym="贛語" data-language-local-name="Ganyu" 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 – Schots-Gaelisch" lang="gd" hreflang="gd" data-title="Seata" data-language-autonym="Gàidhlig" data-language-local-name="Schots-Gaelisch" 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 – Galicisch" lang="gl" hreflang="gl" data-title="Conxunto" data-language-autonym="Galego" data-language-local-name="Galicisch" 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) – Hebreeuws" lang="he" hreflang="he" data-title="קבוצה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebreeuws" 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 – Kroatisch" lang="hr" hreflang="hr" data-title="Skup" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" 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) – Hongaars" lang="hu" hreflang="hu" data-title="Halmaz (matematika)" data-language-autonym="Magyar" data-language-local-name="Hongaars" 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 – Armeens" lang="hy" hreflang="hy" data-title="Բազմություն" data-language-autonym="Հայերեն" data-language-local-name="Armeens" 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) – Indonesisch" lang="id" hreflang="id" data-title="Himpunan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" 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 – IJslands" lang="is" hreflang="is" data-title="Mengi" data-language-autonym="Íslenska" data-language-local-name="IJslands" 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 – Italiaans" lang="it" hreflang="it" data-title="Insieme" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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 – Japans" lang="ja" hreflang="ja" data-title="集合" data-language-autonym="日本語" data-language-local-name="Japans" 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) – Jamaicaans Creools" lang="jam" hreflang="jam" data-title="Set (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaicaans Creools" 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 – Georgisch" lang="ka" hreflang="ka" data-title="სიმრავლე" data-language-autonym="ქართული" data-language-local-name="Georgisch" 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 – Kazachs" lang="kk" hreflang="kk" data-title="Бөлік" data-language-autonym="Қазақша" data-language-local-name="Kazachs" 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 – Koreaans" lang="ko" hreflang="ko" data-title="집합" data-language-autonym="한국어" data-language-local-name="Koreaans" 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 – Koerdisch" lang="ku" hreflang="ku" data-title="Kom" data-language-autonym="Kurdî" data-language-local-name="Koerdisch" 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 – Latijn" lang="la" hreflang="la" data-title="Copia" data-language-autonym="Latina" data-language-local-name="Latijn" 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 – Lombardisch" lang="lmo" hreflang="lmo" data-title="Insemma" data-language-autonym="Lombard" data-language-local-name="Lombardisch" 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 – Litouws" lang="lt" hreflang="lt" data-title="Aibė" data-language-autonym="Lietuvių" data-language-local-name="Litouws" 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 – Lets" lang="lv" hreflang="lv" data-title="Kopa" data-language-autonym="Latviešu" data-language-local-name="Lets" 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 – Macedonisch" lang="mk" hreflang="mk" data-title="Множество" data-language-autonym="Македонски" data-language-local-name="Macedonisch" 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 – Mongools" lang="mn" hreflang="mn" data-title="Олонлог" data-language-autonym="Монгол" data-language-local-name="Mongools" 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 – Maleis" lang="ms" hreflang="ms" data-title="Set" data-language-autonym="Bahasa Melayu" data-language-local-name="Maleis" 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 – Birmaans" lang="my" hreflang="my" data-title="အစု" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmaans" 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 – Erzja" lang="myv" hreflang="myv" data-title="Вейссаевкс" data-language-autonym="Эрзянь" data-language-local-name="Erzja" 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) – Nedersaksisch" lang="nds" hreflang="nds" data-title="Koppel (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Nedersaksisch" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Mengd" title="Mengd – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Mengd" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - 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 – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Mengde" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - 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 – Occitaans" lang="oc" hreflang="oc" data-title="Ensemble" data-language-autonym="Occitan" data-language-local-name="Occitaans" 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 – Pools" lang="pl" hreflang="pl" data-title="Zbiór" data-language-autonym="Polski" data-language-local-name="Pools" 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 – Piëmontees" lang="pms" hreflang="pms" data-title="Ansem" data-language-autonym="Piemontèis" data-language-local-name="Piëmontees" 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 – Portugees" lang="pt" hreflang="pt" data-title="Conjunto" data-language-autonym="Português" data-language-local-name="Portugees" 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 – Roemeens" lang="ro" hreflang="ro" data-title="Mulțime" data-language-autonym="Română" data-language-local-name="Roemeens" 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 – Russisch" lang="ru" hreflang="ru" data-title="Множество" data-language-autonym="Русский" data-language-local-name="Russisch" 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 – Siciliaans" lang="scn" hreflang="scn" data-title="Nzemi" data-language-autonym="Sicilianu" data-language-local-name="Siciliaans" 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 – Servo-Kroatisch" lang="sh" hreflang="sh" data-title="Skup" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Servo-Kroatisch" 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 – Slowaaks" lang="sk" hreflang="sk" data-title="Množina" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" 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 – Sloveens" lang="sl" hreflang="sl" data-title="Množica" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" 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) – Somalisch" lang="so" hreflang="so" data-title="Qaybta (xisaab)" data-language-autonym="Soomaaliga" data-language-local-name="Somalisch" 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 – Albanees" lang="sq" hreflang="sq" data-title="Bashkësitë" data-language-autonym="Shqip" data-language-local-name="Albanees" 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 – Servisch" lang="sr" hreflang="sr" data-title="Скуп" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" 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 – Zweeds" lang="sv" hreflang="sv" data-title="Mängd" data-language-autonym="Svenska" data-language-local-name="Zweeds" 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) – Silezisch" lang="szl" hreflang="szl" data-title="Mynga (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silezisch" 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-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pangkat_(matematika)" title="Pangkat (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Pangkat (matematika)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCme" title="Küme – Turks" lang="tr" hreflang="tr" data-title="Küme" data-language-autonym="Türkçe" data-language-local-name="Turks" 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 – Oekraïens" lang="uk" hreflang="uk" data-title="Множина" data-language-autonym="Українська" data-language-local-name="Oekraïens" 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) – Oezbeeks" lang="uz" hreflang="uz" data-title="Toʻplam (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Oezbeeks" 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) – Vietnamees" lang="vi" hreflang="vi" data-title="Tập hợp (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" 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-Vlaams" lang="vls" hreflang="vls" data-title="Verzoamelienge" data-language-autonym="West-Vlams" data-language-local-name="West-Vlaams" 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） – Wuyu" lang="wuu" hreflang="wuu" data-title="集合（数学）" data-language-autonym="吴语" data-language-local-name="Wuyu" 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 – Kalmuks" lang="xal" hreflang="xal" data-title="Олн" data-language-autonym="Хальмг" data-language-local-name="Kalmuks" 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 – Mingreels" lang="xmf" hreflang="xmf" data-title="მიარე" data-language-autonym="მარგალური" data-language-local-name="Mingreels" 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) – Jiddisch" lang="yi" hreflang="yi" data-title="געזעמל (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" 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) – Chinees" lang="zh" hreflang="zh" data-title="集合 (数学)" data-language-autonym="中文" data-language-local-name="Chinees" 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 – Klassiek Chinees" lang="lzh" hreflang="lzh" data-title="集" data-language-autonym="文言" data-language-local-name="Klassiek Chinees" 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2023 12:36</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Verzameling_(wiskunde)&action=edit&oldid=65357424" title="Verzameling (wiskunde)">bewerken</a></span><span class="mw-diff-timestamp" data-timestamp="2023-08-22T11:36:33Z"></span></strong></div><div id="mw-diff-otitle2"><a href="/wiki/Gebruiker:ChristiaanPR" class="mw-userlink" title="Gebruiker:ChristiaanPR" data-mw-revid="65357424"><bdi>ChristiaanPR</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/Overleg_gebruiker:ChristiaanPR" class="mw-usertoollinks-talk" title="Overleg gebruiker:ChristiaanPR">overleg</a> | <a href="/wiki/Speciaal:Bijdragen/ChristiaanPR" class="mw-usertoollinks-contribs" title="Speciaal:Bijdragen/ChristiaanPR">bijdragen</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Uitgebreid_bevestigde_gebruikers" class="mw-redirect" title="Wikipedia:Uitgebreid bevestigde gebruikers">uitgebreid bevestigde gebruikers</a>, <a href="/wiki/Wikipedia:Uitgezonderden_van_IP-adresblokkades" title="Wikipedia:Uitgezonderden van IP-adresblokkades">Uitgezonderden van IP-adresblokkades</a></div><div class="mw-diff-usereditcount"><span>36.371</span> bewerkingen</div></div></div><div id="mw-diff-otitle3"> <span class="comment comment--without-parentheses"><span class="autocomment"><a href="#Operaties">→<bdi dir="ltr">Operaties</bdi></a>: </span> cirkelverwijzing</span></div><div id="mw-diff-otitle5"></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Verzameling_(wiskunde)&diff=prev&oldid=65357424" title="Verzameling (wiskunde)" id="differences-prevlink">← Oudere bewerking</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Verzameling_(wiskunde)&oldid=68463623" title="Verzameling (wiskunde)">Versie van 2 dec 2024 08:06</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Verzameling_(wiskunde)&action=edit&oldid=68463623" title="Verzameling (wiskunde)">bewerken</a></span><span class="mw-diff-timestamp" data-timestamp="2024-12-02T07:06:46Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Verzameling_(wiskunde)&action=edit&undoafter=65357424&undo=68463623" title="Met "ongedaan maken" draait u deze bewerking terug en komt in het bewerkingsvenster. U kunt in de bewerkingssamenvatting een reden opgeven.">ongedaan maken</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/Gebruiker:Bob.v.R" class="mw-userlink" title="Gebruiker:Bob.v.R" data-mw-revid="68463623"><bdi>Bob.v.R</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/Overleg_gebruiker:Bob.v.R" class="mw-usertoollinks-talk" title="Overleg gebruiker:Bob.v.R">overleg</a> | <a href="/wiki/Speciaal:Bijdragen/Bob.v.R" class="mw-usertoollinks-contribs" title="Speciaal:Bijdragen/Bob.v.R">bijdragen</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Uitgebreid_bevestigde_gebruikers" class="mw-redirect" title="Wikipedia:Uitgebreid bevestigde gebruikers">uitgebreid bevestigde gebruikers</a>, <a href="/wiki/Wikipedia:Uitgezonderden_van_IP-adresblokkades" title="Wikipedia:Uitgezonderden van IP-adresblokkades">Uitgezonderden van IP-adresblokkades</a></div><div class="mw-diff-usereditcount"><span>66.013</span> bewerkingen</div></div></div><div id="mw-diff-ntitle3"><abbr class="minoredit" title="Dit is een kleine bewerking">k</abbr> <span class="comment comment--without-parentheses"><span class="autocomment"><a href="#Getallenverzamelingen">→<bdi dir="ltr">Getallenverzamelingen</bdi></a>: </span> stijl</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"><a href="/w/index.php?title=Verzameling_(wiskunde)&diff=next&oldid=68463623" title="Verzameling (wiskunde)" id="differences-nextlink">Nieuwere bewerking →</a></div></td> </tr><tr><td colspan="4" class="diff-multi" lang="nl">(26 tussenliggende versies door 11 gebruikers niet weergegeven)</td></tr><tr> <td colspan="2" class="diff-lineno">Regel 1:</td> <td colspan="2" class="diff-lineno">Regel 1:</td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>[[Afbeelding:Venn A intersect B.svg|thumb|<del class="diffchange diffchange-inline">De</del> doorsnede <math>A\cap B</math> van twee verzamelingen <math>A</math> en <math>B</math>]]</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>[[Afbeelding:Venn A intersect B.svg|thumb|<ins class="diffchange diffchange-inline">Venndiagram van de</ins> doorsnede <math>A\cap B</math> van twee verzamelingen <math>A</math> en <math>B</math>]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>In de [[wiskunde]] is een '''verzameling''' een abstract [[Wiskundig object|object]] dat het totaal voorstelt van verschillende objecten, die [[Element (wiskunde)|elementen]] van de verzameling<del class="diffchange diffchange-inline"> genoemd</del> worden. Het begrip verzameling is een wiskundig basisbegrip. Dat wil zeggen dat het niet verder gereduceerd (herleid) kan worden tot<del class="diffchange diffchange-inline"> een samenstel van</del> andere, nog fundamentelere theoretische wiskundige begrippen<del class="diffchange diffchange-inline"> (axioma's)</del>, maar dat het zelf [[axioma]]tisch gedefinieerd moet worden. Verzamelingen vormen het studieobject van de [[verzamelingenleer]].</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>In de [[wiskunde]] is een '''verzameling''' een abstract [[Wiskundig object|object]] dat het totaal voorstelt van verschillende objecten, die<ins class="diffchange diffchange-inline"> de</ins> [[Element (wiskunde)|elementen]] van de verzameling worden<ins class="diffchange diffchange-inline"> genoemd</ins>. Het begrip verzameling is een wiskundig basisbegrip. Dat wil zeggen dat het niet verder gereduceerd (herleid) kan worden tot andere, nog fundamentelere theoretische wiskundige begrippen, maar dat het zelf [[axioma]]tisch gedefinieerd moet worden. Verzamelingen vormen het studieobject van de [[verzamelingenleer]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>De verzameling behoort tot de fundamentele concepten van de <del class="diffchange diffchange-inline">[[</del>wiskunde<del class="diffchange diffchange-inline">]]</del>. De grondslag voor dit wiskundige concept werd aan het einde van de negentiende eeuw gelegd door de Duitse wiskundige [[Georg Cantor]]. Hij noemde een verzameling informeel: "een veelheid aan elementen, die volgens een bepaalde definitie bij elkaar horen, en daardoor een geheel vormen".</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>De verzameling behoort tot de fundamentele concepten van de wiskunde. De grondslag voor dit wiskundige concept werd aan het einde van de negentiende eeuw gelegd door de Duitse wiskundige [[Georg Cantor]]. Hij noemde een verzameling informeel: "een veelheid aan elementen, die volgens een bepaalde definitie bij elkaar horen, en daardoor een geheel vormen".</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>De verzamelingenleer is inmiddels alomtegenwoordig in de wiskunde<del class="diffchange diffchange-inline">,</del> en vormt een basis van waaruit bijna de <del class="diffchange diffchange-inline">gehele</del> wiskunde kan worden afgeleid. In het wiskundeonderwijs aan de middelbare scholen worden elementaire onderwerpen als [[venndiagram]]men onderwezen, als aanschouwelijke voorstellingen van verzamelingen<del class="diffchange diffchange-inline">. Meer geavanceerde concepten komen in een universitaire studie wiskunde aan de orde</del>.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>De verzamelingenleer is inmiddels alomtegenwoordig in de wiskunde en vormt een basis van waaruit bijna de <ins class="diffchange diffchange-inline">hele</ins> wiskunde kan worden afgeleid. In het wiskundeonderwijs aan de middelbare scholen worden elementaire onderwerpen als [[venndiagram]]men onderwezen, als aanschouwelijke voorstellingen van verzamelingen.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Twee verzamelingen zijn volgens het [[gelijkheidsaxioma]] identiek als ze dezelfde elementen bevatten. Een verzameling zonder element noemt men een [[lege verzameling]]. Bij de beschrijving van een verzameling gaat het uitsluitend om de vraag welke elementen in de verzameling zijn opgenomen<del class="diffchange diffchange-inline">.</del> <del class="diffchange diffchange-inline">Elementen</del> <del class="diffchange diffchange-inline">komen</del> <del class="diffchange diffchange-inline">daarom</del> <del class="diffchange diffchange-inline">slechts</del> <del class="diffchange diffchange-inline">één</del> <del class="diffchange diffchange-inline">keer</del> in <del class="diffchange diffchange-inline">een</del> <del class="diffchange diffchange-inline">verzameling</del> <del class="diffchange diffchange-inline">voor</del>.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Twee verzamelingen zijn volgens het [[gelijkheidsaxioma]] identiek als ze dezelfde elementen bevatten. Een verzameling zonder element noemt men een [[lege verzameling]]. Bij de beschrijving van een verzameling gaat het uitsluitend om de vraag welke elementen in de verzameling zijn opgenomen<ins class="diffchange diffchange-inline">,</ins> <ins class="diffchange diffchange-inline">niet</ins> <ins class="diffchange diffchange-inline">om</ins> <ins class="diffchange diffchange-inline">de</ins> <ins class="diffchange diffchange-inline">vraag</ins> <ins class="diffchange diffchange-inline">hoe</ins> <ins class="diffchange diffchange-inline">vaak en</ins> in <ins class="diffchange diffchange-inline">welke volgorde ze</ins> <ins class="diffchange diffchange-inline">erin</ins> <ins class="diffchange diffchange-inline">voorkomen</ins>.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>De [[mandelbrotverzameling]] is een bekend voorbeeld van een wiskundige verzameling, en bestaat uit die [[Complex getal|complexe getallen]] die, nadat er herhaald dezelfde [[Operatie (wiskunde)|bewerking]] op is uitgevoerd, naar een eindige waarde [[Iteratie|itereren]].</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>De [[mandelbrotverzameling]] is een bekend voorbeeld van een wiskundige verzameling, en bestaat uit die [[Complex getal|complexe getallen]] die, nadat er herhaald dezelfde [[Operatie (wiskunde)|bewerking]] op is uitgevoerd, naar een eindige waarde [[Iteratie|itereren]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Definitie ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Definitie ==</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Hier wordt alleen een globaal overzicht gegeven van het concept verzameling. Dit overzicht is erop gericht om met verzamelingen te kunnen werken en belangrijke begrippen als [[<del class="diffchange diffchange-inline">afbeelding</del> (wiskunde)|afbeeldingen]], [[Functie (wiskunde)|functies]], [[Getal (wiskunde)|getallen]] en [[Relatie (wiskunde)|relaties]] te kunnen definiëren.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Hier wordt alleen een globaal overzicht gegeven van het concept verzameling. Dit overzicht is erop gericht om met verzamelingen te kunnen werken en belangrijke begrippen als [[<ins class="diffchange diffchange-inline">Afbeelding</ins> (wiskunde)|afbeeldingen]], [[Functie (wiskunde)|functies]], [[Getal (wiskunde)|getallen]] en [[Relatie (wiskunde)|relaties]] te kunnen definiëren.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Georg Cantor gaf aan het begin van zijn ''Beiträge zur Begründung der transfiniten Mengenlehre'':<ref>Geciteerd in Dauben, pag. 170</ref> de volgende definitie van een verzameling:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Georg Cantor gaf aan het begin van zijn ''Beiträge zur Begründung der transfiniten Mengenlehre'':<ref>Geciteerd in Dauben, pag. 170</ref> de volgende definitie van een verzameling:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Cquote|Met een verzameling bedoelen we elke collectie <math>M</math> uit een geheel van concrete, afzonderlijke objecten <math>m</math>, die de elementen van <math>M</math> worden genoemd, van onze [[perceptie]] [Anschauung] of van ons denken.}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{Cquote|Met een verzameling bedoelen we elke collectie <math>M</math> uit een geheel van concrete, afzonderlijke objecten <math>m</math>, die de elementen van <math>M</math> worden genoemd, van onze [[<ins class="diffchange diffchange-inline">Waarneming (perceptie)|</ins>perceptie]] [Anschauung] of van ons denken.}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>De elementen of leden van een verzameling kunnen bijvoorbeeld zijn: getallen, letters van het alfabet, andere verzamelingen en zo verder. Een verzameling wordt gewoonlijk aangeduid door een [[hoofdletter]]. De verzamelingen <math>A</math> en <math>B</math> zijn aan elkaar [[Gelijkheid (verzamelingenleer)|gelijk]] als zij dezelfde elementen hebben.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>De elementen of leden van een verzameling kunnen bijvoorbeeld zijn: getallen, letters van het alfabet, andere verzamelingen en zo verder. Een verzameling wordt gewoonlijk aangeduid door een [[<ins class="diffchange diffchange-inline">Kapitaal (typografie)|</ins>hoofdletter]]. De verzamelingen <math>A</math> en <math>B</math> zijn aan elkaar [[Gelijkheid (verzamelingenleer)|gelijk]] als zij dezelfde elementen hebben.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Zoals hieronder wordt besproken, bleek de hierboven gegeven definitie ontoereikend voor de [[Formalisme (wiskunde)|formele wiskunde]]. In plaats daarvan wordt het begrip 'verzameling' in de [[axiomatische verzamelingenleer]] als een ongedefinieerde primitieve<del class="diffchange diffchange-inline"> (</del>Engels: <del class="diffchange diffchange-inline">'</del>primitive notion<del class="diffchange diffchange-inline">')</del> genomen, en worden haar eigenschappen gedefinieerd door de [[Zermelo-Fraenkel-verzamelingenleer|axioma's van Zermelo-Fraenkel]]. De twee meest fundamentele eigenschappen zijn dat een verzameling elementen <del class="diffchange diffchange-inline">"heeft"</del> en dat twee verzamelingen dan en slechts dan aan elkaar gelijk zijn, als deze dezelfde elementen hebben.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Zoals hieronder wordt besproken, bleek de hierboven gegeven definitie ontoereikend voor de [[Formalisme (wiskunde)|formele wiskunde]]. In plaats daarvan wordt het begrip 'verzameling' in de [[axiomatische verzamelingenleer]] als een ongedefinieerde primitieve<ins class="diffchange diffchange-inline"><ref>[[</ins>Engels<ins class="diffchange diffchange-inline">]]</ins>: primitive notion<ins class="diffchange diffchange-inline"></ref></ins> genomen, en worden haar eigenschappen gedefinieerd door de [[Zermelo-Fraenkel-verzamelingenleer|axioma's van Zermelo-Fraenkel]]. De twee meest fundamentele eigenschappen zijn dat een verzameling<ins class="diffchange diffchange-inline"> door de</ins> elementen <ins class="diffchange diffchange-inline">er in is gedefinieerd</ins> en dat twee verzamelingen<ins class="diffchange diffchange-inline"> [[Dan en slechts</ins> <ins class="diffchange diffchange-inline">dan als|</ins>dan en slechts dan<ins class="diffchange diffchange-inline">]]</ins> aan elkaar gelijk zijn, als deze dezelfde elementen hebben.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_87_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_18_0_rhs"></a>Men dient voorzichtig te zijn met verbale beschrijvingen van verzamelingen, omdat deze gemakkelijk tot [[Paradox (logica)#<ins class="diffchange diffchange-inline">Wiskundige</ins> paradoxen|paradoxen]] kunnen leiden. De [[axiomatische verzamelingenleer]] is geconstrueerd om deze paradoxen te vermijden.</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_22_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_19_0_lhs"></a>== Beschrijving<del class="diffchange diffchange-inline"> van verzamelingen</del> ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Zie hoofdartikel|Element (wiskunde)}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_19_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_22_0_rhs"></a>== Beschrijving ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>In het dagelijkse spraakgebruik komt het begrip 'verzameling' ook voor: met "bestek" wordt in een huishouden de verzameling lepels, vorken en messen bedoeld, het "servies" van oma is een verzameling borden, schalen .... Een "pak" speelkaarten is een verzameling speelkaarten.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>In het dagelijkse spraakgebruik komt het begrip 'verzameling' ook voor: met "bestek" wordt in een huishouden de verzameling lepels, vorken en messen bedoeld, het "servies" van oma is een verzameling borden, schalen .... Een "pak" speelkaarten is een verzameling speelkaarten.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Er zijn twee manieren om de elementen van een verzameling vast te leggen. Eén manier is door een beschrijving, waarbij gebruik wordt gemaakt van een regel of een [[Semantiek|semantische]] beschrijving van de elementen:</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Er zijn twee manieren om de <ins class="diffchange diffchange-inline">[[Element (wiskunde)|</ins>elementen<ins class="diffchange diffchange-inline">]]</ins> van een verzameling vast te leggen. Eén manier is door een beschrijving, waarbij gebruik wordt gemaakt van een regel of een [[Semantiek|semantische]] beschrijving van de elementen:</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>: <math>A</math> is de verzameling waarvan de elementen de eerste vier positieve [[Getal (wiskunde)|getallen]] zijn.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>: <math>A</math> is de verzameling waarvan de elementen de eerste vier positieve [[Getal (wiskunde)|getallen]] zijn.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>: <math>B</math> is de verzameling van alle kleuren van de [[Vlag van Nederland|Nederlandse vlag]]: <math>B=\{x\mid x \text{ is een kleur van de Nederlandse vlag}\}</math>. In plaats van de verticale streep schrijft men ook wel een dubbelepunt: <math>B=\{x: x \text{ is een kleur van de Nederlandse vlag}\}</math>.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>: <math>B</math> is de verzameling van alle kleuren van de [[Vlag van Nederland|Nederlandse vlag]]: <math>B=\{x\mid x \text{ is een kleur van de Nederlandse vlag}\}</math>. In plaats van de verticale streep schrijft men ook wel een dubbelepunt: <math>B=\{x: x \text{ is een kleur van de Nederlandse vlag}\}</math>.</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Regel 41:</td> <td colspan="2" class="diff-lineno">Regel 41:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>: <math>\text{groen} \notin B</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>: <math>\text{groen} \notin B</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Twee verzamelingen zijn aan elkaar gelijk, als ze dezelfde elementen bevatten. Dat twee verzamelingen <math>A</math> en <math>B</math> aan elkaar gelijk zijn, noteert men<del class="diffchange diffchange-inline"> eenvoudigweg</del> als <math>A=B</math>.<del class="diffchange diffchange-inline"> Formeel:</del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Twee verzamelingen zijn aan elkaar gelijk, als ze dezelfde elementen bevatten<ins class="diffchange diffchange-inline">. Bijvoorbeeld <math>\{4, 2, 1, 3\} = \{1, 2, 3, 4\}</math></ins>. Dat twee verzamelingen <math>A</math> en <math>B</math> aan elkaar gelijk zijn, noteert men als <math>A=B</math>.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>: <math>A = B<del class="diffchange diffchange-inline"></math></del> <del class="diffchange diffchange-inline">betekent</del> <del class="diffchange diffchange-inline">dat</del> <del class="diffchange diffchange-inline">voor alle <math></del>x<del class="diffchange diffchange-inline"></math></del> <del class="diffchange diffchange-inline">geldt:</del> <del class="diffchange diffchange-inline"><math></del>x\in A \Longleftrightarrow x\in B</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>: <math>A = B<ins class="diffchange diffchange-inline">\</ins> <ins class="diffchange diffchange-inline">\Longleftrightarrow\</ins> <ins class="diffchange diffchange-inline">\forall</ins> x<ins class="diffchange diffchange-inline">\</ins> <ins class="diffchange diffchange-inline">\left(</ins> x<ins class="diffchange diffchange-inline"> </ins>\in A \Longleftrightarrow x<ins class="diffchange diffchange-inline"> </ins>\in B<ins class="diffchange diffchange-inline"> \right)</ins></math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Ook geldt omgekeerd:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>: Als voor alle <math>x</math> geldt: <math>x\in A \Longleftrightarrow x\in B</math>, dan is <math>A=B</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Anders dan bij een [[multiset]] komt elk element van een verzameling maar één keer voor als element van de verzameling, ook al wordt een element meer keren genoemd. Zo is de verzameling letters <math>\{a,b,a,c,a\}</math> dezelfde als de verzameling <math>\{a,b,c\}</math> en de verzameling <math>\{b,a,c,c\}</math>. Ieder element van een verzameling <math>A</math> blijft onder alle [[Algebra van verzamelingen|bewerkingen]] op <math>A</math> uniek. De volgorde waarin de elementen van een verzameling worden opgesomd, telt niet, dit in tegenstelling tot bij een [[Rij (wiskunde)|rij]] of een [[tupel]]. Elementen staan in een rij opeenvolgend opgesomd en mogen in tegenstelling tot in een verzameling wel meer dan één keer in een rij voorkomen.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Anders dan bij een [[multiset]] komt elk element van een verzameling maar één keer voor als element van de verzameling, ook al wordt een element meer keren genoemd. Zo is de verzameling letters <math>\{a,b,a,c,a\}</math> dezelfde als de verzameling <math>\{a,b,c\}</math> en de verzameling <math>\{b,a,c,c\}</math>. Ieder element van een verzameling <math>A</math> blijft onder alle [[Algebra van verzamelingen|bewerkingen]] op <math>A</math> uniek. De volgorde waarin de elementen van een verzameling worden opgesomd, telt niet, dit in tegenstelling tot bij een [[Rij (wiskunde)|rij]] of een [[tupel]]. Elementen staan in een rij opeenvolgend opgesomd en mogen in tegenstelling tot in een verzameling wel meer dan één keer in een rij voorkomen.</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Regel 52:</td> <td colspan="2" class="diff-lineno">Regel 50:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>De lege verzameling, die geen elementen heeft, wordt met het symbool ∅ genoteerd. Minder gebruikelijk is de notatie {}.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>De lege verzameling, die geen elementen heeft, wordt met het symbool ∅ genoteerd. Minder gebruikelijk is de notatie {}.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Het aantal elementen in een verzameling noemt men de [[#<del class="diffchange diffchange-inline">kardinaliteit</del>|kardinaliteit]] van de verzameling.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Het aantal elementen in een verzameling noemt men de [[#<ins class="diffchange diffchange-inline">Kardinaliteit</ins>|kardinaliteit]] van de verzameling.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">=</del>== Deelverzamelingen <del class="diffchange diffchange-inline">=</del>==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== Deelverzamelingen ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_36_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_32_0_rhs"></a><div style="float:right;margin:1em;">[[<ins class="diffchange diffchange-inline">Afbeelding</ins>:Venn A subset B.svg|150px|center]]<div class="center"><small> <math>A</math> is een deelverzameling van <math>B</math></small></div></div></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Zie hoofdartikel|Deelverzameling}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Als elk element van de verzameling <math>A</math> ook element is van de verzameling <math>B</math>, zegt men dat <math>A</math> een [[deelverzameling]] is van <math>B</math>. Dit wordt genoteerd als <math>A\subseteq B</math> of als <math>A\subset B</math>, en uitgesproken als <math>A</math> is een deel(verzameling) van <math>B</math>, of als <math>A</math> wordt door <math>B</math> omvat. In plaats daarvan kan ook worden geschreven: <math>B \supseteq A</math>, of <math>B \supset A</math> zeg: <math>B</math> omvat <math>A</math>, <math>B</math> sluit <math>A</math> in, of <math>B</math> is een [[superset]] van <math>A</math>. De [[Relatie (wiskunde)|relatie]] tussen verzamelingen die wordt vastgelegd door ⊆ wordt inclusie of omvatting genoemd.</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_32_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_36_0_lhs"></a><div style="float:right;margin:1em;">[[<del class="diffchange diffchange-inline">Bestand</del>:Venn A subset B.svg|150px|center]]<div class="center"><small> <math>A</math> is een deelverzameling van <math>B</math></small></div></div></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Als<del class="diffchange diffchange-inline"> elk element van de verzameling</del> <math>A</math> <del class="diffchange diffchange-inline">ook</del> <del class="diffchange diffchange-inline">element</del> is van<del class="diffchange diffchange-inline"> de verzameling</del> <math>B</math>, <del class="diffchange diffchange-inline">zegt</del> <del class="diffchange diffchange-inline">men</del> <del class="diffchange diffchange-inline">dat</del> <del class="diffchange diffchange-inline"><math>A</math> een deelverzameling</del> is<del class="diffchange diffchange-inline"> van <math>B</math>. Dit</del> wordt<del class="diffchange diffchange-inline"> genoteerd als</del> <math>A<del class="diffchange diffchange-inline">\subseteq B</del></math><del class="diffchange diffchange-inline"> of als <math>A\subset B</math>, en uitgesproken als <math>A</math> is</del> een <del class="diffchange diffchange-inline">deel(verzameling) van <math>B</math>,</del> of <del class="diffchange diffchange-inline">als</del> <del class="diffchange diffchange-inline"><math>A</math></del> <del class="diffchange diffchange-inline">wordt door</del> <math>B</math> <del class="diffchange diffchange-inline">omvat</del>. <del class="diffchange diffchange-inline">In</del> <del class="diffchange diffchange-inline">plaats</del> <del class="diffchange diffchange-inline">daarvan</del> <del class="diffchange diffchange-inline">kan</del> <del class="diffchange diffchange-inline">ook worden geschreven:</del> <math><del class="diffchange diffchange-inline">B</del> \<del class="diffchange diffchange-inline">supseteq</del> <del class="diffchange diffchange-inline">A</del></math>, of <math>B \<del class="diffchange diffchange-inline">supset</del> A</math><del class="diffchange diffchange-inline"> zeg</del>:<del class="diffchange diffchange-inline"> <math>B</math> omvat <math>A</math>, <math>B</math> sluit <math>A</math> in, of</del> <math>B</math> is een <del class="diffchange diffchange-inline">[[</del>superset<del class="diffchange diffchange-inline">]]</del> van <math>A</math><del class="diffchange diffchange-inline">. De [[Relatie (wiskunde)|relatie]] tussen verzamelingen die wordt vastgelegd door ⊆ wordt inclusie of omvatting genoemd</del>.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Als <math>A</math> <ins class="diffchange diffchange-inline">een</ins> <ins class="diffchange diffchange-inline">deelverzameling</ins> is van <math>B</math>, <ins class="diffchange diffchange-inline">maar</ins> <ins class="diffchange diffchange-inline">niet</ins> <ins class="diffchange diffchange-inline">daaraan</ins> <ins class="diffchange diffchange-inline">gelijk</ins> is<ins class="diffchange diffchange-inline">,</ins> wordt <math>A</math> een <ins class="diffchange diffchange-inline">echte</ins> of <ins class="diffchange diffchange-inline">strikte</ins> <ins class="diffchange diffchange-inline">deelverzameling</ins> <ins class="diffchange diffchange-inline">van</ins> <math>B</math> <ins class="diffchange diffchange-inline">genoemd</ins>. <ins class="diffchange diffchange-inline">Dit</ins> <ins class="diffchange diffchange-inline">wordt</ins> <ins class="diffchange diffchange-inline">wel</ins> <ins class="diffchange diffchange-inline">genoteerd</ins> <ins class="diffchange diffchange-inline">als</ins> <math><ins class="diffchange diffchange-inline">A</ins> \<ins class="diffchange diffchange-inline">subsetneq</ins> <ins class="diffchange diffchange-inline">B</ins></math>, of <math>B \<ins class="diffchange diffchange-inline">supsetneq</ins> A</math>: <math>B</math> is een <ins class="diffchange diffchange-inline">strikte </ins>superset van <math>A</math>.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Als <math>A</math> een deelverzameling is van <math>B</math>, maar niet daaraan gelijk is, wordt <math>A</math> een ''echte of strikte deelverzameling'' van <math>B</math> genoemd. Dit wordt wel genoteerd als <math>A \subsetneq B</math>, of <math>B \supsetneq A</math>: <math>B</math> is een strikte superset van <math>A</math>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Voorbeeld:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Voorbeeld:</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Regel 71:</td> <td colspan="2" class="diff-lineno">Regel 67:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>De lege verzameling is een deelverzameling van elke verzameling en elke verzameling is een deelverzameling van zichzelf:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>De lege verzameling is een deelverzameling van elke verzameling en elke verzameling is een deelverzameling van zichzelf:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* <math>\varnothing\subseteq A</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <math>\varnothing<ins class="diffchange diffchange-inline"> </ins>\subseteq A</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* <math>A\subseteq A</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <math>A<ins class="diffchange diffchange-inline"> </ins>\subseteq A</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Een vanzelfsprekende identiteit, die vaak kan worden gebruikt om aan te tonen dat twee ogenschijnlijk verschillende verzamelingen toch aan elkaar gelijk zijn:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Een vanzelfsprekende identiteit, die vaak kan worden gebruikt om aan te tonen dat twee ogenschijnlijk verschillende verzamelingen toch aan elkaar gelijk zijn:</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* <math>A=B</math> dan en slechts dan als <math>A\subseteq B</math> en <math>B\subseteq A</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* <math>A=B</math> dan en slechts dan als <math>A\subseteq B</math> en <math>B\subseteq A</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Kardinaliteit</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== <ins class="diffchange diffchange-inline">Bewerkingen</ins> ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_17_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_44_0_rhs"></a>* De <ins class="diffchange diffchange-inline">[[Vereniging (verzamelingenleer)|</ins>vereniging<ins class="diffchange diffchange-inline">]]</ins> van twee verzamelingen <math>A</math> en <math>B</math> wordt gevormd door de elementen die in <math>A</math> of in <math>B</math> (of in beide) zitten. Notatie: <math>A\cup B</math>.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Zie hoofdartikel|Kardinaliteit}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_69_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_1_lhs"></a>De kardinaliteit <math>|A|</math> van een verzameling <math>A</math> is "het aantal elementen van <math>A</math>". Aangezien bijvoorbeeld de Nederlandse vlag drie kleuren kent, is de kardinaliteit van de verzameling <math>B=\{\text{kleuren van de Nederlandse vlag}\}</math> gelijk aan <math>|B|=3</math>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_72_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_3_lhs"></a>De [[lege verzameling]] <math>\varnothing</math> heeft kardinaliteit 0. Hoewel het misschien triviaal lijkt, is de lege verzameling, net zoals het getal [[0 (getal)|nul]], belangrijk in de wiskunde<del class="diffchange diffchange-inline">;</del> <del class="diffchange diffchange-inline">het</del> bestaan van de lege verzameling is zelfs een van de fundamentele concepten uit de [[axiomatische verzamelingenleer]].</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_76_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_5_lhs"></a>Sommige verzamelingen hebben een [[Oneindige verzameling|oneindige]] kardinaliteit. De verzameling <math>\N</math> van de [[Natuurlijk getal|natuurlijke getallen]] is bijvoorbeeld oneindig. Men kan echter aantonen dat sommige oneindige kardinaliteiten groter zijn dan andere. De verzameling van de [[Reëel getal|reële getallen]] bijvoorbeeld heeft een grotere kardinaliteit dan de verzameling van de natuurlijke getallen. Het kan worden aangetoond dat de kardinaliteit van <del class="diffchange diffchange-inline">(</del>dat wil zeggen: het aantal punten op<del class="diffchange diffchange-inline">)</del> een [[<del class="diffchange diffchange-inline">rechte</del> lijn]] dezelfde is als de kardinaliteit van enig [[lijnstuk]] van die lijn, dezelfde als die van het gehele [[Vlak (meetkunde)|vlak]] en ook dezelfde als die van enige eindig-dimensionale [[euclidische ruimte]].</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>=== Machtsverzamelingen ===</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Zie hoofdartikel|Machtsverzameling}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_81_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_10_lhs"></a>De machtsverzameling van een verzameling <math>A</math> is de verzameling van alle deelverzamelingen van <math>A</math>. Daartoe behoort de verzameling <math>A</math> zelf en de lege verzameling. Als een [[eindige verzameling]] <math>A</math> een [[kardinaliteit]] <math>n</math> heeft, is de kardinaliteit van de machtsverzameling van <math>A</math> gelijk aan <math>2^n</math>. De machtsverzameling wordt genoteerd als <math>\mathcal{P}(A)</math> of als <math>2^A</math>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_85_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_12_lhs"></a>Als <math>A</math> een oneindige <del class="diffchange diffchange-inline">(</del>[[Aftelbare verzameling|aftelbare]] dan wel [[<del class="diffchange diffchange-inline">Overaftelbaarheid|</del>overaftelbare<del class="diffchange diffchange-inline">]])</del> verzameling is, is de machtsverzameling van <math>A</math> altijd overaftelbaar. Als <math>A</math> bovendien een verzameling is, dan is er nooit een [[bijectie]] van <math>A</math> op <math>\mathcal{P}(A)</math> mogelijk. Met andere woorden: de machtsverzameling van <math>A</math> is altijd strikt "groter" dan <math>A</math> zelf.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_89_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_14_lhs"></a>De machtsverzameling van de verzameling {1, 2, 3} is bijvoorbeeld { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅ }. De kardinaliteit van de oorspronkelijke verzameling is 3 en de kardinaliteit van de machtsverzameling is 2<sup>3</sup> = 8. Deze relatie is een van de redenen voor de terminologie <del class="diffchange diffchange-inline">''</del>machtsverzameling<del class="diffchange diffchange-inline">''</del>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Operaties ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_44_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_46_17_lhs"></a>* De vereniging van twee verzamelingen <math>A</math> en <math>B</math> wordt gevormd door de elementen die in <math>A</math> of in <math>B</math> (of in beide) zitten. Notatie: <math>A\cup B</math>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* De [[Doorsnede (verzamelingenleer)|doorsnede]] van twee verzamelingen <math>A</math> en <math>B</math> wordt gevormd door de verzameling van gemeenschappelijke elementen, dus alle elementen die zowel in <math>A</math> als in <math>B</math> zitten. Notatie: <math>A\cap B</math>.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* De [[Doorsnede (verzamelingenleer)|doorsnede]] van twee verzamelingen <math>A</math> en <math>B</math> wordt gevormd door de verzameling van gemeenschappelijke elementen, dus alle elementen die zowel in <math>A</math> als in <math>B</math> zitten. Notatie: <math>A\cap B</math>.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* <del class="diffchange diffchange-inline">Een</del> <del class="diffchange diffchange-inline">verzameling</del> <del class="diffchange diffchange-inline">is een deel</del> van <del class="diffchange diffchange-inline">het</del> <del class="diffchange diffchange-inline">universum</del> <math><del class="diffchange diffchange-inline">U</del></math><del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">waarmee in dit verband wordt bedoeld de verzameling met alle mogelijke relevante elementen. De complementaire verzameling van een verzameling</del> <math><del class="diffchange diffchange-inline">A</del></math> <del class="diffchange diffchange-inline">is</del> <del class="diffchange diffchange-inline">dan</del> <del class="diffchange diffchange-inline">de verzameling van</del> alle elementen <del class="diffchange diffchange-inline">in</del> <math><del class="diffchange diffchange-inline">U</del></math> die niet in <math><del class="diffchange diffchange-inline">A</del></math> zitten<del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">notatie</del>: <math>A<del class="diffchange diffchange-inline">^c=</del>\<del class="diffchange diffchange-inline">{x\in</del> <del class="diffchange diffchange-inline">U\mid x\notin A\}</del></math><del class="diffchange diffchange-inline">.</del> <del class="diffchange diffchange-inline"><math>A^c</math> wordt in het algemeen als het complement van</del> <math>A<del class="diffchange diffchange-inline"></math></del> <del class="diffchange diffchange-inline">aangeduid.</del> <del class="diffchange diffchange-inline">Andere notaties voor het complement zijn <math>\bar{A}</math> en <math>A'</del></math>.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* <ins class="diffchange diffchange-inline">Het</ins> <ins class="diffchange diffchange-inline">[[Verschil</ins> <ins class="diffchange diffchange-inline">(verzamelingenleer)|verschil]]</ins> van <ins class="diffchange diffchange-inline">twee</ins> <ins class="diffchange diffchange-inline">verzamelingen</ins> <math><ins class="diffchange diffchange-inline">A</ins></math> <ins class="diffchange diffchange-inline">en</ins> <math><ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">word</ins> <ins class="diffchange diffchange-inline">gevormd</ins> <ins class="diffchange diffchange-inline">door</ins> alle elementen <ins class="diffchange diffchange-inline">van</ins> <math><ins class="diffchange diffchange-inline">A</ins></math> die niet in <math><ins class="diffchange diffchange-inline">B</ins></math> zitten<ins class="diffchange diffchange-inline">.</ins> <ins class="diffchange diffchange-inline">Notatie</ins>: <math>A\<ins class="diffchange diffchange-inline">setminus</ins> <ins class="diffchange diffchange-inline">B</ins></math> <ins class="diffchange diffchange-inline">of</ins> <math>A <ins class="diffchange diffchange-inline">-</ins> <ins class="diffchange diffchange-inline">B</ins></math>.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_74_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_49_0_rhs"></a><ins class="diffchange diffchange-inline">* </ins>Het relatieve <ins class="diffchange diffchange-inline">[[Complement (verzamelingenleer)|</ins>complement<ins class="diffchange diffchange-inline">]]</ins> van <math>B</math> ten opzichte van <math>A</math> is de verzameling van de elementen van <math>A</math> die niet tot <math>B</math> behoren. Het wordt genoteerd als:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_74_1_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_49_1_rhs"></a><ins class="diffchange diffchange-inline">:</ins>: <math>A<ins class="diffchange diffchange-inline"> </ins>\setminus B = \{x<ins class="diffchange diffchange-inline"> </ins>\in A<ins class="diffchange diffchange-inline"> </ins>\mid x<ins class="diffchange diffchange-inline"> </ins>\notin B\}</math></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>: Lees: <math>A</math> met daaruit weggelaten <math>B</math>. Het relatieve complement wordt ook wel genoteerd als: <math>A-B</math>. Kortheidshalve wordt genoteerd: <math>B^c = \{x \mid x \notin B\}</math>, waarbij niet wordt vermeld ten opzichte van welke verzameling <math>A</math> het complement wordt genomen. Voor <math>A</math> kan hierbij iedere verzameling worden genomen, zolang <math>B \subseteq A</math>.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Er gelden de volgende eigenschappen:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Er gelden de volgende eigenschappen:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:{| class="wikitable"</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>:<ins class="diffchange diffchange-inline"> </ins>{| class="wikitable"</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>! Eigenschap || Doorsnede || Vereniging</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>! Eigenschap || Doorsnede || Vereniging</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>|-</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>|-</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Regel 125:</td> <td colspan="2" class="diff-lineno">Regel 106:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>|}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>|}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Een [[Partitie (verzamelingenleer)|partitie]] is een opdeling van een verzameling in niet-lege, onderling disjuncte, deelverzamelingen, die wel <del class="diffchange diffchange-inline">''</del>blokken<del class="diffchange diffchange-inline">''</del> worden genoemd. Bijvoorbeeld: als <math>A=\{1,2,3,4,5,6,7,8\}</math>, dan vormen de deelverzamelingen {1,3}, {2,4,5,7} en {6,8} een partitie van <math>A</math> met drie blokken.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Een [[Partitie (verzamelingenleer)|partitie]] is een opdeling van een verzameling in niet-lege, onderling disjuncte, deelverzamelingen, die wel blokken worden genoemd. Bijvoorbeeld: als <math>A=\{1,2,3,4,5,6,7,8\}</math>, dan vormen de deelverzamelingen {1,3}, {2,4,5,7} en {6,8} een partitie van <math>A</math> met drie blokken.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>De deelverzamelingen van een gegeven verzameling vormen een [[booleaanse algebra]] onder doorsnede en vereniging.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>De deelverzamelingen van een gegeven verzameling vormen een [[booleaanse algebra]] onder doorsnede en vereniging.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Bekende</del> <del class="diffchange diffchange-inline">verzamelingen</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== <ins class="diffchange diffchange-inline">Wetten</ins> <ins class="diffchange diffchange-inline">van De Morgan</ins> ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_82_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_56_0_rhs"></a>De [[wetten van De Morgan]] luiden:</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_95_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_57_0_lhs"></a>Voorbeelden van getallenverzamelingen zijn:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_83_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_58_0_rhs"></a>* <math>(A<ins class="diffchange diffchange-inline"> </ins>\cup B)^c = A^c<ins class="diffchange diffchange-inline"> </ins>\cap B^c</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div># De [[Natuurlijk getal|natuurlijke getallen]] die in het algemeen aantallen voorstellen en gesloten zijn onder optelling en vermenigvuldiging.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_83_1_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_60_0_rhs"></a>* <math>(A<ins class="diffchange diffchange-inline"> </ins>\cap B)^c = A^c<ins class="diffchange diffchange-inline"> </ins>\cup B^c</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div># De [[Geheel getal|gehele getallen]], die ook gesloten zijn onder aftrekking</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_83_2_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_62_0_rhs"></a>* <math>A<ins class="diffchange diffchange-inline"> </ins>\setminus(B\cup C) = (A<ins class="diffchange diffchange-inline"> </ins>\setminus B)\cap (A\setminus C)</math></div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_99_1_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_63_0_lhs"></a># De [[Rationaal getal|rationale getallen]], die bestaan uit de gehele getallen en de [[Breuk (wiskunde)|breuken]].</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_83_3_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_64_0_rhs"></a>* <math>A<ins class="diffchange diffchange-inline"> </ins>\setminus(B\cap C) = (A<ins class="diffchange diffchange-inline"> </ins>\setminus B)\cup (A\setminus C)</math></div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_99_2_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_65_0_lhs"></a># De [[Reëel getal|reële getallen]], waaronder ook de [[Transcendent getal|transcendente getallen]] vallen.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_99_3_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_66_0_lhs"></a># De [[Complex getal|complexe getallen]] verschijnen als oplossing van vergelijkingen als <math>x^2+1=0</math>.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Venndiagrammen</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== <ins class="diffchange diffchange-inline">Kardinaliteit</ins> ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_1_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_69_0_rhs"></a>De <ins class="diffchange diffchange-inline">[[</ins>kardinaliteit<ins class="diffchange diffchange-inline">]]</ins> <math>|A|</math> van een verzameling <math>A</math> is "het aantal elementen van <math>A</math>". Aangezien bijvoorbeeld de Nederlandse vlag drie kleuren kent, is de kardinaliteit van de verzameling <math>B=\{\text{kleuren van de Nederlandse vlag}\}</math> gelijk aan <math>|B|=3</math>.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Met behulp van [[venndiagram]]men, genoemd naar [[John Venn]], kunnen verzamelingen aanschouwelijk voorgesteld worden. In de bovenstaande afbeelding van een venndiagram is de [[Doorsnede (verzamelingenleer)|doorsnede]] van twee verzamelingen <math>A</math> en <math>B</math> lichtpaars weergegeven.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_3_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_72_0_rhs"></a>De [[lege verzameling]] <math>\varnothing</math> heeft kardinaliteit 0. Hoewel het misschien triviaal lijkt, is de lege verzameling, net zoals het getal [[0 (getal)|nul]], belangrijk in de wiskunde<ins class="diffchange diffchange-inline">.</ins> <ins class="diffchange diffchange-inline">Het</ins> bestaan van de lege verzameling is zelfs een van de fundamentele concepten uit de [[axiomatische verzamelingenleer]].</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Relatief complement ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_49_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_74_0_lhs"></a>Het <del class="diffchange diffchange-inline">''</del>relatieve complement<del class="diffchange diffchange-inline">''</del> van <math>B</math> ten opzichte van <math>A</math> is<del class="diffchange diffchange-inline"><ref>[https://web.archive.org/web/20170921143826/https://mod-est.tbm.tudelft.nl/wiki/index.php/Verzameling Verzameling], reader Systeemmodellering, TU Delft</ref></del> de verzameling van de elementen van <math>A</math> die niet tot <math>B</math> behoren. Het wordt genoteerd als:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_49_1_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_74_1_lhs"></a>: <math>A\setminus B = \{x\in A\mid x\notin B\}</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_5_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_76_0_rhs"></a>Sommige verzamelingen hebben een [[Oneindige verzameling|oneindige]] kardinaliteit. De verzameling <math>\N</math> van de [[Natuurlijk getal|natuurlijke getallen]] is bijvoorbeeld oneindig. Men kan echter aantonen dat sommige oneindige kardinaliteiten groter zijn dan andere. De verzameling van de [[Reëel getal|reële getallen]] bijvoorbeeld heeft een grotere kardinaliteit dan de verzameling van de natuurlijke getallen. Het kan worden aangetoond dat de kardinaliteit van<ins class="diffchange diffchange-inline">,</ins> dat wil zeggen: het aantal punten op<ins class="diffchange diffchange-inline">,</ins> een [[<ins class="diffchange diffchange-inline">Lijn</ins> <ins class="diffchange diffchange-inline">(meetkunde)|</ins>lijn]] dezelfde is als de kardinaliteit van enig [[lijnstuk]] van die lijn, dezelfde als die van het gehele [[Vlak (meetkunde)|vlak]] en ook dezelfde als die van enige eindig-dimensionale [[euclidische ruimte]].</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>(lees: ''A'' met daaruit weggelaten ''B'').</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Het relatieve complement wordt ook wel genoteerd als: <math>A-B</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Wetten van De Morgan</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><ins class="diffchange diffchange-inline">=</ins>== <ins class="diffchange diffchange-inline">Machtsverzamelingen</ins> <ins class="diffchange diffchange-inline">=</ins>==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_10_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_81_0_rhs"></a>De <ins class="diffchange diffchange-inline">[[</ins>machtsverzameling<ins class="diffchange diffchange-inline">]]</ins> van een verzameling <math>A</math> is de verzameling van alle deelverzamelingen van <math>A</math>. Daartoe behoort de verzameling <math>A</math> zelf en de lege verzameling. Als een [[eindige verzameling]] <math>A</math> een [[kardinaliteit]] <math>n</math> heeft, is de kardinaliteit van de machtsverzameling van <math>A</math> gelijk aan <math>2^n</math>. De machtsverzameling wordt genoteerd als <math>\mathcal{P}(A)</math> of als <math>2^A</math>.</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_56_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_82_0_lhs"></a>De [[wetten van De Morgan]] luiden:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_58_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_83_0_lhs"></a>* <math>(A\cup B)^c = A^c\cap B^c</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_60_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_83_1_lhs"></a>* <math>(A\cap B)^c = A^c\cup B^c</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_62_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_83_2_lhs"></a>* <math>A\setminus(B\cup C) = (A\setminus B)\cap (A\setminus C)</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_64_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_83_3_lhs"></a>* <math>A\setminus(B\cap C) = (A\setminus B)\cup (A\setminus C)</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_12_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_85_0_rhs"></a>Als <math>A</math> een oneindige<ins class="diffchange diffchange-inline">, een</ins> [[Aftelbare verzameling|aftelbare]] dan wel [[overaftelbare verzameling<ins class="diffchange diffchange-inline">]]</ins> is, is de machtsverzameling van <math>A</math> altijd overaftelbaar. Als <math>A</math> bovendien een verzameling is, dan is er nooit een [[bijectie]] van <math>A</math> op <math>\mathcal{P}(A)</math> mogelijk. Met andere woorden: de machtsverzameling van <math>A</math> is altijd strikt "groter" dan <math>A</math> zelf.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Modellering ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_18_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_87_0_lhs"></a>Men dient voorzichtig te zijn met verbale beschrijvingen van verzamelingen, omdat deze gemakkelijk tot [[Paradox (logica)#<del class="diffchange diffchange-inline">wiskundige</del> paradoxen|paradoxen]] kunnen leiden. De [[axiomatische verzamelingenleer]] is geconstrueerd om deze paradoxen te vermijden.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_46_14_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_89_0_rhs"></a>De machtsverzameling van de verzameling {1, 2, 3} is bijvoorbeeld { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅ }. De kardinaliteit van de oorspronkelijke verzameling is 3 en de kardinaliteit van de machtsverzameling is 2<sup>3</sup> = 8. Deze relatie is een van de redenen voor de terminologie machtsverzameling.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Relaties met andere takken van de wiskunde ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_101_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_91_0_lhs"></a>Vrijwel alle andere takken van de wiskunde worden gebaseerd op de [[verzamelingenleer]]. Zo is bijvoorbeeld in de [[kansrekening]] de uitkomstenruimte de universele verzameling van alle mogelijkheden en zijn de gebeurtenissen de (deel)verzamelingen. Andere elementaire begrippen in de wiskunde, zoals [[Functie (wiskunde)|functies]] en [[Rij (wiskunde)|<del class="diffchange diffchange-inline">rij</del>]]<del class="diffchange diffchange-inline">en</del>, worden ook in termen van verzamelingen gedefinieerd. </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== Getallenverzamelingen ==</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_103_1_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_94_0_lhs"></a>De [[ordetheorie]] houdt zich bezig met de verschillende manieren om de elementen van een verzameling te ordenen. Een [[Tweeplaatsige relatie|relatie]] legt daartoe de volgorde tussen de elementen van de verzameling vast en geeft zo aan welke van de elementen opvolger is van het andere.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_57_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_95_0_rhs"></a>Voorbeelden van getallenverzamelingen zijn:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div># [[Natuurlijk getal|Natuurlijke getallen]]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Zie ook|Zie ook [[Bovengrens en ondergrens]].}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div># [[Geheel getal|Gehele getallen]]</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_63_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_99_1_rhs"></a># De [[Rationaal getal|rationale getallen]], die bestaan uit de gehele getallen en de [[Breuk (wiskunde)|breuken]].</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_65_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_99_2_rhs"></a># De [[Reëel getal|reële getallen]], waaronder ook de [[Transcendent getal|transcendente getallen]] vallen.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_66_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_99_3_rhs"></a># De [[Complex getal|complexe getallen]] verschijnen als oplossing van vergelijkingen als <math>x^2+1=0</math>.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Toepassingen ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Toepassingen ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_91_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_101_0_rhs"></a>Vrijwel alle andere takken van de wiskunde worden gebaseerd op de [[verzamelingenleer]]. Zo is bijvoorbeeld in de [[kansrekening]] de uitkomstenruimte de <ins class="diffchange diffchange-inline">[[</ins>universele verzameling<ins class="diffchange diffchange-inline">]]</ins> van alle mogelijkheden en zijn de gebeurtenissen de (deel)verzamelingen. Andere elementaire begrippen in de wiskunde, zoals [[Functie (wiskunde)|functies]] en [[Rij (wiskunde)|<ins class="diffchange diffchange-inline">rijen</ins>]], worden ook in termen van verzamelingen gedefinieerd. </div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_103_3_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_102_0_lhs"></a>* Er wordt in de [[topologie]] verschil tussen [[<del class="diffchange diffchange-inline">open</del> verzameling]]<del class="diffchange diffchange-inline">en</del> en [[gesloten verzameling]]en gemaakt.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_94_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_103_1_rhs"></a>De [[ordetheorie]] houdt zich bezig met de verschillende manieren om de elementen van een verzameling te ordenen. Een [[Tweeplaatsige relatie|relatie]] legt daartoe de volgorde tussen de elementen van de verzameling<ins class="diffchange diffchange-inline"> in een [[Rij (wiskunde)|rij]]</ins> vast en geeft zo aan welke van de elementen opvolger is van het andere<ins class="diffchange diffchange-inline">. Dat begint met het bepalen van een [[bovengrens en ondergrens]] van de verzameling</ins>.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_102_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_103_3_rhs"></a>* Er wordt in de [[topologie]] verschil tussen [[<ins class="diffchange diffchange-inline">Open</ins> verzameling<ins class="diffchange diffchange-inline">|open</ins>]] en [[gesloten verzameling]]en gemaakt.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* Verzamelingen zijn in de [[informatica]] geïmplementeerd en heten daar ook weer [[Verzameling (informatica)|verzamelingen]].</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* Verzamelingen zijn in de [[informatica]] geïmplementeerd en heten daar ook weer [[Verzameling (informatica)|verzamelingen]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker"><a class="mw-diff-movedpara-right" title="Deze alinea is verplaatst. Klik om naar de oude locatie te springen." href="#movedpara_112_0_lhs">⚫</a></td> <td class="diff-addedline diff-side-added"><div><a name="movedpara_105_0_rhs"></a>{{Appendix|<ins class="diffchange diffchange-inline">|</ins></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Literatuur ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>; voetnoten</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{References}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>; literatuur</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* {{en}} {{aut|JW Dauben}}. Georg Cantor: His Mathematics and Philosophy of the Infinite, 1979. {{ISBN|978-0-691-02447-9}}.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* {{en}} {{aut|JW Dauben}}. Georg Cantor: His Mathematics and Philosophy of the Infinite, 1979. {{ISBN|978-0-691-02447-9}}.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>; websites</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* [[MathWorld]]. [https://mathworld.wolfram.com/Set.html Set].</div></td> </tr> <tr> <td class="diff-marker"><a class="mw-diff-movedpara-left" title="Deze alinea is verplaatst. Klik om naar de nieuwe locatie te springen." href="#movedpara_105_0_rhs">⚫</a></td> <td class="diff-deletedline diff-side-deleted"><div><a name="movedpara_112_0_lhs"></a>{{Appendix|<del class="diffchange diffchange-inline">Voetnoten}}</del></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{Commonscat}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{Commonscat}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Versie van 2 dec 2024 08:06</h2> <div class="mw-content-ltr mw-parser-output" lang="nl" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Bestand:Venn_A_intersect_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/260px-Venn_A_intersect_B.svg.png" decoding="async" width="260" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/390px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/520px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a><figcaption>Venndiagram van de doorsnede <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27b38cf9eac6060e67b61f66cd9beec5067f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cap B}"></span> van twee verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></figcaption></figure> <p>In de <a href="/wiki/Wiskunde" title="Wiskunde">wiskunde</a> is een <b>verzameling</b> een abstract <a href="/wiki/Wiskundig_object" class="mw-redirect" title="Wiskundig object">object</a> dat het totaal voorstelt van verschillende objecten, die de <a href="/wiki/Element_(wiskunde)" title="Element (wiskunde)">elementen</a> van de verzameling worden genoemd. Het begrip verzameling is een wiskundig basisbegrip. Dat wil zeggen dat het niet verder gereduceerd (herleid) kan worden tot andere, nog fundamentelere theoretische wiskundige begrippen, maar dat het zelf <a href="/wiki/Axioma" title="Axioma">axiomatisch</a> gedefinieerd moet worden. Verzamelingen vormen het studieobject van de <a href="/wiki/Verzamelingenleer" title="Verzamelingenleer">verzamelingenleer</a>. </p><p>De verzameling behoort tot de fundamentele concepten van de wiskunde. De grondslag voor dit wiskundige concept werd aan het einde van de negentiende eeuw gelegd door de Duitse wiskundige <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>. Hij noemde een verzameling informeel: "een veelheid aan elementen, die volgens een bepaalde definitie bij elkaar horen, en daardoor een geheel vormen". </p><p>De verzamelingenleer is inmiddels alomtegenwoordig in de wiskunde en vormt een basis van waaruit bijna de hele wiskunde kan worden afgeleid. In het wiskundeonderwijs aan de middelbare scholen worden elementaire onderwerpen als <a href="/wiki/Venndiagram" title="Venndiagram">venndiagrammen</a> onderwezen, als aanschouwelijke voorstellingen van verzamelingen. </p><p>Twee verzamelingen zijn volgens het <a href="/wiki/Gelijkheidsaxioma" title="Gelijkheidsaxioma">gelijkheidsaxioma</a> identiek als ze dezelfde elementen bevatten. Een verzameling zonder element noemt men een <a href="/wiki/Lege_verzameling" title="Lege verzameling">lege verzameling</a>. Bij de beschrijving van een verzameling gaat het uitsluitend om de vraag welke elementen in de verzameling zijn opgenomen, niet om de vraag hoe vaak en in welke volgorde ze erin voorkomen. </p><p>De <a href="/wiki/Mandelbrotverzameling" title="Mandelbrotverzameling">mandelbrotverzameling</a> is een bekend voorbeeld van een wiskundige verzameling, en bestaat uit die <a href="/wiki/Complex_getal" title="Complex getal">complexe getallen</a> die, nadat er herhaald dezelfde <a href="/wiki/Operatie_(wiskunde)" title="Operatie (wiskunde)">bewerking</a> op is uitgevoerd, naar een eindige waarde <a href="/wiki/Iteratie" title="Iteratie">itereren</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitie">Definitie</h2></div> <p>Hier wordt alleen een globaal overzicht gegeven van het concept verzameling. Dit overzicht is erop gericht om met verzamelingen te kunnen werken en belangrijke begrippen als <a href="/wiki/Afbeelding_(wiskunde)" title="Afbeelding (wiskunde)">afbeeldingen</a>, <a href="/wiki/Functie_(wiskunde)" title="Functie (wiskunde)">functies</a>, <a href="/wiki/Getal_(wiskunde)" title="Getal (wiskunde)">getallen</a> en <a href="/wiki/Relatie_(wiskunde)" title="Relatie (wiskunde)">relaties</a> te kunnen definiëren. </p><p>Georg Cantor gaf aan het begin van zijn <i>Beiträge zur Begründung der transfiniten Mengenlehre</i>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> de volgende definitie van een verzameling: <style data-mw-deduplicate="TemplateStyles:r58633982">.mw-parser-output .cquote{margin:1.5em 0;padding:0 50px;display:table;position:relative;border-left:none}.mw-parser-output .cquote>:nth-last-child(2){margin-bottom:0}.mw-parser-output .cquote-cite{position:relative;margin:20px -40px 0;text-align:right;font-size:90%}.mw-parser-output .cquote-cite-leeg{margin-top:0}.mw-parser-output .cquote>:first-child::before,.mw-parser-output .cquote-cite::after{color:#B2B7F2;font-size:42px;font-family:"Times New Roman",Times,serif;font-weight:bold;position:absolute}.mw-parser-output .cquote>:first-child::before{content:"“";left:10px;top:-19px}.mw-parser-output .cquote-cite::after{content:"”";right:0;top:-53px;height:0}.mw-parser-output .cquote-cite-leeg::after{top:-33px}</style> </p> <blockquote class="cquote"> <p>Met een verzameling bedoelen we elke collectie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> uit een geheel van concrete, afzonderlijke objecten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, die de elementen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> worden genoemd, van onze <a href="/wiki/Waarneming_(perceptie)" title="Waarneming (perceptie)">perceptie</a> [Anschauung] of van ons denken. </p> <div class="cquote-cite cquote-cite-leeg"></div> </blockquote> <p>De elementen of leden van een verzameling kunnen bijvoorbeeld zijn: getallen, letters van het alfabet, andere verzamelingen en zo verder. Een verzameling wordt gewoonlijk aangeduid door een <a href="/wiki/Kapitaal_(typografie)" title="Kapitaal (typografie)">hoofdletter</a>. De verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> zijn aan elkaar <a href="/wiki/Gelijkheid_(verzamelingenleer)" title="Gelijkheid (verzamelingenleer)">gelijk</a> als zij dezelfde elementen hebben. </p><p>Zoals hieronder wordt besproken, bleek de hierboven gegeven definitie ontoereikend voor de <a href="/wiki/Formalisme_(wiskunde)" title="Formalisme (wiskunde)">formele wiskunde</a>. In plaats daarvan wordt het begrip 'verzameling' in de <a href="/wiki/Axiomatische_verzamelingenleer" title="Axiomatische verzamelingenleer">axiomatische verzamelingenleer</a> als een ongedefinieerde primitieve<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> genomen, en worden haar eigenschappen gedefinieerd door de <a href="/wiki/Zermelo-Fraenkel-verzamelingenleer" title="Zermelo-Fraenkel-verzamelingenleer">axioma's van Zermelo-Fraenkel</a>. De twee meest fundamentele eigenschappen zijn dat een verzameling door de elementen er in is gedefinieerd en dat twee verzamelingen <a href="/wiki/Dan_en_slechts_dan_als" title="Dan en slechts dan als">dan en slechts dan</a> aan elkaar gelijk zijn, als deze dezelfde elementen hebben. </p><p>Men dient voorzichtig te zijn met verbale beschrijvingen van verzamelingen, omdat deze gemakkelijk tot <a href="/wiki/Paradox_(logica)#Wiskundige_paradoxen" title="Paradox (logica)">paradoxen</a> kunnen leiden. De <a href="/wiki/Axiomatische_verzamelingenleer" title="Axiomatische verzamelingenleer">axiomatische verzamelingenleer</a> is geconstrueerd om deze paradoxen te vermijden. </p> <div class="mw-heading mw-heading2"><h2 id="Beschrijving">Beschrijving</h2></div> <p>In het dagelijkse spraakgebruik komt het begrip 'verzameling' ook voor: met "bestek" wordt in een huishouden de verzameling lepels, vorken en messen bedoeld, het "servies" van oma is een verzameling borden, schalen .... Een "pak" speelkaarten is een verzameling speelkaarten. </p><p>Er zijn twee manieren om de <a href="/wiki/Element_(wiskunde)" title="Element (wiskunde)">elementen</a> van een verzameling vast te leggen. Eén manier is door een beschrijving, waarbij gebruik wordt gemaakt van een regel of een <a href="/wiki/Semantiek" title="Semantiek">semantische</a> beschrijving van de elementen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is de verzameling waarvan de elementen de eerste vier positieve <a href="/wiki/Getal_(wiskunde)" title="Getal (wiskunde)">getallen</a> zijn.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is de verzameling van alle kleuren van de <a href="/wiki/Vlag_van_Nederland" title="Vlag van Nederland">Nederlandse vlag</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 B=\{x\mid x{\text{ is een kleur van de Nederlandse vlag}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is een kleur van de Nederlandse vlag</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{x\mid x{\text{ is een kleur van de Nederlandse vlag}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797166eb6623a0b300c2a9b10c657cd4f2f51ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.441ex; height:2.843ex;" alt="{\displaystyle B=\{x\mid x{\text{ is een kleur van de Nederlandse vlag}}\}}"></span>. In plaats van de verticale streep schrijft men ook wel een dubbelepunt: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{x:x{\text{ is een kleur van de Nederlandse vlag}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is een kleur van de Nederlandse vlag</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{x:x{\text{ is een kleur van de Nederlandse vlag}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e52f0fa10037dce230f4b08035cacb51bd9ed91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.441ex; height:2.843ex;" alt="{\displaystyle B=\{x:x{\text{ is een kleur van de Nederlandse vlag}}\}}"></span>.</dd></dl> <p>De tweede manier is door opsomming, dat is wanneer elk element van de verzameling expliciet wordt genoemd. De elementen van de verzameling worden hierbij tussen <a href="/wiki/Accolade_(leesteken)" title="Accolade (leesteken)">accolades</a> geplaatst: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{4,2,1,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{4,2,1,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f31b274af6691b14f143abccd4c1d8c1673fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.918ex; height:2.843ex;" alt="{\displaystyle A=\{4,2,1,3\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{{\text{rood, wit, blauw}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>rood, wit, blauw</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{{\text{rood, wit, blauw}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a7db22c68a46f1f1c017a0dbde1362c97c6f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.473ex; height:2.843ex;" alt="{\displaystyle B=\{{\text{rood, wit, blauw}}\}}"></span></dd></dl> <p>Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> een element is van de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, wordt dit genoteerd als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bcc9b2afb295d4234bc294860cd0c63bcad2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle x\in A}"></span>. Is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> géén element van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, dan wordt dit wel aangeduid door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\notin A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\notin A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8152431575305d6a6145adf9b279891a65923eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.913ex; height:2.676ex;" alt="{\displaystyle x\notin A}"></span>. </p><p>Met betrekking tot de verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{4,2,1,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{4,2,1,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f31b274af6691b14f143abccd4c1d8c1673fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.918ex; height:2.843ex;" alt="{\displaystyle A=\{4,2,1,3\}}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{{\text{rood,wit,blauw}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>rood,wit,blauw</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{{\text{rood,wit,blauw}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ffd85c819f9ec8606f03c5cfb7076225cb2e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.312ex; height:2.843ex;" alt="{\displaystyle B=\{{\text{rood,wit,blauw}}\}}"></span> bijvoorbeeld, zoals hierboven gedefinieerd, geldt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752a0e5b73d1b52e13aa59056b872cae13c109a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.746ex; height:2.176ex;" alt="{\displaystyle 4\in A}"></span></dd></dl> <p>en </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{groen}}\notin B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>groen</mtext> </mrow> <mo>∉</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{groen}}\notin B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7ef1619f34b4d005fcbfe095fdede878f35a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.166ex; height:2.676ex;" alt="{\displaystyle {\text{groen}}\notin B}"></span></dd></dl> <p>Twee verzamelingen zijn aan elkaar gelijk, als ze dezelfde elementen bevatten. Bijvoorbeeld <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{4,2,1,3\}=\{1,2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{4,2,1,3\}=\{1,2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b42d8864341a1008d33aa06ae34204d5ece197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.252ex; height:2.843ex;" alt="{\displaystyle \{4,2,1,3\}=\{1,2,3,4\}}"></span>. Dat twee verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> aan elkaar gelijk zijn, noteert men als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045cafe35b1e9c9ac889481fd7178d6f59a77fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A=B}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B\ \Longleftrightarrow \ \forall x\ \left(x\in A\Longleftrightarrow x\in B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mtext> </mtext> <mo stretchy="false">⟺</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">⟺</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B\ \Longleftrightarrow \ \forall x\ \left(x\in A\Longleftrightarrow x\in B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f703c26d65aa2eb942b240c85773e7e68ed9401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.228ex; height:2.843ex;" alt="{\displaystyle A=B\ \Longleftrightarrow \ \forall x\ \left(x\in A\Longleftrightarrow x\in B\right)}"></span></dd></dl> <p>Anders dan bij een <a href="/wiki/Multiset" title="Multiset">multiset</a> komt elk element van een verzameling maar één keer voor als element van de verzameling, ook al wordt een element meer keren genoemd. Zo is de verzameling letters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b,a,c,a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b,a,c,a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020d38105887f248b5e2f11b1908079c532ed9da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.154ex; height:2.843ex;" alt="{\displaystyle \{a,b,a,c,a\}}"></span> dezelfde als de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e9bc621ced3f02e87b1c40be37867929142bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle \{a,b,c\}}"></span> en de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b,a,c,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{b,a,c,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b93e957d89b1dbb69f715f2458b2ebb4950761c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.668ex; height:2.843ex;" alt="{\displaystyle \{b,a,c,c\}}"></span>. Ieder element van een verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> blijft onder alle <a href="/wiki/Algebra_van_verzamelingen" title="Algebra van verzamelingen">bewerkingen</a> op <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> uniek. De volgorde waarin de elementen van een verzameling worden opgesomd, telt niet, dit in tegenstelling tot bij een <a href="/wiki/Rij_(wiskunde)" title="Rij (wiskunde)">rij</a> of een <a href="/wiki/Tupel" title="Tupel">tupel</a>. Elementen staan in een rij opeenvolgend opgesomd en mogen in tegenstelling tot in een verzameling wel meer dan één keer in een rij voorkomen. </p><p>Een verzameling objecten in het dagelijks leven, bijvoorbeeld een platenverzameling, of de spullen in een tas, kan identieke objecten bevatten, waarbij de multipliciteit vaak relevant is, en moet dan als een multiset worden beschreven, niet als verzameling. </p><p>De lege verzameling, die geen elementen heeft, wordt met het symbool ∅ genoteerd. Minder gebruikelijk is de notatie {}. </p><p>Het aantal elementen in een verzameling noemt men de <a href="#Kardinaliteit">kardinaliteit</a> van de verzameling. </p> <div class="mw-heading mw-heading2"><h2 id="Deelverzamelingen">Deelverzamelingen</h2></div> <div style="float:right;margin:1em;"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Bestand:Venn_A_subset_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/150px-Venn_A_subset_B.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/225px-Venn_A_subset_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/300px-Venn_A_subset_B.svg.png 2x" data-file-width="155" data-file-height="155" /></a><figcaption></figcaption></figure><div class="center"><small> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is een deelverzameling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></small></div></div> <p>Als elk element van de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ook element is van de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, zegt men dat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> een <a href="/wiki/Deelverzameling" title="Deelverzameling">deelverzameling</a> is van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Dit wordt genoteerd als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}"></span> of als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></span>, en uitgesproken als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is een deel(verzameling) van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, of als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> wordt door <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> omvat. In plaats daarvan kan ook worden geschreven: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊇</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fd7d8e0fa00d29c0d6a35ab2c3d4cd636bd136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\supseteq A}"></span>, of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊃</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supset A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450398271587fcd521f7313ee3ebfdb5023e1c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle B\supset A}"></span> zeg: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> omvat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> sluit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in, of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is een <a href="/wiki/Superset" title="Superset">superset</a> van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. De <a href="/wiki/Relatie_(wiskunde)" title="Relatie (wiskunde)">relatie</a> tussen verzamelingen die wordt vastgelegd door ⊆ wordt inclusie of omvatting genoemd. </p><p>Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> een deelverzameling is van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, maar niet daaraan gelijk is, wordt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> een echte of strikte deelverzameling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> genoemd. Dit wordt wel genoteerd als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf81e9a4a81df2d596b4db1cde6b9bdf82c73db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B}"></span>, of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supsetneq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊋</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supsetneq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6f9bc7f3f838bcc1b8136530626324a40bad17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle B\supsetneq A}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is een strikte superset van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p><p>Voorbeeld: </p> <ul><li>De verzameling van alle mannen is een strikte deelverzameling van de verzameling van alle mensen.</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 \{1,3\}\subseteq \{1,2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,3\}\subseteq \{1,2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e18c350e769f61c5c9ce8c92bba394d54b5fbe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.859ex; height:2.843ex;" alt="{\displaystyle \{1,3\}\subseteq \{1,2,3,4\}}"></span>, maar ook <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,3\}\subset \{1,2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,3\}\subset \{1,2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f939503daa20a005d11af108adfc07893c2b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.859ex; height:2.843ex;" alt="{\displaystyle \{1,3\}\subset \{1,2,3,4\}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4\}\subseteq \{1,2,3,4\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4\}\subseteq \{1,2,3,4\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9805912e7dbfff7bb063a88e9eb27c700550d256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.252ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4\}\subseteq \{1,2,3,4\}}"></span></li></ul> <p>De <a href="/wiki/Uitdrukking_(wiskunde)" title="Uitdrukking (wiskunde)">uitdrukkingen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊃</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supset A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450398271587fcd521f7313ee3ebfdb5023e1c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle B\supset A}"></span> worden door verschillende auteurs verschillend gebruikt: sommigen gebruiken deze relatie in de betekenis van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}"></span> (respectievelijk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊇</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fd7d8e0fa00d29c0d6a35ab2c3d4cd636bd136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\supseteq A}"></span>), terwijl anderen er <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf81e9a4a81df2d596b4db1cde6b9bdf82c73db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B}"></span> (respectievelijk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supsetneq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊋</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supsetneq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6f9bc7f3f838bcc1b8136530626324a40bad17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle B\supsetneq A}"></span>) mee bedoelen. </p><p>De lege verzameling is een deelverzameling van elke verzameling en elke verzameling is een deelverzameling van zichzelf: </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 \varnothing \subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b947b700faaf478c312531745e80bf69ed50d493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.65ex; height:2.343ex;" alt="{\displaystyle \varnothing \subseteq A}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ce5093be9e30238b83393aed738eafd3a43030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.585ex; height:2.343ex;" alt="{\displaystyle A\subseteq A}"></span></li></ul> <p>Een vanzelfsprekende identiteit, die vaak kan worden gebruikt om aan te tonen dat twee ogenschijnlijk verschillende verzamelingen toch aan elkaar gelijk zijn: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045cafe35b1e9c9ac889481fd7178d6f59a77fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A=B}"></span> dan en slechts dan als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bewerkingen">Bewerkingen</h2></div> <ul><li>De <a href="/wiki/Vereniging_(verzamelingenleer)" title="Vereniging (verzamelingenleer)">vereniging</a> van twee verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> wordt gevormd door de elementen die in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> (of in beide) zitten. Notatie: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}"></span>.</li> <li>De <a href="/wiki/Doorsnede_(verzamelingenleer)" title="Doorsnede (verzamelingenleer)">doorsnede</a> van twee verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> wordt gevormd door de verzameling van gemeenschappelijke elementen, dus alle elementen die zowel in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> als in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> zitten. Notatie: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27b38cf9eac6060e67b61f66cd9beec5067f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cap B}"></span>.</li> <li>Het <a href="/wiki/Verschil_(verzamelingenleer)" title="Verschil (verzamelingenleer)">verschil</a> van twee verzamelingen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> word gevormd door alle elementen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> die niet in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> zitten. Notatie: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef797ed5deb971321592e34281d9fac27c3249d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle A\setminus B}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc58c452f31f578fdf98cafc1c53fe98a0c0975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A-B}"></span>.</li> <li>Het relatieve <a href="/wiki/Complement_(verzamelingenleer)" title="Complement (verzamelingenleer)">complement</a> van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> ten opzichte van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is de verzameling van de elementen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> die niet tot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> behoren. Het wordt genoteerd als:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=\{x\in A\mid x\notin B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∣</mo> <mi>x</mi> <mo>∉</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B=\{x\in A\mid x\notin B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/debe8b87e053686ac9c485464f055fa5dbedc7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.91ex; height:2.843ex;" alt="{\displaystyle A\setminus B=\{x\in A\mid x\notin B\}}"></span></dd></dl></dd> <dd>Lees: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> met daaruit weggelaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Het relatieve complement wordt ook wel genoteerd als: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc58c452f31f578fdf98cafc1c53fe98a0c0975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A-B}"></span>. Kortheidshalve wordt genoteerd: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{c}=\{x\mid x\notin B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>∉</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{c}=\{x\mid x\notin B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0615bc1ad9195ac773bc1aaf0807c79709bf065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.333ex; height:2.843ex;" alt="{\displaystyle B^{c}=\{x\mid x\notin B\}}"></span>, waarbij niet wordt vermeld ten opzichte van welke verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> het complement wordt genomen. Voor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> kan hierbij iedere verzameling worden genomen, zolang <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}"></span>.</dd></dl> <p>Er gelden de volgende eigenschappen: </p> <dl><dd><table class="wikitable"> <tbody><tr> <th>Eigenschap</th> <th>Doorsnede</th> <th>Vereniging </th></tr> <tr> <td><a href="/wiki/Commutativiteit" title="Commutativiteit">commutatief</a> </td> <td>A ∩ B = B ∩ A </td> <td>A ∪ B = B ∪ A </td></tr> <tr> <td><a href="/wiki/Associativiteit_(wiskunde)" title="Associativiteit (wiskunde)">associatief</a> </td> <td>A ∩ (B ∩ C) = (A ∩ B) ∩ C </td> <td>A ∪ (B ∪ C) = (A ∪ B) ∪ C </td></tr> <tr> <td><a href="/wiki/Distributiviteit" title="Distributiviteit">distributief</a> </td> <td>A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) </td> <td>A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) </td></tr> <tr> <td><a href="/wiki/Neutraal_element" title="Neutraal element">neutraal element</a> </td> <td>A ∩ <i>U</i> = A voor alle A </td> <td>A ∪ ∅ = A voor alle A </td></tr> <tr> <td>'kleinste' en 'grootste'<br /> verzameling </td> <td>A ∩ ∅ = ∅ voor alle A </td> <td>A ∪ <i>U</i> = <i>U</i> voor alle A </td></tr></tbody></table></dd></dl> <p>Een <a href="/wiki/Partitie_(verzamelingenleer)" title="Partitie (verzamelingenleer)">partitie</a> is een opdeling van een verzameling in niet-lege, onderling disjuncte, deelverzamelingen, die wel blokken worden genoemd. Bijvoorbeeld: als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{1,2,3,4,5,6,7,8\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{1,2,3,4,5,6,7,8\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88aa9dac2cfe63514d161d2e1a964c769df975b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.704ex; height:2.843ex;" alt="{\displaystyle A=\{1,2,3,4,5,6,7,8\}}"></span>, dan vormen de deelverzamelingen {1,3}, {2,4,5,7} en {6,8} een partitie van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> met drie blokken. </p><p>De deelverzamelingen van een gegeven verzameling vormen een <a href="/wiki/Booleaanse_algebra" title="Booleaanse algebra">booleaanse algebra</a> onder doorsnede en vereniging. </p> <div class="mw-heading mw-heading2"><h2 id="Wetten_van_De_Morgan">Wetten van De Morgan</h2></div> <p>De <a href="/wiki/Wetten_van_De_Morgan" title="Wetten van De Morgan">wetten van De Morgan</a> luiden: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\cup B)^{c}=A^{c}\cap B^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cup B)^{c}=A^{c}\cap B^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531b94f9c1e7930c6217d246705b50734333dee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.92ex; height:2.843ex;" alt="{\displaystyle (A\cup B)^{c}=A^{c}\cap B^{c}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\cap B)^{c}=A^{c}\cup B^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>∪</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cap B)^{c}=A^{c}\cup B^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d72c214bafdab9ee788629cd2356c3081679326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.92ex; height:2.843ex;" alt="{\displaystyle (A\cap B)^{c}=A^{c}\cup B^{c}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6eec34036c269c1d841f48e1706692a6c9b1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.566ex; height:2.843ex;" alt="{\displaystyle A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad16772dbc38576aa521b69205e72af74fc975d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.566ex; height:2.843ex;" alt="{\displaystyle A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Kardinaliteit">Kardinaliteit</h2></div> <p>De <a href="/wiki/Kardinaliteit" title="Kardinaliteit">kardinaliteit</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648fce92f29d925f04d39244ccfe435320dfc6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle |A|}"></span> van een verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is "het aantal elementen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>". Aangezien bijvoorbeeld de Nederlandse vlag drie kleuren kent, is de kardinaliteit van de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{{\text{kleuren van de Nederlandse vlag}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>kleuren van de Nederlandse vlag</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{{\text{kleuren van de Nederlandse vlag}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c789c471d8d6a5925cfc035f56bb7df08fc4ebe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.507ex; height:2.843ex;" alt="{\displaystyle B=\{{\text{kleuren van de Nederlandse vlag}}\}}"></span> gelijk aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |B|=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |B|=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7cb3cbb85fc3cf95a0543ffd7ab01dd1c74da97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.319ex; height:2.843ex;" alt="{\displaystyle |B|=3}"></span>. </p><p>De <a href="/wiki/Lege_verzameling" title="Lege verzameling">lege verzameling</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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> heeft kardinaliteit 0. Hoewel het misschien triviaal lijkt, is de lege verzameling, net zoals het getal <a href="/wiki/0_(getal)" title="0 (getal)">nul</a>, belangrijk in de wiskunde. Het bestaan van de lege verzameling is zelfs een van de fundamentele concepten uit de <a href="/wiki/Axiomatische_verzamelingenleer" title="Axiomatische verzamelingenleer">axiomatische verzamelingenleer</a>. </p><p>Sommige verzamelingen hebben een <a href="/wiki/Oneindige_verzameling" title="Oneindige verzameling">oneindige</a> kardinaliteit. De verzameling <span class="mwe-math-element"><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> van de <a href="/wiki/Natuurlijk_getal" title="Natuurlijk getal">natuurlijke getallen</a> is bijvoorbeeld oneindig. Men kan echter aantonen dat sommige oneindige kardinaliteiten groter zijn dan andere. De verzameling van de <a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">reële getallen</a> bijvoorbeeld heeft een grotere kardinaliteit dan de verzameling van de natuurlijke getallen. Het kan worden aangetoond dat de kardinaliteit van, dat wil zeggen: het aantal punten op, een <a href="/wiki/Lijn_(meetkunde)" title="Lijn (meetkunde)">lijn</a> dezelfde is als de kardinaliteit van enig <a href="/wiki/Lijnstuk" title="Lijnstuk">lijnstuk</a> van die lijn, dezelfde als die van het gehele <a href="/wiki/Vlak_(meetkunde)" title="Vlak (meetkunde)">vlak</a> en ook dezelfde als die van enige eindig-dimensionale <a href="/wiki/Euclidische_ruimte" title="Euclidische ruimte">euclidische ruimte</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Machtsverzamelingen">Machtsverzamelingen</h3></div> <p>De <a href="/wiki/Machtsverzameling" title="Machtsverzameling">machtsverzameling</a> van een verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is de verzameling van alle deelverzamelingen van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Daartoe behoort de verzameling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> zelf en de lege verzameling. Als een <a href="/wiki/Eindige_verzameling" title="Eindige verzameling">eindige verzameling</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> een <a href="/wiki/Kardinaliteit" title="Kardinaliteit">kardinaliteit</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> heeft, is de kardinaliteit van de machtsverzameling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> gelijk aan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span>. De machtsverzameling wordt genoteerd als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1757ec21abe0a22f8e91b51fe3e6ac4ea63a9122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(A)}"></span> of als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6d28a1b787f8c321de35ccc9305fd6cbda9934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.676ex;" alt="{\displaystyle 2^{A}}"></span>. </p><p>Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> een oneindige, een <a href="/wiki/Aftelbare_verzameling" title="Aftelbare verzameling">aftelbare</a> dan wel <a href="/wiki/Overaftelbare_verzameling" title="Overaftelbare verzameling">overaftelbare verzameling</a> is, is de machtsverzameling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> altijd overaftelbaar. Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> bovendien een verzameling is, dan is er nooit een <a href="/wiki/Bijectie" title="Bijectie">bijectie</a> van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> op <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1757ec21abe0a22f8e91b51fe3e6ac4ea63a9122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(A)}"></span> mogelijk. Met andere woorden: de machtsverzameling van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is altijd strikt "groter" dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> zelf. </p><p>De machtsverzameling van de verzameling {1, 2, 3} is bijvoorbeeld { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅ }. De kardinaliteit van de oorspronkelijke verzameling is 3 en de kardinaliteit van de machtsverzameling is 2<sup>3</sup> = 8. Deze relatie is een van de redenen voor de terminologie machtsverzameling. </p> <div class="mw-heading mw-heading2"><h2 id="Getallenverzamelingen">Getallenverzamelingen</h2></div> <p>Voorbeelden van getallenverzamelingen zijn: </p> <ol><li><a href="/wiki/Natuurlijk_getal" title="Natuurlijk getal">Natuurlijke getallen</a></li> <li><a href="/wiki/Geheel_getal" title="Geheel getal">Gehele getallen</a></li> <li>De <a href="/wiki/Rationaal_getal" title="Rationaal getal">rationale getallen</a>, die bestaan uit de gehele getallen en de <a href="/wiki/Breuk_(wiskunde)" title="Breuk (wiskunde)">breuken</a>.</li> <li>De <a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">reële getallen</a>, waaronder ook de <a href="/wiki/Transcendent_getal" title="Transcendent getal">transcendente getallen</a> vallen.</li> <li>De <a href="/wiki/Complex_getal" title="Complex getal">complexe getallen</a> verschijnen als oplossing van vergelijkingen als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0}"></span>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Toepassingen">Toepassingen</h2></div> <p>Vrijwel alle andere takken van de wiskunde worden gebaseerd op de <a href="/wiki/Verzamelingenleer" title="Verzamelingenleer">verzamelingenleer</a>. Zo is bijvoorbeeld in de <a href="/wiki/Kansrekening" title="Kansrekening">kansrekening</a> de uitkomstenruimte de <a href="/wiki/Universele_verzameling" title="Universele verzameling">universele verzameling</a> van alle mogelijkheden en zijn de gebeurtenissen de (deel)verzamelingen. Andere elementaire begrippen in de wiskunde, zoals <a href="/wiki/Functie_(wiskunde)" title="Functie (wiskunde)">functies</a> en <a href="/wiki/Rij_(wiskunde)" title="Rij (wiskunde)">rijen</a>, worden ook in termen van verzamelingen gedefinieerd. </p><p>De <a href="/wiki/Ordetheorie" title="Ordetheorie">ordetheorie</a> houdt zich bezig met de verschillende manieren om de elementen van een verzameling te ordenen. Een <a href="/wiki/Tweeplaatsige_relatie" title="Tweeplaatsige relatie">relatie</a> legt daartoe de volgorde tussen de elementen van de verzameling in een <a href="/wiki/Rij_(wiskunde)" title="Rij (wiskunde)">rij</a> vast en geeft zo aan welke van de elementen opvolger is van het andere. Dat begint met het bepalen van een <a href="/wiki/Bovengrens_en_ondergrens" title="Bovengrens en ondergrens">bovengrens en ondergrens</a> van de verzameling. </p> <ul><li>Er wordt in de <a href="/wiki/Topologie" title="Topologie">topologie</a> verschil tussen <a href="/wiki/Open_verzameling" title="Open verzameling">open</a> en <a href="/wiki/Gesloten_verzameling" title="Gesloten verzameling">gesloten verzamelingen</a> gemaakt.</li> <li>Verzamelingen zijn in de <a href="/wiki/Informatica" title="Informatica">informatica</a> geïmplementeerd en heten daar ook weer <a href="/wiki/Verzameling_(informatica)" title="Verzameling (informatica)">verzamelingen</a>.</li></ul> <div class="toccolours appendix" role="presentation" style="font-size:90%; margin:1em 0 -0.5em; clear:both;"> <div></div> <dl><dt>voetnoten</dt></dl> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Geciteerd in Dauben, pag. 170</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="/wiki/Engels" title="Engels">Engels</a>: primitive notion</span> </li> </ol></div></div> <dl><dt>literatuur</dt></dl> <ul><li><style data-mw-deduplicate="TemplateStyles:r67679320">.mw-parser-output .taalaanduiding{font-family:sans-serif;font-size:85%;cursor:help;color:var(--color-subtle,#555)}.mw-parser-output .taalaanduiding span{border-bottom:1px dotted var(--color-subtle,#555)}</style><span class="taalaanduiding" title="Taal: Engels">(<span>en</span>) </span> <span style="font-variant:small-caps;">JW Dauben</span>. Georg Cantor: His Mathematics and Philosophy of the Infinite, 1979. <span class="ISBN"><a href="/wiki/Speciaal:Boekbronnen/978-0-691-02447-9" title="Speciaal:Boekbronnen/978-0-691-02447-9">ISBN 978-0-691-02447-9</a></span>.</li></ul> <dl><dt>websites</dt></dl> <ul><li><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Set.html">Set</a>.</li></ul> </div> <div class="interProject commons mw-list-item" style="display:none;"><a href="https://commons.wikimedia.org/wiki/Category:Verzameling_(wiskunde)#mw-subcategories" class="extiw" title="commons:Category:Verzameling (wiskunde)">Mediabestanden</a></div> <div class="interProjectTemplate interProject-groot toccolours" style="display:flex; gap:1em; align-items:center; clear:both; margin:1em 0 -0.5em 0;"> <div style="min-width:max-content;"><span class="noviewer noresize" typeof="mw:File"><a href="/wiki/Bestand:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/25px-Commons-logo.svg.png" decoding="async" width="25" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/38px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/50px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div>Zie de categorie <i><b><a href="https://commons.wikimedia.org/wiki/Category:Sets_(mathematics)#mw-subcategories" class="extiw" title="commons:Category:Sets (mathematics)">Sets (mathematics)</a></b></i> van <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a> voor mediabestanden over dit onderwerp.</div> </div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Verzameling_(wiskunde)&oldid=68463623">https://nl.wikipedia.org/w/index.php?title=Verzameling_(wiskunde)&oldid=68463623</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorie</a>: <ul><li><a href="/wiki/Categorie:Verzamelingenleer" title="Categorie:Verzamelingenleer">Verzamelingenleer</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Deze pagina is voor het laatst bewerkt op 2 dec 2024 om 08:06.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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