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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Zaloguj się</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strony dla anonimowych edytorów <a href="/wiki/Pomoc:Pierwsze_kroki" aria-label="Dowiedz się więcej na temat edytowania"><span>dowiedz się więcej</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Specjalna:M%C3%B3j_wk%C5%82ad" title="Lista edycji wykonanych z tego adresu IP [y]" accesskey="y"><span>Edycje</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Specjalna:Moja_dyskusja" title="Dyskusja użytkownika dla tego adresu IP [n]" accesskey="n"><span>Dyskusja</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="Witryna"> <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="Spis treści" 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">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</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">Początek</div> </a> </li> <li id="toc-Definicje_formalne" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicje_formalne"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicje formalne</span> </div> </a> <button aria-controls="toc-Definicje_formalne-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>Przełącz podsekcję Definicje formalne</span> </button> <ul id="toc-Definicje_formalne-sublist" class="vector-toc-list"> <li id="toc-Przykłady" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Przykłady"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Przykłady</span> </div> </a> <ul id="toc-Przykłady-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poprawność_definicji_sumy_zbiorów" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poprawność_definicji_sumy_zbiorów"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Poprawność definicji sumy zbiorów</span> </div> </a> <ul id="toc-Poprawność_definicji_sumy_zbiorów-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Własności" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Własności"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Własności</span> </div> </a> <button aria-controls="toc-Własności-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>Przełącz podsekcję Własności</span> </button> <ul id="toc-Własności-sublist" class="vector-toc-list"> <li id="toc-Operacje_skończone" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operacje_skończone"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Operacje skończone</span> </div> </a> <ul id="toc-Operacje_skończone-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operacje_nieskończone" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operacje_nieskończone"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Operacje nieskończone</span> </div> </a> <ul id="toc-Operacje_nieskończone-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suma_a_obrazy_i_przeciwobrazy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suma_a_obrazy_i_przeciwobrazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Suma a obrazy i przeciwobrazy</span> </div> </a> <ul id="toc-Suma_a_obrazy_i_przeciwobrazy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linki_zewnętrzne" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linki_zewnętrzne"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Linki zewnętrzne</span> </div> </a> <ul id="toc-Linki_zewnętrzne-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="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Spis treści" > <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="Przełącz stan spisu treści" > <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">Przełącz stan spisu treści</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Suma zbiorów</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Przejdź do artykułu w innym języku. Treść dostępna w 62 językach" > <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-62" 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">62 języki</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%8D%E1%88%85%E1%8B%B5_%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ውህድ ስብስብ – amharski" lang="am" hreflang="am" data-title="ውህድ ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="amharski" 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/%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A7%D8%AA)" title="اتحاد (نظرية المجموعات) – arabski" lang="ar" hreflang="ar" data-title="اتحاد (نظرية المجموعات)" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%E2%80%99%D1%8F%D0%B4%D0%BD%D0%B0%D0%BD%D0%BD%D0%B5_%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0%D1%9E" title="Аб’яднанне мностваў – białoruski" lang="be" hreflang="be" data-title="Аб’яднанне мностваў" data-language-autonym="Беларуская" data-language-local-name="białoruski" 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%90%D0%B1%E2%80%99%D1%8F%D0%B4%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5_%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0%D1%9E" title="Аб’яднаньне мностваў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Аб’яднаньне мностваў" 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%9E%D0%B1%D0%B5%D0%B4%D0%B8%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D1%82%D0%B0)" title="Обединение (теория на множествата) – bułgarski" lang="bg" hreflang="bg" data-title="Обединение (теория на множествата)" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Uni%C3%B3" title="Unió – kataloński" lang="ca" hreflang="ca" data-title="Unió" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</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%D1%81%D0%B5%D0%BD_%D0%BF%C4%95%D1%80%D0%BB%D0%B5%D1%88%C4%95%D0%B2%C4%95" title="Йышсен пĕрлешĕвĕ – czuwaski" lang="cv" hreflang="cv" data-title="Йышсен пĕрлешĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="czuwaski" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sjednocen%C3%AD" title="Sjednocení – czeski" lang="cs" hreflang="cs" data-title="Sjednocení" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Uniad_set" title="Uniad set – walijski" lang="cy" hreflang="cy" data-title="Uniad set" data-language-autonym="Cymraeg" data-language-local-name="walijski" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Vereinigungsmenge" title="Vereinigungsmenge – niemiecki" lang="de" hreflang="de" data-title="Vereinigungsmenge" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/%C3%9Chend" title="Ühend – estoński" lang="et" hreflang="et" data-title="Ühend" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%88%CE%BD%CF%89%CF%83%CE%B7_%CF%83%CF%85%CE%BD%CF%8C%CE%BB%CF%89%CE%BD" title="Ένωση συνόλων – grecki" lang="el" hreflang="el" data-title="Ένωση συνόλων" data-language-autonym="Ελληνικά" data-language-local-name="grecki" 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/Union_(set_theory)" title="Union (set theory) – angielski" lang="en" hreflang="en" data-title="Union (set theory)" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Uni%C3%B3n_de_conjuntos" title="Unión de conjuntos – hiszpański" lang="es" hreflang="es" data-title="Unión de conjuntos" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kuna%C4%B5o" title="Kunaĵo – esperanto" lang="eo" hreflang="eo" data-title="Kunaĵo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bilketa_(multzo-teoria)" title="Bilketa (multzo-teoria) – baskijski" lang="eu" hreflang="eu" data-title="Bilketa (multzo-teoria)" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%D8%A7%D8%AC%D8%AA%D9%85%D8%A7%D8%B9_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87%E2%80%8C%D9%87%D8%A7)" title="اجتماع (نظریه مجموعه‌ها) – perski" lang="fa" hreflang="fa" data-title="اجتماع (نظریه مجموعه‌ها)" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Union_(math%C3%A9matiques)" title="Union (mathématiques) – francuski" lang="fr" hreflang="fr" data-title="Union (mathématiques)" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Aontas_(tacartheoiric)" title="Aontas (tacartheoiric) – irlandzki" lang="ga" hreflang="ga" data-title="Aontas (tacartheoiric)" data-language-autonym="Gaeilge" data-language-local-name="irlandzki" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Uni%C3%B3n_(conxuntos)" title="Unión (conxuntos) – galicyjski" lang="gl" hreflang="gl" data-title="Unión (conxuntos)" data-language-autonym="Galego" data-language-local-name="galicyjski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9D%D0%B8%D0%B8%D0%BB%D0%B2%D0%B0%D1%80" title="Ниилвар – kałmucki" lang="xal" hreflang="xal" data-title="Ниилвар" data-language-autonym="Хальмг" data-language-local-name="kałmucki" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A9%EC%A7%91%ED%95%A9" title="합집합 – koreański" lang="ko" hreflang="ko" data-title="합집합" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%A1%D5%BE%D5%B8%D6%80%D5%B8%D6%82%D5%B4_(%D5%A2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Միավորում (բազմությունների տեսություն) – ormiański" lang="hy" hreflang="hy" data-title="Միավորում (բազմությունների տեսություն)" data-language-autonym="Հայերեն" data-language-local-name="ormiański" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%98_(%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4)" title="संघ (समुच्चय सिद्धान्त) – hindi" lang="hi" hreflang="hi" data-title="संघ (समुच्चय सिद्धान्त)" 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/Unija_skupova" title="Unija skupova – chorwacki" lang="hr" hreflang="hr" data-title="Unija skupova" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ensemblo-uniono" title="Ensemblo-uniono – ido" lang="io" hreflang="io" data-title="Ensemblo-uniono" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gabungan_(teori_himpunan)" title="Gabungan (teori himpunan) – indonezyjski" lang="id" hreflang="id" data-title="Gabungan (teori himpunan)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezyjski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Union_(theoria_de_insimules)" title="Union (theoria de insimules) – interlingua" lang="ia" hreflang="ia" data-title="Union (theoria de insimules)" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sammengi" title="Sammengi – islandzki" lang="is" hreflang="is" data-title="Sammengi" data-language-autonym="Íslenska" data-language-local-name="islandzki" 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/Unione_(insiemistica)" title="Unione (insiemistica) – włoski" lang="it" hreflang="it" data-title="Unione (insiemistica)" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%97%D7%95%D7%93_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="איחוד (מתמטיקה) – hebrajski" lang="he" hreflang="he" data-title="איחוד (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D1%96%D1%80%D1%96%D0%BA%D1%82%D1%96%D1%80%D1%83" title="Біріктіру – kazachski" lang="kk" hreflang="kk" data-title="Біріктіру" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Union_(insemma)" title="Union (insemma) – lombardzki" lang="lmo" hreflang="lmo" data-title="Union (insemma)" data-language-autonym="Lombard" data-language-local-name="lombardzki" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Uni%C3%B3_(halmazelm%C3%A9let)" title="Unió (halmazelmélet) – węgierski" lang="hu" hreflang="hu" data-title="Unió (halmazelmélet)" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D1%98%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0)" title="Унија (теорија на множества) – macedoński" lang="mk" hreflang="mk" data-title="Унија (теорија на множества)" data-language-autonym="Македонски" data-language-local-name="macedoński" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nah mw-list-item"><a href="https://nah.wikipedia.org/wiki/Tlacet%C4%ABliztli_(tlap%C5%8Dhualmatiliztli)" title="Tlacetīliztli (tlapōhualmatiliztli) – Nahuatl" lang="nah" hreflang="nah" data-title="Tlacetīliztli (tlapōhualmatiliztli)" data-language-autonym="Nāhuatl" data-language-local-name="Nahuatl" class="interlanguage-link-target"><span>Nāhuatl</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vereniging_(verzamelingenleer)" title="Vereniging (verzamelingenleer) – niderlandzki" lang="nl" hreflang="nl" data-title="Vereniging (verzamelingenleer)" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%92%8C%E9%9B%86%E5%90%88" title="和集合 – japoński" lang="ja" hreflang="ja" data-title="和集合" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Union_(mengdel%C3%A6re)" title="Union (mengdelære) – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Union (mengdelære)" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Union_i_matematikk" title="Union i matematikk – norweski (nynorsk)" lang="nn" hreflang="nn" data-title="Union i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norweski (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Makoo_Tuutotaa" title="Makoo Tuutotaa – oromo" lang="om" hreflang="om" data-title="Makoo Tuutotaa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Union" title="Union – piemoncki" lang="pms" hreflang="pms" data-title="Union" data-language-autonym="Piemontèis" data-language-local-name="piemoncki" 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/Uni%C3%A3o_(matem%C3%A1tica)" title="União (matemática) – portugalski" lang="pt" hreflang="pt" data-title="União (matemática)" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Reuniune_(matematic%C4%83)" title="Reuniune (matematică) – rumuński" lang="ro" hreflang="ro" data-title="Reuniune (matematică)" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%8A%D0%B5%D0%B4%D0%B8%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2" title="Объединение множеств – rosyjski" lang="ru" hreflang="ru" data-title="Объединение множеств" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Bashkimi_(teoria_e_bashk%C3%ABsive)" title="Bashkimi (teoria e bashkësive) – albański" lang="sq" hreflang="sq" data-title="Bashkimi (teoria e bashkësive)" data-language-autonym="Shqip" data-language-local-name="albański" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zjednotenie_(matematika)" title="Zjednotenie (matematika) – słowacki" lang="sk" hreflang="sk" data-title="Zjednotenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="słowacki" 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/Unija_mno%C5%BEic" title="Unija množic – słoweński" lang="sl" hreflang="sl" data-title="Unija množic" data-language-autonym="Slovenščina" data-language-local-name="słoweński" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D1%98%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%81%D0%BA%D1%83%D0%BF%D0%BE%D0%B2%D0%B0)" title="Унија (теорија скупова) – serbski" lang="sr" hreflang="sr" data-title="Унија (теорија скупова)" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yhdiste_(matematiikka)" title="Yhdiste (matematiikka) – fiński" lang="fi" hreflang="fi" data-title="Yhdiste (matematiikka)" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Union_(matematik)" title="Union (matematik) – szwedzki" lang="sv" hreflang="sv" data-title="Union (matematik)" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Samahan_(matematika)" title="Samahan (matematika) – tagalski" lang="tl" hreflang="tl" data-title="Samahan (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalski" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81_(%E0%AE%95%E0%AE%A3%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81)" title="சேர்ப்பு (கணக் கோட்பாடு) – tamilski" lang="ta" hreflang="ta" data-title="சேர்ப்பு (கணக் கோட்பாடு)" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A2%E0%B8%B9%E0%B9%80%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B8%99" title="ยูเนียน – tajski" lang="th" hreflang="th" data-title="ยูเนียน" data-language-autonym="ไทย" data-language-local-name="tajski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birle%C5%9Fme_%C3%B6zelli%C4%9Fi_(k%C3%BCme_teorisi)" title="Birleşme özelliği (küme teorisi) – turecki" lang="tr" hreflang="tr" data-title="Birleşme özelliği (küme teorisi)" data-language-autonym="Türkçe" data-language-local-name="turecki" 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%9E%D0%B1%27%D1%94%D0%B4%D0%BD%D0%B0%D0%BD%D0%BD%D1%8F_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD" title="Об'єднання множин – ukraiński" lang="uk" hreflang="uk" data-title="Об'єднання множин" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_h%E1%BB%A3p" title="Phép hợp – wietnamski" lang="vi" hreflang="vi" data-title="Phép hợp" data-language-autonym="Tiếng Việt" data-language-local-name="wietnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Hulk%C3%B5_kogo" title="Hulkõ kogo – võro" lang="vro" hreflang="vro" data-title="Hulkõ kogo" data-language-autonym="Võro" data-language-local-name="võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BD%B5%E9%9B%86" title="併集 – chiński klasyczny" lang="lzh" hreflang="lzh" data-title="併集" data-language-autonym="文言" data-language-local-name="chiński klasyczny" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B9%B6%E9%9B%86" title="并集 – wu" lang="wuu" hreflang="wuu" data-title="并集" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BD%B5%E9%9B%86" title="併集 – kantoński" lang="yue" hreflang="yue" data-title="併集" data-language-autonym="粵語" data-language-local-name="kantoński" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B6%E9%9B%86" title="并集 – chiński" lang="zh" hreflang="zh" data-title="并集" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q185359#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Przestrzenie nazw"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pl" dir="ltr"><p><b>Suma zbiorów</b> (rzadko: unia zbiorów) – działanie <a href="/wiki/Algebra_zbior%C3%B3w_(nauka)" title="Algebra zbiorów (nauka)">algebry zbiorów</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicje_formalne">Definicje formalne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=1" title="Edytuj sekcję: Definicje formalne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Definicje formalne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Plik:Venn_A_union_B.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Venn_A_union_B.svg/250px-Venn_A_union_B.svg.png" decoding="async" width="250" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Venn_A_union_B.svg/375px-Venn_A_union_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Venn_A_union_B.svg/500px-Venn_A_union_B.svg.png 2x" data-file-width="271" data-file-height="186" /></a><figcaption>Suma zbiorów <i>A</i> i <i>B</i></figcaption></figure> <p><b>Sumą zbiorów</b> nazywa się <a href="/wiki/Zbi%C3%B3r" title="Zbiór">zbiór</a> złożony ze wszystkich elementów należących do <b>któregokolwiek</b> z sumowanych zbiorów (i niezawierający innych elementów)<sup id="cite_ref-CITEREFRasiowa197512_1-0" class="reference"><a href="#cite_note-CITEREFRasiowa197512-1">[1]</a></sup><sup id="cite_ref-CITEREFKuratowski198020_2-0" class="reference"><a href="#cite_note-CITEREFKuratowski198020-2">[2]</a></sup><sup id="cite_ref-CITEREFKuratowskiMostowski19526_3-0" class="reference"><a href="#cite_note-CITEREFKuratowskiMostowski19526-3">[3]</a></sup>. </p><p>Suma zbiorów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> i <span class="mwe-math-element"><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> jest oznaczana symbolem <span class="mwe-math-element"><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> (rzadziej <span class="mwe-math-element"><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/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" 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><sup id="cite_ref-CITEREFKuratowskiMostowski19526_3-1" class="reference"><a href="#cite_note-CITEREFKuratowskiMostowski19526-3">[3]</a></sup>). Tak więc: </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 x\in (A\cup B)\Leftrightarrow (x\in A)\lor (x\in B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (A\cup B)\Leftrightarrow (x\in A)\lor (x\in B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dafc589e81aad0d1d942b128bc1024a8714a24a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.732ex; height:2.843ex;" alt="{\displaystyle x\in (A\cup B)\Leftrightarrow (x\in A)\lor (x\in B)}" /></span><sup id="cite_ref-CITEREFRasiowa197512_1-1" class="reference"><a href="#cite_note-CITEREFRasiowa197512-1">[1]</a></sup><sup id="cite_ref-CITEREFKuratowski198020_2-1" class="reference"><a href="#cite_note-CITEREFKuratowski198020-2">[2]</a></sup><sup id="cite_ref-CITEREFKuratowskiMostowski19526_3-2" class="reference"><a href="#cite_note-CITEREFKuratowskiMostowski19526-3">[3]</a></sup>,</dd></dl> <p>co można równoważnie zapisać jako </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\cup B=\{x\in \Omega :x\in A\vee x\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</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\cup B=\{x\in \Omega :x\in A\vee x\in B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47268703276de2f31e43e9ff979a7e42e93cfba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.729ex; height:2.843ex;" alt="{\displaystyle A\cup B=\{x\in \Omega :x\in A\vee x\in B\}}" /></span><sup id="cite_ref-CITEREFLeitner199938_4-0" class="reference"><a href="#cite_note-CITEREFLeitner199938-4">[4]</a></sup><sup id="cite_ref-CITEREFRossWright199625_5-0" class="reference"><a href="#cite_note-CITEREFRossWright199625-5">[5]</a></sup>,</dd></dl> <p>gdzie <span class="mwe-math-element"><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\subset \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊂</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\subset \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fe09addbb8835ed80856fc17b15224525ba8df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.318ex; height:2.509ex;" alt="{\displaystyle A,B\subset \Omega }" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }" /></span> jest zbiorem wszystkich rozważanych obiektów zwanym <i>przestrzenią</i><sup id="cite_ref-CITEREFKuratowskiMostowski195218_6-0" class="reference"><a href="#cite_note-CITEREFKuratowskiMostowski195218-6">[6]</a></sup><sup id="cite_ref-CITEREFRasiowa197521_7-0" class="reference"><a href="#cite_note-CITEREFRasiowa197521-7">[7]</a></sup> lub <a href="/wiki/Uniwersum_(matematyka)" title="Uniwersum (matematyka)"><i>uniwersum</i></a><sup id="cite_ref-CITEREFRossWright199627_8-0" class="reference"><a href="#cite_note-CITEREFRossWright199627-8">[8]</a></sup>. </p><p>Suma jest zdefiniowana również dla większej ilości zbiorów: sumę <a href="/wiki/Rodzina_zbior%C3%B3w" title="Rodzina zbiorów">rodziny zbiorów</a> (zwaną też <b>sumą uogólnioną</b>) definiujemy jako zbiór elementów, które należą do przynajmniej jednego ze zbiorów z tej rodziny. Tak więc suma rodziny zbiorów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}" /></span> to </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 \bigcup {\mathfrak {A}}=\{x\in \Omega :(\exists A\in {\mathfrak {A}})(x\in A)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup {\mathfrak {A}}=\{x\in \Omega :(\exists A\in {\mathfrak {A}})(x\in A)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c6773ff5f2a936dc937f95ec71b0535c980828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:34.923ex; height:3.843ex;" alt="{\displaystyle \bigcup {\mathfrak {A}}=\{x\in \Omega :(\exists A\in {\mathfrak {A}})(x\in A)\}}" /></span><sup id="cite_ref-CITEREFKuratowskiMostowski195244_9-0" class="reference"><a href="#cite_note-CITEREFKuratowskiMostowski195244-9">[9]</a></sup>.</dd></dl> <p>Podobnie dla indeksowanej rodziny zbiorów <span class="mwe-math-element"><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_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcea4e7a3f02fab22acf9253d5cdb350e19de76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.077ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in I}}" /></span> definiujemy </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 \bigcup _{i\in I}A_{i}=\{a\in \Omega :(\exists i\in I)(a\in A_{i})\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{i\in I}A_{i}=\{a\in \Omega :(\exists i\in I)(a\in A_{i})\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400fde0d49ac42e026d44892e1fe89f6ddf01908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.607ex; height:5.676ex;" alt="{\displaystyle \bigcup _{i\in I}A_{i}=\{a\in \Omega :(\exists i\in I)(a\in A_{i})\},}" /></span></dd></dl> <p>co jest równoważne </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\in \bigcup _{i\in I}A_{i}\Leftrightarrow \exists i\in I\,(a\in A_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">⇔</mo> <mi mathvariant="normal">∃</mi> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \bigcup _{i\in I}A_{i}\Leftrightarrow \exists i\in I\,(a\in A_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8fb59fd36f8d349381d5906bfcc25c06a75711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.113ex; height:5.676ex;" alt="{\displaystyle a\in \bigcup _{i\in I}A_{i}\Leftrightarrow \exists i\in I\,(a\in A_{i})}" /></span><sup id="cite_ref-CITEREFRasiowa197552_10-0" class="reference"><a href="#cite_note-CITEREFRasiowa197552-10">[10]</a></sup><sup id="cite_ref-CITEREFKuratowski198043_11-0" class="reference"><a href="#cite_note-CITEREFKuratowski198043-11">[11]</a></sup>.</dd></dl> <p>Należy zauważyć, że poza <a href="/wiki/Teoria_mnogo%C5%9Bci" title="Teoria mnogości">teorią mnogości</a> matematycy używają raczej sum rodzin indeksowanych niż sum zbiorów zbiorów. Jedne mogą zostać zredukowane do drugich, np. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{i\in I}A_{i}=\bigcup \{A_{i}:i\in I\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{i\in I}A_{i}=\bigcup \{A_{i}:i\in I\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6223f4ce611355eeb04117f437c9ab72ca970210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.458ex; height:5.676ex;" alt="{\displaystyle \bigcup _{i\in I}A_{i}=\bigcup \{A_{i}:i\in I\},}" /></span> a użycie zapisu indeksowanego jest często bardziej czytelne. </p> <div class="mw-heading mw-heading3"><h3 id="Przykłady"><span id="Przyk.C5.82ady"></span>Przykłady</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=2" title="Edytuj sekcję: Przykłady" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Przykłady"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /></span> będzie zbiorem <a href="/wiki/Liczby_wymierne" title="Liczby wymierne">liczb wymiernych</a> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {I} \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2ae10a689f9746796e2a41bef7bf39bbb019e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle \mathbb {I} \mathbb {Q} }" /></span> niech będzie zbiorem <a href="/wiki/Liczby_niewymierne" title="Liczby niewymierne">liczb niewymiernych</a>. Wówczas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a955aa619d986d8c8e0ce962de0b40c4c546dbf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.104ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} }" /></span> jest zbiorem wszystkich <a href="/wiki/Liczby_rzeczywiste" title="Liczby rzeczywiste">liczb rzeczywistych</a>, tzn. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} =\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} =\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a6374008e541af4f3ffe9000f3bfe3925ad84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.88ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} \cup \mathbb {I} \mathbb {Q} =\mathbb {R} }" /></span><sup id="cite_ref-CITEREFRasiowa197512_1-2" class="reference"><a href="#cite_note-CITEREFRasiowa197512-1">[1]</a></sup>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,2)\cup [1,3]=(0,3],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,2)\cup [1,3]=(0,3],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94aa4a6b5364d8183ed600a571765d8de91cebab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.059ex; height:2.843ex;" alt="{\displaystyle (0,2)\cup [1,3]=(0,3],}" /></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 \bigcup \limits _{n\in \mathbb {N} }\left(1,2-{\frac {1}{n+1}}\right)=(1,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{n\in \mathbb {N} }\left(1,2-{\frac {1}{n+1}}\right)=(1,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae445995e18ff3968ad893f6ee9b748a2588e568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.777ex; height:6.843ex;" alt="{\displaystyle \bigcup \limits _{n\in \mathbb {N} }\left(1,2-{\frac {1}{n+1}}\right)=(1,2)}" /></span></li> <li>Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34aa92fbdb716183c034a2cfc30dafbaa51cfcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}" /></span> będzie rodziną wszystkich otwartych <a href="/wiki/Przedzia%C5%82_(matematyka)" title="Przedział (matematyka)">przedziałów</a> o końcach wymiernych zawartych w odcinku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\sqrt {2}},{\sqrt {5}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\sqrt {2}},{\sqrt {5}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965a4988f51bab7462038a0104fc114cca73f2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.429ex; height:3.176ex;" alt="{\displaystyle [{\sqrt {2}},{\sqrt {5}}).}" /></span> Wówczas</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup {\mathfrak {A}}=({\sqrt {2}},{\sqrt {5}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup {\mathfrak {A}}=({\sqrt {2}},{\sqrt {5}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4e78b2a63aa0fa790d26575ff7264f52d47a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.423ex; height:3.843ex;" alt="{\displaystyle \bigcup {\mathfrak {A}}=({\sqrt {2}},{\sqrt {5}}).}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Poprawność_definicji_sumy_zbiorów"><span id="Poprawno.C5.9B.C4.87_definicji_sumy_zbior.C3.B3w"></span>Poprawność definicji sumy zbiorów</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=3" title="Edytuj sekcję: Poprawność definicji sumy zbiorów" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Poprawność definicji sumy zbiorów"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>W powyższej definicji zakłada się, że dodawane zbiory są podzbiorami pewnego zbioru Ω zwanego przestrzenią. Definicja sumy dwóch zbiorów jest więc pewnym dwuargumentowym działaniem określonym na <a href="/wiki/Zbi%C3%B3r_pot%C4%99gowy" title="Zbiór potęgowy">zbiorze potęgowym</a> pewnego ustalonego zbioru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega {:}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>:</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega {:}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e42fba6c8fa68aa22cc0cda68512ef8923096" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \Omega {:}}" /></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 \cup :{\mathcal {P}}(\Omega )\times {\mathcal {P}}(\Omega ){\mathcal {\to }}{\mathcal {P}}(\Omega ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> <mo>:</mo> <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 mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>×</mo> <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 mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </mrow> <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 mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup :{\mathcal {P}}(\Omega )\times {\mathcal {P}}(\Omega ){\mathcal {\to }}{\mathcal {P}}(\Omega ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c52c9e45a60ca6001609ae58636932a402d877c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.871ex; height:2.843ex;" alt="{\displaystyle \cup :{\mathcal {P}}(\Omega )\times {\mathcal {P}}(\Omega ){\mathcal {\to }}{\mathcal {P}}(\Omega ).}" /></span></dd></dl> <p>Poprawność zdefiniowanego działania tj. istnienie jednoznacznego wyniku dla dowolnych dwóch argumentów wynika np. z <a href="/wiki/Aksjomat_podzbior%C3%B3w" title="Aksjomat podzbiorów">aksjomatu podzbiorów</a>. </p><p>Takie rozumienie definicji sumy wzmacniają <a href="/wiki/Diagram_Venna" title="Diagram Venna">diagramy Venna</a>, w których zbiory obrazowane są owalami rozgraniczającymi elementy przestrzeni Ω na te, które należą do danego zbioru, od tych, które do niego nie należą. </p><p>Opuszczenie warunku, aby dodawane zbiory były podzbiorami pewnego wspólnego zbioru, prowadzi do poważnych problemów teoriomnogościowych. Dodawanie zbiorów byłoby wówczas dwuargumentowym działaniem określonym na zbiorze wszystkich zbiorów, co oznacza odwołanie się do nieistniejącego obiektu (patrz <a href="/wiki/Paradoks_zbioru_wszystkich_zbior%C3%B3w" title="Paradoks zbioru wszystkich zbiorów">paradoks zbioru wszystkich zbiorów</a>). Z kolei definicja postaci <span class="mwe-math-element"><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=\{x:x\in A\vee x\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</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\cup B=\{x:x\in A\vee x\in B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9acd36ac0e7fa2743e63e6bb8b6690ac1102f565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.21ex; height:2.843ex;" alt="{\displaystyle A\cup B=\{x:x\in A\vee x\in B\}}" /></span> jest konstruowaniem zbioru poprzez podanie formuły, którą muszą spełniać jego elementy, co jest metodą, której należy unikać <a href="/wiki/Aksjomaty_Zermela-Fraenkla" title="Aksjomaty Zermela-Fraenkla">aksjomatycznej teorii mnogości</a>. Ostatecznie oznacza to nieistnienie dwuargumentowego działania dodawania zbiorów, o których nie ma dodatkowych założeń, a dla stwierdzenia istnienia sumy dwóch danych zbiorów należy powołać na <a href="/wiki/Aksjomat_sumy" title="Aksjomat sumy">aksjomat sumy</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Własności"><span id="W.C5.82asno.C5.9Bci"></span>Własności</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=4" title="Edytuj sekcję: Własności" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Własności"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Operacje_skończone"><span id="Operacje_sko.C5.84czone"></span>Operacje skończone</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=5" title="Edytuj sekcję: Operacje skończone" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Operacje skończone"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dla dowolnych zbiorów <span class="mwe-math-element"><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"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </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/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}" /></span> zachodzą następujące równości: </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 \bigcup \varnothing =\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mi class="MJX-variant">∅</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \varnothing =\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f03e6c969b987e9998419e103e6bcaa2a0f503e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.683ex; height:3.843ex;" alt="{\displaystyle \bigcup \varnothing =\varnothing }" /></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 \bigcup \{A\}=A=A\cup A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>∪</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \{A\}=A=A\cup A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d95cdfc45955de2a78611b3d176d7f6b3fd61e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.658ex; height:3.843ex;" alt="{\displaystyle \bigcup \{A\}=A=A\cup A}" /></span><sup id="cite_ref-CITEREFRasiowa197513_12-0" class="reference"><a href="#cite_note-CITEREFRasiowa197513-12">[12]</a></sup>     (<a href="/wiki/Idempotentno%C5%9B%C4%87" title="Idempotentność">idempotentność</a>)</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 \bigcup \{A,B\}=A\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \{A,B\}=A\cup B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b215b81991b9cad752e85f4332906eaa3e2ca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.636ex; height:3.843ex;" alt="{\displaystyle \bigcup \{A,B\}=A\cup B}" /></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 \varnothing \cup A=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>∪</mo> <mi>A</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \cup A=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708de667b9bbbba96d957d5750874ac37b7d47f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.975ex; height:2.176ex;" alt="{\displaystyle \varnothing \cup A=A}" /></span><sup id="cite_ref-CITEREFRasiowa197513_12-1" class="reference"><a href="#cite_note-CITEREFRasiowa197513-12">[12]</a></sup>     (<a href="/wiki/Zbi%C3%B3r_pusty" title="Zbiór pusty">zbiór pusty</a> jest <a href="/wiki/Element_neutralny" title="Element neutralny">elementem neutralnym</a> sumowania zbiorów)</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\cup B)\cup C=A\cup (B\cup C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cup B)\cup C=A\cup (B\cup C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d45d350edbde3ac8cfbd65af4013b5f5bc9559b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.594ex; height:2.843ex;" alt="{\displaystyle (A\cup B)\cup C=A\cup (B\cup C)}" /></span><sup id="cite_ref-CITEREFRasiowa197513_12-2" class="reference"><a href="#cite_note-CITEREFRasiowa197513-12">[12]</a></sup>     (<a href="/wiki/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka)">łączność</a>)</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\cup B=B\cup A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mo>∪</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B=B\cup A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fdb5dfc00d1e5850310af370e406c817267287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.278ex; height:2.176ex;" alt="{\displaystyle A\cup B=B\cup A}" /></span><sup id="cite_ref-CITEREFRasiowa197513_12-3" class="reference"><a href="#cite_note-CITEREFRasiowa197513-12">[12]</a></sup>     (<a href="/wiki/Przemienno%C5%9B%C4%87" title="Przemienność">przemienność</a>)</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)\cup C=(A\cup C)\cap (B\cup C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cap B)\cup C=(A\cup C)\cap (B\cup C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d8fe8892c74e04f8bc8b7e82379ee168902133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.753ex; height:2.843ex;" alt="{\displaystyle (A\cap B)\cup C=(A\cup C)\cap (B\cup C)}" /></span> oraz <span class="mwe-math-element"><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)\cap C=(A\cap C)\cup (B\cap C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cup B)\cap C=(A\cap C)\cup (B\cap C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d10919cb5277c9a7f31d51c4fec3b4910c7d32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.753ex; height:2.843ex;" alt="{\displaystyle (A\cup B)\cap C=(A\cap C)\cup (B\cap C)}" /></span><sup id="cite_ref-CITEREFRasiowa197517_13-0" class="reference"><a href="#cite_note-CITEREFRasiowa197517-13">[13]</a></sup>     (<a href="/wiki/Rozdzielno%C5%9B%C4%87" title="Rozdzielność">rozdzielność</a> każdego z dwóch działań, <a href="/wiki/Cz%C4%99%C5%9B%C4%87_wsp%C3%B3lna" title="Część wspólna">przekroju</a> i sumy, względem drugiego)</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 C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9645ff17cf12f478ebd4eeb16087a867b2ea9622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.589ex; height:2.843ex;" alt="{\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B)}" /></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62256681e906daba7376fb07b3ce0d060ee66777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.589ex; height:2.843ex;" alt="{\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B)}" /></span><sup id="cite_ref-CITEREFRasiowa197519_14-0" class="reference"><a href="#cite_note-CITEREFRasiowa197519-14">[14]</a></sup>     (<a href="/wiki/Prawa_De_Morgana" title="Prawa De Morgana">prawo De Morgana</a>).</li></ul> <p>Ponadto: </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\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> wtedy i tylko wtedy, gdy <span class="mwe-math-element"><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=B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B=B.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ede7f95cd817386b1fcf8c87e170b90d52391489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.599ex; height:2.176ex;" alt="{\displaystyle A\cup B=B.}" /></span></li> <li>Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> będzie niepustym zbiorem 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 {\mathcal {P}}(\mathbf {U} )}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(\mathbf {U} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53317fc98a93d2f181f82cdae0b6d119004837e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.57ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(\mathbf {U} )}" /></span> niech będzie <a href="/wiki/Zbi%C3%B3r_pot%C4%99gowy" title="Zbiór potęgowy">rodziną wszystkich podzbiorów</a> zbioru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a305ef479ab152035f334467a2c314baa23eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" /></span> Wówczas</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {P}}(\mathbf {U} ),\cup ,\cap ,\setminus ,\varnothing ,\mathbf {U} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>∪</mo> <mo>,</mo> <mo>∩</mo> <mo>,</mo> <mo class="MJX-variant">∖</mo> <mo>,</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {P}}(\mathbf {U} ),\cup ,\cap ,\setminus ,\varnothing ,\mathbf {U} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3204f998e703c0f4d99950147b2b656828a8dde7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.676ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {P}}(\mathbf {U} ),\cup ,\cap ,\setminus ,\varnothing ,\mathbf {U} )}" /></span></dd> <dd>jest zupełną <a href="/wiki/Algebra_Boole%E2%80%99a" title="Algebra Boole’a">algebrą Boole’a</a>.</dd></dl> <ul><li>Za pomocą sumy i <a href="/wiki/R%C3%B3%C5%BCnica_symetryczna_zbior%C3%B3w" title="Różnica symetryczna zbiorów">różnicy symetrycznej</a> można wyrazić <a href="/wiki/Cz%C4%99%C5%9B%C4%87_wsp%C3%B3lna" title="Część wspólna">iloczyn</a> i <a href="/wiki/R%C3%B3%C5%BCnica_zbior%C3%B3w" title="Różnica zbiorów">różnicę zbiorów</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=(A\cup B){\dot {-}}(A{\dot {-}}B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>−</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>−</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B=(A\cup B){\dot {-}}(A{\dot {-}}B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f162cd7a4dcac27606a98edd6deac987e8a238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.02ex; height:3.009ex;" alt="{\displaystyle A\cap B=(A\cup B){\dot {-}}(A{\dot {-}}B)}" /></span> oraz <span class="mwe-math-element"><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=A{\dot {-}}(A\cap 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> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>−</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B=A{\dot {-}}(A\cap B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e38946f2bd7867db7655c4d5cbc3c935515cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.251ex; height:3.009ex;" alt="{\displaystyle A\setminus B=A{\dot {-}}(A\cap B)}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Operacje_nieskończone"><span id="Operacje_niesko.C5.84czone"></span>Operacje nieskończone</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=6" title="Edytuj sekcję: Operacje nieskończone" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=6" title="Edytuj kod źródłowy sekcji: Operacje nieskończone"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Własności sumy skończenie wielu zbiorów uogólniają się na sumę rodzin indeksowanych zbiorów. Niech <span class="mwe-math-element"><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_{i}:i\in I\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A_{i}:i\in I\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574e983357455adefda2f8e51099c05944f70a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.267ex; height:2.843ex;" alt="{\displaystyle \{A_{i}:i\in I\},}" /></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_{i}:i\in I\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{B_{i}:i\in I\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e7a25d9f89c8c38b6f5831ffa4bdb0413577d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.641ex; height:2.843ex;" alt="{\displaystyle \{B_{i}:i\in I\}}" /></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{C_{j,k}:j\in J\ \wedge \ k\in K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>:</mo> <mi>j</mi> <mo>∈</mo> <mi>J</mi> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mi>k</mi> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{C_{j,k}:j\in J\ \wedge \ k\in K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa227d9ed501439138427d78d10311f1d7a8b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.279ex; height:3.009ex;" alt="{\displaystyle \{C_{j,k}:j\in J\ \wedge \ k\in K\}}" /></span> będą indeksowanymi rodzinami zbiorów. Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /></span> będzie zbiorem. Wówczas </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 \bigcup \limits _{i\in I}(A_{i}\cup B_{i})=\bigcup \limits _{i\in I}A_{i}\cup \bigcup \limits _{i\in I}B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∪</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∪</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{i\in I}(A_{i}\cup B_{i})=\bigcup \limits _{i\in I}A_{i}\cup \bigcup \limits _{i\in I}B_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0afd32d1b4d90a6687dded0088f986ae47776e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.805ex; height:5.676ex;" alt="{\displaystyle \bigcup \limits _{i\in I}(A_{i}\cup B_{i})=\bigcup \limits _{i\in I}A_{i}\cup \bigcup \limits _{i\in I}B_{i}}" /></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 \bigcup \limits _{i\in I}(A_{i}\cap B_{i})\subseteq \bigcup \limits _{i\in I}A_{i}\cap \bigcup \limits _{i\in I}B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊆</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{i\in I}(A_{i}\cap B_{i})\subseteq \bigcup \limits _{i\in I}A_{i}\cap \bigcup \limits _{i\in I}B_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811f042e3df9e72c7e9b03f8da784e3733db6856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.805ex; height:5.676ex;" alt="{\displaystyle \bigcup \limits _{i\in I}(A_{i}\cap B_{i})\subseteq \bigcup \limits _{i\in I}A_{i}\cap \bigcup \limits _{i\in I}B_{i}}" /></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 D\cap \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cap D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>∩</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\cap \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cap D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3a284e3afc17460719736f2a2b0cb32fd85621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.557ex; height:5.676ex;" alt="{\displaystyle D\cap \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cap D)}" /></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 D\cup \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cup D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>∪</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∪</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\cup \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cup D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bef244b3662e3f4509d52099652d843e242ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.557ex; height:5.676ex;" alt="{\displaystyle D\cup \bigcup \limits _{i\in I}A_{i}=\bigcup \limits _{i\in I}(A_{i}\cup D)}" /></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 D\setminus \bigcup \limits _{i\in I}A_{i}=\bigcap \limits _{i\in I}D\setminus A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo class="MJX-variant">∖</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>D</mi> <mo class="MJX-variant">∖</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\setminus \bigcup \limits _{i\in I}A_{i}=\bigcap \limits _{i\in I}D\setminus A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e76f00bc6e7f77d034680bedceca3177f068c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.359ex; height:5.676ex;" alt="{\displaystyle D\setminus \bigcup \limits _{i\in I}A_{i}=\bigcap \limits _{i\in I}D\setminus A_{i}}" /></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 \bigcup \limits _{j\in J}\bigcup \limits _{k\in K}C_{j,k}=\bigcup \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{j\in J}\bigcup \limits _{k\in K}C_{j,k}=\bigcup \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10920e21ac3d45ffa42383bde2ee62f6af7adfff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:24.892ex; height:5.843ex;" alt="{\displaystyle \bigcup \limits _{j\in J}\bigcup \limits _{k\in K}C_{j,k}=\bigcup \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}" /></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 \bigcup \limits _{j\in J}\bigcap \limits _{k\in K}C_{j,k}\subseteq \bigcap \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <munder> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>⊆</mo> <munder> <mo movablelimits="false">⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup \limits _{j\in J}\bigcap \limits _{k\in K}C_{j,k}\subseteq \bigcap \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1f53097bc21c8b3c530ca7b98570aaba17ec25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:24.892ex; height:5.843ex;" alt="{\displaystyle \bigcup \limits _{j\in J}\bigcap \limits _{k\in K}C_{j,k}\subseteq \bigcap \limits _{k\in K}\bigcup \limits _{j\in J}C_{j,k}}" /></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Suma_a_obrazy_i_przeciwobrazy">Suma a obrazy i przeciwobrazy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=7" title="Edytuj sekcję: Suma a obrazy i przeciwobrazy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=7" title="Edytuj kod źródłowy sekcji: Suma a obrazy i przeciwobrazy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dla dowolnej funkcji <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\longrightarrow Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⟶</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\longrightarrow Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc1d9f5c39fc63e86686b3c47d6648f4f7405d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.809ex; height:2.509ex;" alt="{\displaystyle f\colon X\longrightarrow Y,}" /></span> dla dowolnej rodziny indeksowanej <span class="mwe-math-element"><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_{i}:i\in I\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A_{i}:i\in I\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281c6b78c86b6c92ac20c8044889aa73a59c9dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.62ex; height:2.843ex;" alt="{\displaystyle \{A_{i}:i\in I\}}" /></span> <a href="/wiki/Podzbi%C3%B3r" title="Podzbiór">podzbiorów</a> zbioru <span class="mwe-math-element"><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> <mo>,</mo> </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/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></span> oraz dla dowolnej rodziny indeksowanej <span class="mwe-math-element"><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_{j}:j\in J\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>:</mo> <mi>j</mi> <mo>∈</mo> <mi>J</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{B_{j}:j\in J\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8febb1e68f15050434a2079407afa83e6c77f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.206ex; height:3.009ex;" alt="{\displaystyle \{B_{j}:j\in J\}}" /></span> podzbiorów zbioru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" /></span> prawdziwe są następujące dwa stwierdzenia: </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 f\left(\bigcup \limits _{i\in I}A_{i}\right)=\bigcup \limits _{i\in I}f(A_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(\bigcup \limits _{i\in I}A_{i}\right)=\bigcup \limits _{i\in I}f(A_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67fcd74c490bbfa6ae1223c556518aed13183ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.556ex; height:7.509ex;" alt="{\displaystyle f\left(\bigcup \limits _{i\in I}A_{i}\right)=\bigcup \limits _{i\in I}f(A_{i})}" /></span> (czyli <a href="/wiki/Obraz_(matematyka)" title="Obraz (matematyka)">obraz</a> sumy jest sumą obrazów).</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 f^{-1}\left(\bigcup \limits _{j\in J}B_{j}\right)=\bigcup \limits _{j\in J}f^{-1}(B_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="false">⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\left(\bigcup \limits _{j\in J}B_{j}\right)=\bigcup \limits _{j\in J}f^{-1}(B_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a941c8f69a4580d5d0bc92ff298a488ead1ba7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.033ex; height:7.676ex;" alt="{\displaystyle f^{-1}\left(\bigcup \limits _{j\in J}B_{j}\right)=\bigcup \limits _{j\in J}f^{-1}(B_{j})}" /></span> (inaczej mówiąc, <a href="/wiki/Przeciwobraz" title="Przeciwobraz">przeciwobraz</a> sumy jest sumą przeciwobrazów);</li></ul> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=8" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=8" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint plainlinks" cellpadding="4" role="presentation"> <tbody><tr> <td style="vertical-align:middle; text-align:center; width:30px;"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/28px-Wikibooks-logo.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/42px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/56px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> </td> <td style="line-height:normal; vertical-align:middle; text-align:center; flex:unset;">Zobacz podręcznik w Wikibooks: <b><a href="https://pl.wikibooks.org/wiki/Matematyka_dla_liceum" class="extiw" title="b:Matematyka dla liceum">Matematyka dla liceum</a> – <a href="https://pl.wikibooks.org/wiki/Matematyka_dla_liceum/Liczby_i_ich_zbiory/Dzia%C5%82ania_na_zbiorach#Suma_zbiorów" class="extiw" title="b:Matematyka dla liceum/Liczby i ich zbiory/Działania na zbiorach">Liczby i ich zbiory</a></b> </td></tr></tbody></table> <ul><li><a href="/wiki/Zasada_w%C5%82%C4%85cze%C5%84_i_wy%C5%82%C4%85cze%C5%84" title="Zasada włączeń i wyłączeń">zasada włączeń i wyłączeń</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=9" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=9" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-CITEREFRasiowa197512-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-CITEREFRasiowa197512_1-0">a</a></sup> <sup><a href="#cite_ref-CITEREFRasiowa197512_1-1">b</a></sup> <sup><a href="#cite_ref-CITEREFRasiowa197512_1-2">c</a></sup></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 12.</span> </li> <li id="cite_note-CITEREFKuratowski198020-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-CITEREFKuratowski198020_2-0">a</a></sup> <sup><a href="#cite_ref-CITEREFKuratowski198020_2-1">b</a></sup></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFKuratowski1980">Kuratowski 1980 ↓</a></span>, s. 20.</span> </li> <li id="cite_note-CITEREFKuratowskiMostowski19526-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-CITEREFKuratowskiMostowski19526_3-0">a</a></sup> <sup><a href="#cite_ref-CITEREFKuratowskiMostowski19526_3-1">b</a></sup> <sup><a href="#cite_ref-CITEREFKuratowskiMostowski19526_3-2">c</a></sup></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFKuratowskiMostowski1952">Kuratowski i Mostowski 1952 ↓</a></span>, s. 6.</span> </li> <li id="cite_note-CITEREFLeitner199938-4"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFLeitner199938_4-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFLeitner1999">Leitner 1999 ↓</a></span>, s. 38.</span> </li> <li id="cite_note-CITEREFRossWright199625-5"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRossWright199625_5-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRossWright1996">Ross i Wright 1996 ↓</a></span>, s. 25.</span> </li> <li id="cite_note-CITEREFKuratowskiMostowski195218-6"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFKuratowskiMostowski195218_6-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFKuratowskiMostowski1952">Kuratowski i Mostowski 1952 ↓</a></span>, s. 18.</span> </li> <li id="cite_note-CITEREFRasiowa197521-7"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRasiowa197521_7-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 21.</span> </li> <li id="cite_note-CITEREFRossWright199627-8"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRossWright199627_8-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRossWright1996">Ross i Wright 1996 ↓</a></span>, s. 27.</span> </li> <li id="cite_note-CITEREFKuratowskiMostowski195244-9"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFKuratowskiMostowski195244_9-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFKuratowskiMostowski1952">Kuratowski i Mostowski 1952 ↓</a></span>, s. 44.</span> </li> <li id="cite_note-CITEREFRasiowa197552-10"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRasiowa197552_10-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 52.</span> </li> <li id="cite_note-CITEREFKuratowski198043-11"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFKuratowski198043_11-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFKuratowski1980">Kuratowski 1980 ↓</a></span>, s. 43.</span> </li> <li id="cite_note-CITEREFRasiowa197513-12"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-CITEREFRasiowa197513_12-0">a</a></sup> <sup><a href="#cite_ref-CITEREFRasiowa197513_12-1">b</a></sup> <sup><a href="#cite_ref-CITEREFRasiowa197513_12-2">c</a></sup> <sup><a href="#cite_ref-CITEREFRasiowa197513_12-3">d</a></sup></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 13.</span> </li> <li id="cite_note-CITEREFRasiowa197517-13"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRasiowa197517_13-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 17.</span> </li> <li id="cite_note-CITEREFRasiowa197519-14"><span class="mw-cite-backlink"><a href="#cite_ref-CITEREFRasiowa197519_14-0">↑</a></span> <span class="reference-text"> <span class="harvard-citation"><a href="#CITEREFRasiowa1975">Rasiowa 1975 ↓</a></span>, s. 19.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=10" title="Edytuj sekcję: Bibliografia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=10" title="Edytuj kod źródłowy sekcji: Bibliografia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book" id="CITEREFKuratowskiMostowski1952"><a href="/wiki/Kazimierz_Kuratowski" title="Kazimierz Kuratowski">Kazimierz Kuratowski</a>, <a href="/wiki/Andrzej_Mostowski" class="mw-redirect" title="Andrzej Mostowski">Andrzej Mostowski</a>: <i><a rel="nofollow" class="external text" href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.dl-catalog-234246e7-642c-4afd-8764-5edfcc26fad4">Teoria mnogości</a></i>. Warszawa: Polskie Towarzystwo Matematyczne, 1952, seria: Monografie matematyczne, t. 27. <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/250182901">250182901</a>. [dostęp 2016-09-23].<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teoria+mnogo%C5%9Bci&rft.aulast=Kuratowski&rft.aufirst=Kazimierz&rft.pub=Polskie+Towarzystwo+Matematyczne&rft.place=Warszawa&rft.series=Monografie+matematyczne%2C+t.+27&rft_id=info:oclcnum/250182901&rft_id=http%3A%2F%2Fpldml.icm.edu.pl%2Fpldml%2Felement%2Fbwmeta1.element.dl-catalog-234246e7-642c-4afd-8764-5edfcc26fad4"></span></cite></li> <li><cite class="citation book" id="CITEREFKuratowski1980"><a href="/wiki/Kazimierz_Kuratowski" title="Kazimierz Kuratowski">Kazimierz Kuratowski</a>: <i>Wstęp do teorii mnogości i topologii</i>. Wyd. 8. Warszawa: PWN, 1980, seria: <a href="/wiki/Biblioteka_Matematyczna" title="Biblioteka Matematyczna">Biblioteka Matematyczna</a>, t. 9. <a href="/wiki/Specjalna:Ksi%C4%85%C5%BCki/8301013729" title="Specjalna:Książki/8301013729">ISBN <span class="isbn">83-01-01372-9</span></a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wst%C4%99p+do+teorii+mnogo%C5%9Bci+i+topologii&rft.aulast=Kuratowski&rft.aufirst=Kazimierz&rft.edition=8&rft.pub=PWN&rft.place=Warszawa&rft.series=%5B%5BBiblioteka+Matematyczna%5D%5D%2C+t.+9&rft.isbn=83-01-01372-9"></span></cite></li> <li><cite class="citation book" id="CITEREFLeitner1999">Roman Leitner: <i>Zarys matematyki wyższej dla studentów</i>. Wyd. 11. Cz. 1. Warszawa: WNT, 1999. <a href="/wiki/Specjalna:Ksi%C4%85%C5%BCki/8320423953" title="Specjalna:Książki/8320423953">ISBN <span class="isbn">83-204-2395-3</span></a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zarys+matematyki+wy%C5%BCszej+dla+student%C3%B3w&rft.aulast=Leitner&rft.aufirst=Roman&rft.edition=11&rft.pub=WNT&rft.place=Warszawa&rft.isbn=83-204-2395-3"></span></cite></li> <li><cite class="citation book" id="CITEREFRasiowa1975"><a href="/wiki/Helena_Rasiowa" title="Helena Rasiowa">Helena Rasiowa</a>: <i>Wstęp do matematyki współczesnej</i>. Wyd. 5. Warszawa: PWN, 1975, seria: <a href="/wiki/Biblioteka_Matematyczna" title="Biblioteka Matematyczna">Biblioteka Matematyczna</a>, t. 30. <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/749626864">749626864</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wst%C4%99p+do+matematyki+wsp%C3%B3%C5%82czesnej&rft.aulast=Rasiowa&rft.aufirst=Helena&rft.edition=5&rft.pub=PWN&rft.place=Warszawa&rft.series=%5B%5BBiblioteka+Matematyczna%5D%5D%2C+t.+30&rft_id=info:oclcnum/749626864"></span></cite></li> <li><cite class="citation book" id="CITEREFRossWright1996">Kenneth A. Ross, Charles R.B Wright: <i>Matematyka dyskretna</i>. E. Sepko-Guzicka (tłum.), W. Guzicki (tłum.), P. Zakrzewski (tłum.). Warszawa: <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a>, 1996. <a href="/wiki/Specjalna:Ksi%C4%85%C5%BCki/8301121297" title="Specjalna:Książki/8301121297">ISBN <span class="isbn">83-01-12129-7</span></a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matematyka+dyskretna&rft.aulast=Ross&rft.aufirst=Kenneth+A.&rft.pub=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft.place=Warszawa&rft.isbn=83-01-12129-7"></span></cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Linki_zewnętrzne"><span id="Linki_zewn.C4.99trzne"></span>Linki zewnętrzne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Suma_zbior%C3%B3w&veaction=edit&section=11" title="Edytuj sekcję: Linki zewnętrzne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Suma_zbior%C3%B3w&action=edit&section=11" title="Edytuj kod źródłowy sekcji: Linki zewnętrzne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Otwarty_dost%C4%99p" title="publikacja w otwartym dostępie – możesz ją przeczytać"><img alt="publikacja w otwartym dostępie – możesz ją przeczytać" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/8px-Open_Access_logo_green_alt2.svg.png" decoding="async" width="8" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/12px-Open_Access_logo_green_alt2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/16px-Open_Access_logo_green_alt2.svg.png 2x" data-file-width="640" data-file-height="1000" /></a></span> <i><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Union_of_sets">Union of sets</a></i> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>, Encyclopedia of Mathematics, encyclopediaofmath.org [dostęp 2024-09-19].</li></ul> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r75675918">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output table.navbox:not(.pionowy){width:100%}.mw-parser-output .navbox+.navbox{border-top:0;margin-top:0}.mw-parser-output .navbox.pionowy{width:250px;float:right;clear:right;margin:0 0 0.4em 1.4em}.mw-parser-output .navbox.pionowy .before,.mw-parser-output .navbox.pionowy .after{padding:0.5em 0;text-align:center}.mw-parser-output .navbox>.caption,.mw-parser-output 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.opis.a3,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a3 .opis{background:#764617}}</style><ul class="tnavbar noprint plainlinks hlist"><li><a href="/wiki/Szablon:Algebra_zbior%C3%B3w" title="Szablon:Algebra zbiorów"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/w/index.php?title=Dyskusja_szablonu:Algebra_zbior%C3%B3w&action=edit&redlink=1" class="new" title="Dyskusja szablonu:Algebra zbiorów (strona nie istnieje)"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Algebra_zbior%C3%B3w&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Algebra_zbior%C3%B3w_(nauka)" title="Algebra zbiorów (nauka)">Algebra zbiorów</a></div><div class="mw-collapsible-content flex"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row"><a href="/wiki/Dzia%C5%82anie_algebraiczne" title="Działanie algebraiczne">działania</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Dzia%C5%82anie_jednoargumentowe" title="Działanie jednoargumentowe">jednoargumentowe</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Dope%C5%82nienie_zbioru" title="Dopełnienie zbioru">dopełnienie</a></li> <li><a href="/wiki/Zbi%C3%B3r_pot%C4%99gowy" title="Zbiór potęgowy">zbiór potęgowy</a></li></ul> </td></tr><tr class="a1_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Dzia%C5%82anie_dwuargumentowe" title="Działanie dwuargumentowe">dwuargumentowe</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a class="mw-selflink selflink">suma</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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }" /></span></li> <li><a href="/wiki/Cz%C4%99%C5%9B%C4%87_wsp%C3%B3lna" title="Część wspólna">przekrój</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 \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" /></span></li> <li><a href="/wiki/R%C3%B3%C5%BCnica_zbior%C3%B3w" title="Różnica zbiorów">różnica</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 \setminus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">∖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \setminus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e20e45087a97f0448fc3d4bc27b060084830f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \setminus }" /></span></li> <li><a href="/wiki/R%C3%B3%C5%BCnica_symetryczna_zbior%C3%B3w" title="Różnica symetryczna zbiorów">różnica symetryczna</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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /></span></li> <li><a href="/wiki/Iloczyn_kartezja%C5%84ski" title="Iloczyn kartezjański">iloczyn kartezjański</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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" /></span></li> <li><a href="/wiki/Suma_roz%C5%82%C4%85czna" title="Suma rozłączna">suma rozłączna</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 \sqcup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sqcup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1596aedf354da694149e44ce2bf53ede54eca8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \sqcup }" /></span></li></ul> </td></tr></tbody></table></td></tr><tr class="a2"><th class="navbox-group opis" scope="row">własności<br />działań</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a2_1"><th class="navbox-group opis" scope="row">indywidualne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li>prawa <a href="/wiki/Przemienno%C5%9B%C4%87" title="Przemienność">przemienności</a>, <a href="/wiki/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka)">łączności</a> i <a href="/wiki/Idempotentno%C5%9B%C4%87" title="Idempotentność">idempotencji</a></li> <li>elementy <a href="/wiki/Element_neutralny" title="Element neutralny">neutralne</a> i <a href="/wiki/Element_absorbuj%C4%85cy" title="Element absorbujący">absorbujące</a></li></ul> </td></tr><tr class="a2_2"><th class="navbox-group opis" scope="row">związki między działaniami</th><td class="navbox-list spis hlist navbox-even"> <ul><li>prawa <a href="/wiki/Rozdzielno%C5%9B%C4%87" title="Rozdzielność">rozdzielności</a> i <a href="/wiki/Prawa_De_Morgana" title="Prawa De Morgana">De Morgana</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a3"><th class="navbox-group opis" scope="row">powiązane<br /><a href="/wiki/Relacja_(matematyka)" title="Relacja (matematyka)">relacje</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li>przecinanie <ul><li>zawieranie, in. inkluzja: <a href="/wiki/Podzbi%C3%B3r" title="Podzbiór">podzbiór</a> i nadzbiór</li></ul></li> <li><a href="/wiki/Zbiory_roz%C5%82%C4%85czne" title="Zbiory rozłączne">rozłączność</a></li></ul> </td></tr><tr class="a4"><th class="navbox-group opis" scope="row">tworzone<br /><a href="/wiki/Algebra_og%C3%B3lna" title="Algebra ogólna">struktury<br />algebraiczne</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a4_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Grupoid" title="Grupoid">grupoid</a> (magma)</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/P%C3%B3%C5%82grupa" title="Półgrupa">półgrupa</a> <ul><li><a href="/wiki/Monoid" title="Monoid">monoid</a></li> <li><a href="/wiki/Pas_(teoria_p%C3%B3%C5%82grup)" title="Pas (teoria półgrup)">pas</a></li> <li><a href="/wiki/Grupa_przemienna" title="Grupa przemienna">grupa przemienna</a></li></ul></li></ul> </td></tr><tr class="a4_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Krata_(matematyka)#Półkraty" title="Krata (matematyka)">półkrata</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Krata_(matematyka)" title="Krata (matematyka)">krata</a> <ul><li><a href="/wiki/Algebra_Boole%E2%80%99a" title="Algebra Boole’a">algebra Boole’a</a></li></ul></li></ul> </td></tr><tr class="a4_3"><th class="navbox-group opis" scope="row"><a href="/wiki/P%C3%B3%C5%82pier%C5%9Bcie%C5%84" title="Półpierścień">półpierścień</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Pier%C5%9Bcie%C5%84_(matematyka)" title="Pierścień (matematyka)">pierścień</a> <ul><li><a href="/wiki/Pier%C5%9Bcie%C5%84_zbior%C3%B3w" title="Pierścień zbiorów">pierścień zbiorów</a></li></ul></li></ul> </td></tr></tbody></table></td></tr><tr class="a5"><th class="navbox-group opis" scope="row">inne <a href="/wiki/Rodzina_zbior%C3%B3w" title="Rodzina zbiorów">rodziny</a><br />zdefiniowane<br />działaniami</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a5_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Pokrycie_zbioru" title="Pokrycie zbioru">pokrycie zbioru</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Rozbicie_zbioru" title="Rozbicie zbioru">rozbicie zbioru</a> <ul><li><a href="/wiki/Dychotomia" title="Dychotomia">dychotomia</a></li></ul></li></ul> </td></tr><tr class="a5_2"><th class="navbox-group opis" scope="row"><a href="/wiki/%CE%A0-uk%C5%82ad" title="Π-układ">π-układ</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Przestrze%C5%84_topologiczna" title="Przestrzeń topologiczna">topologia</a></li></ul> </td></tr><tr class="a5_3"><th class="navbox-group opis" scope="row">definiowane różnicami</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/%CE%A3-pier%C5%9Bcie%C5%84" title="Σ-pierścień">σ-pierścień</a></li> <li><a href="/wiki/Cia%C5%82o_zbior%C3%B3w" title="Ciało zbiorów">ciało zbiorów</a> <ul><li><a href="/wiki/Przestrze%C5%84_mierzalna#Definicje" title="Przestrzeń mierzalna">σ-ciało</a></li></ul></li> <li><a href="/wiki/%CE%9B-uk%C5%82ad" title="Λ-układ">λ-układ</a></li></ul> </td></tr><tr class="a5_4"><th class="navbox-group opis" scope="row">pozostałe</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Filtr_(teoria_zbior%C3%B3w)" title="Filtr (teoria zbiorów)">filtr</a> <ul><li><a href="/wiki/Filtr_(matematyka)#Filtr_maksymalny" title="Filtr (matematyka)">ultrafiltr</a></li></ul></li> <li><a href="/wiki/Idea%C5%82_(teoria_mnogo%C5%9Bci)" title="Ideał (teoria mnogości)">ideał</a> <ul><li><a href="/wiki/Idea%C5%82_pierwszy_(teoria_mnogo%C5%9Bci)" title="Ideał pierwszy (teoria mnogości)">ideał pierwszy</a></li></ul></li> <li><a href="/wiki/Klasa_monotoniczna" title="Klasa monotoniczna">klasa monotoniczna</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a6"><th class="navbox-group opis" scope="row"><a href="/wiki/Twierdzenie" title="Twierdzenie">twierdzenia</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Lemat_o_%CF%80-_i_%CE%BB-uk%C5%82adach" title="Lemat o π- i λ-układach">lemat o π- i λ-układach</a></li></ul> </td></tr><tr class="a7"><th class="navbox-group opis" scope="row">powiązane<br />nauki</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a7_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Podstawy_matematyki" title="Podstawy matematyki">podstawy matematyki</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Logika_matematyczna" title="Logika matematyczna">logika matematyczna</a> <ul><li><a href="/wiki/Klasyczny_rachunek_zda%C5%84" title="Klasyczny rachunek zdań">klasyczny rachunek zdań</a></li></ul></li> <li><a href="/wiki/Teoria_mnogo%C5%9Bci" title="Teoria mnogości">teoria mnogości</a></li></ul> </td></tr><tr class="a7_2"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Kombinatoryka" title="Kombinatoryka">kombinatoryka</a></li> <li><a href="/wiki/Teoria_miary" title="Teoria miary">teoria miary</a></li> <li><a href="/wiki/Teoria_prawdopodobie%C5%84stwa" title="Teoria prawdopodobieństwa">probabilistyka</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a8"><th class="navbox-group opis" 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