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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definícia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definícia</span> </div> </a> <ul id="toc-Definícia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Označovanie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Označovanie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Označovanie</span> </div> </a> <ul id="toc-Označovanie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Príklady" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Príklady"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Príklady</span> </div> </a> <button aria-controls="toc-Príklady-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>Přepnout podsekci Príklady</span> </button> <ul id="toc-Príklady-sublist" class="vector-toc-list"> <li id="toc-Používané_Boolové_algebry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Používané_Boolové_algebry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Používané Boolové algebry</span> </div> </a> <ul id="toc-Používané_Boolové_algebry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pozri_aj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pozri_aj"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Pozri aj</span> </div> </a> <ul id="toc-Pozri_aj-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="Obsah" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Prepnúť obsah" > <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">Prepnúť obsah</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">Boolova algebra</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="Prejsť na článok v inom jazyku. Je dostupný v 66 jazykoch" > <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-66" 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">66 jazykov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Boolse_algebra" title="Boolse algebra – afrikánčina" lang="af" hreflang="af" data-title="Boolse algebra" data-language-autonym="Afrikaans" data-language-local-name="afrikánčina" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%A8%D9%88%D9%84" title="جبر بول – arabčina" lang="ar" hreflang="ar" data-title="جبر بول" data-language-autonym="العربية" data-language-local-name="arabčina" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81lxebra_de_Boole" title="Álxebra de Boole – astúrčina" lang="ast" hreflang="ast" data-title="Álxebra de Boole" data-language-autonym="Asturianu" data-language-local-name="astúrčina" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bul_c%C9%99bri_(m%C9%99ntiqi)" title="Bul cəbri (məntiqi) – azerbajdžančina" lang="az" hreflang="az" data-title="Bul cəbri (məntiqi)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžančina" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A8%D9%88%D9%84_%D8%AC%D8%A8%D8%B1%DB%8C" title="بول جبری – South Azerbaijani" lang="azb" hreflang="azb" data-title="بول جبری" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D2%BB%D1%8B" title="Логика алгебраһы – baškirčina" lang="ba" hreflang="ba" data-title="Логика алгебраһы" data-language-autonym="Башҡортса" data-language-local-name="baškirčina" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булева алгебра – bulharčina" lang="bg" hreflang="bg" data-title="Булева алгебра" data-language-autonym="Български" data-language-local-name="bulharčina" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%81%E0%A6%B2%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="বুলিয়ান বীজগণিত – bengálčina" lang="bn" hreflang="bn" data-title="বুলিয়ান বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="bengálčina" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – bosniačtina" lang="bs" hreflang="bs" data-title="Booleova algebra" data-language-autonym="Bosanski" data-language-local-name="bosniačtina" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_de_Boole" title="Àlgebra de Boole – katalánčina" lang="ca" hreflang="ca" data-title="Àlgebra de Boole" data-language-autonym="Català" data-language-local-name="katalánčina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – čeština" lang="cs" hreflang="cs" data-title="Booleova algebra" data-language-autonym="Čeština" data-language-local-name="čeština" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D0%B0%D0%BD%C4%83%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8" title="Каланăлăхсен алгебри – čuvaština" lang="cv" hreflang="cv" data-title="Каланăлăхсен алгебри" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra – dánčina" lang="da" hreflang="da" data-title="Boolesk algebra" data-language-autonym="Dansk" data-language-local-name="dánčina" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Boolesche_Algebra" title="Boolesche Algebra – nemčina" lang="de" hreflang="de" data-title="Boolesche Algebra" data-language-autonym="Deutsch" data-language-local-name="nemčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1_%CE%9C%CF%80%CE%BF%CF%85%CE%BB" title="Άλγεβρα Μπουλ – gréčtina" lang="el" hreflang="el" data-title="Άλγεβρα Μπουλ" data-language-autonym="Ελληνικά" data-language-local-name="gréčtina" 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/Boolean_algebra" title="Boolean algebra – angličtina" lang="en" hreflang="en" data-title="Boolean algebra" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Bulea_algebro" title="Bulea algebro – esperanto" lang="eo" hreflang="eo" data-title="Bulea algebro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_de_Boole" title="Álgebra de Boole – španielčina" lang="es" hreflang="es" data-title="Álgebra de Boole" data-language-autonym="Español" data-language-local-name="španielčina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Boole%27i_algebra" title="Boole'i algebra – estónčina" lang="et" hreflang="et" data-title="Boole'i algebra" data-language-autonym="Eesti" data-language-local-name="estónčina" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Booleren_aljebra" title="Booleren aljebra – baskičtina" lang="eu" hreflang="eu" data-title="Booleren aljebra" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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%AC%D8%A8%D8%B1_%D8%A8%D9%88%D9%84%DB%8C" title="جبر بولی – perzština" lang="fa" hreflang="fa" data-title="جبر بولی" data-language-autonym="فارسی" data-language-local-name="perzština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Boolen_algebra" title="Boolen algebra – fínčina" lang="fi" hreflang="fi" data-title="Boolen algebra" data-language-autonym="Suomi" data-language-local-name="fínčina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Boole_(logique)" title="Algèbre de Boole (logique) – francúzština" lang="fr" hreflang="fr" data-title="Algèbre de Boole (logique)" data-language-autonym="Français" data-language-local-name="francúzština" 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/Ailg%C3%A9abar_Boole" title="Ailgéabar Boole – írčina" lang="ga" hreflang="ga" data-title="Ailgéabar Boole" data-language-autonym="Gaeilge" data-language-local-name="írčina" 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/%C3%81lxebra_de_Boole" title="Álxebra de Boole – galícijčina" lang="gl" hreflang="gl" data-title="Álxebra de Boole" data-language-autonym="Galego" data-language-local-name="galícijčina" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%95%D7%9C%D7%99%D7%90%D7%A0%D7%99%D7%AA" title="אלגברה בוליאנית – hebrejčina" lang="he" hreflang="he" data-title="אלגברה בוליאנית" data-language-autonym="עברית" data-language-local-name="hebrejčina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%82%E0%A4%B2%E0%A5%80%E0%A4%AF_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4_(%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="बूलीय बीजगणित (तर्कशास्त्र) – hindčina" lang="hi" hreflang="hi" data-title="बूलीय बीजगणित (तर्कशास्त्र)" data-language-autonym="हिन्दी" data-language-local-name="hindčina" 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/Booleova_algebra" title="Booleova algebra – chorvátčina" lang="hr" hreflang="hr" data-title="Booleova algebra" data-language-autonym="Hrvatski" data-language-local-name="chorvátčina" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Boole-algebra_(informatika)" title="Boole-algebra (informatika) – maďarčina" lang="hu" hreflang="hu" data-title="Boole-algebra (informatika)" data-language-autonym="Magyar" data-language-local-name="maďarčina" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D6%82%D5%AC%D5%B5%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Բուլյան հանրահաշիվ – arménčina" lang="hy" hreflang="hy" data-title="Բուլյան հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="arménčina" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_Boole" title="Aljabar Boole – indonézština" lang="id" hreflang="id" data-title="Aljabar Boole" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonézština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Booleana_algebro" title="Booleana algebro – ido" lang="io" hreflang="io" data-title="Booleana algebro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_di_Boole" title="Algebra di Boole – taliančina" lang="it" hreflang="it" data-title="Algebra di Boole" data-language-autonym="Italiano" data-language-local-name="taliančina" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Boolean_Logic" title="Boolean Logic – kikujčina" lang="ki" hreflang="ki" data-title="Boolean Logic" data-language-autonym="Gĩkũyũ" data-language-local-name="kikujčina" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%82%E0%B2%B2%E0%B2%BF%E0%B2%AF%E0%B2%A8%E0%B3%8D_%E0%B2%AC%E0%B3%80%E0%B2%9C%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಬೂಲಿಯನ್ ಬೀಜಗಣಿತ – kannadčina" lang="kn" hreflang="kn" data-title="ಬೂಲಿಯನ್ ಬೀಜಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannadčina" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%88_%EB%85%BC%EB%A6%AC" title="불 논리 – kórejčina" lang="ko" hreflang="ko" data-title="불 논리" data-language-autonym="한국어" data-language-local-name="kórejčina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Mantiq%C3%AA_B%C3%BBl%C3%AE" title="Mantiqê Bûlî – kurdčina" lang="ku" hreflang="ku" data-title="Mantiqê Bûlî" data-language-autonym="Kurdî" data-language-local-name="kurdčina" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D1%82%D1%8B%D0%BB%D1%8B%D1%88%D1%82%D0%B0%D1%80_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%81%D1%8B" title="Айтылыштар алгебрасы – kirgizština" lang="ky" hreflang="ky" data-title="Айтылыштар алгебрасы" data-language-autonym="Кыргызча" data-language-local-name="kirgizština" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_Booleana_(logica)" title="Algebra Booleana (logica) – latinčina" lang="la" hreflang="la" data-title="Algebra Booleana (logica)" data-language-autonym="Latina" data-language-local-name="latinčina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/B%C5%ABlio_algebra" title="Būlio algebra – litovčina" lang="lt" hreflang="lt" data-title="Būlio algebra" data-language-autonym="Lietuvių" data-language-local-name="litovčina" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/B%C5%ABla_algebra" title="Būla algebra – lotyština" lang="lv" hreflang="lv" data-title="Būla algebra" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булова алгебра – macedónčina" lang="mk" hreflang="mk" data-title="Булова алгебра" data-language-autonym="Македонски" data-language-local-name="macedónčina" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/%C3%81lgebra_de_Boole" title="Álgebra de Boole – mirandčina" lang="mwl" hreflang="mwl" data-title="Álgebra de Boole" data-language-autonym="Mirandés" data-language-local-name="mirandčina" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%98%E1%80%B0%E1%80%9C%E1%80%AE%E1%80%9A%E1%80%94%E1%80%BA%E1%80%A1%E1%80%80%E1%80%B9%E1%80%81%E1%80%9B%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ဘူလီယန်အက္ခရာသင်္ချာ – barmčina" lang="my" hreflang="my" data-title="ဘူလီယန်အက္ခရာသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="barmčina" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Booleaanse_algebra" title="Booleaanse algebra – holandčina" lang="nl" hreflang="nl" data-title="Booleaanse algebra" data-language-autonym="Nederlands" data-language-local-name="holandčina" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Boolsk_algebra" title="Boolsk algebra – nórčina (nynorsk)" lang="nn" hreflang="nn" data-title="Boolsk algebra" data-language-autonym="Norsk nynorsk" data-language-local-name="nórčina (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Boolsk_algebra" title="Boolsk algebra – nórčina (bokmal)" lang="nb" hreflang="nb" data-title="Boolsk algebra" data-language-autonym="Norsk bokmål" data-language-local-name="nórčina (bokmal)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/%C3%80lgebra_%C3%ABd_Boole" title="Àlgebra ëd Boole – piemonština" lang="pms" hreflang="pms" data-title="Àlgebra ëd Boole" data-language-autonym="Piemontèis" data-language-local-name="piemonština" 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/%C3%81lgebra_booliana" title="Álgebra booliana – portugalčina" lang="pt" hreflang="pt" data-title="Álgebra booliana" data-language-autonym="Português" data-language-local-name="portugalčina" 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/Algebr%C4%83_boolean%C4%83" title="Algebră booleană – rumunčina" lang="ro" hreflang="ro" data-title="Algebră booleană" data-language-autonym="Română" data-language-local-name="rumunčina" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B8" title="Алгебра логики – ruština" lang="ru" hreflang="ru" data-title="Алгебра логики" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bulova_algebra" title="Bulova algebra – srbochorvátčina" lang="sh" hreflang="sh" data-title="Bulova algebra" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvátčina" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Boolean_algebra" title="Boolean algebra – Simple English" lang="en-simple" hreflang="en-simple" data-title="Boolean algebra" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra – slovinčina" lang="sl" hreflang="sl" data-title="Booleova algebra" data-language-autonym="Slovenščina" data-language-local-name="slovinčina" 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%91%D1%83%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Булова алгебра – srbčina" lang="sr" hreflang="sr" data-title="Булова алгебра" data-language-autonym="Српски / srpski" data-language-local-name="srbčina" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra – švédčina" lang="sv" hreflang="sv" data-title="Boolesk algebra" data-language-autonym="Svenska" data-language-local-name="švédčina" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%82%E0%AE%B2%E0%AE%BF%E0%AE%AF_%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="பூலிய இயற்கணிதம் – tamilčina" lang="ta" hreflang="ta" data-title="பூலிய இயற்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="tamilčina" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8_%D0%BC%D0%B0%D0%BD%D1%82%D0%B8%D2%9B" title="Алгебраи мантиқ – tadžičtina" lang="tg" hreflang="tg" data-title="Алгебраи мантиқ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžičtina" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B5%E0%B8%8A%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%9A%E0%B8%B9%E0%B8%A5" title="พีชคณิตแบบบูล – thajčina" lang="th" hreflang="th" data-title="พีชคณิตแบบบูล" data-language-autonym="ไทย" data-language-local-name="thajčina" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alhebrang_Boolean" title="Alhebrang Boolean – tagalčina" lang="tl" hreflang="tl" data-title="Alhebrang Boolean" data-language-autonym="Tagalog" data-language-local-name="tagalčina" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boole_cebiri" title="Boole cebiri – turečtina" lang="tr" hreflang="tr" data-title="Boole cebiri" data-language-autonym="Türkçe" data-language-local-name="turečtina" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B8" title="Алгебра логіки – ukrajinčina" lang="uk" hreflang="uk" data-title="Алгебра логіки" data-language-autonym="Українська" data-language-local-name="ukrajinčina" 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/%C4%90%E1%BA%A1i_s%E1%BB%91_Boole" title="Đại số Boole – vietnamčina" lang="vi" hreflang="vi" data-title="Đại số Boole" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamčina" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数 – čínština (wu)" lang="wuu" hreflang="wuu" data-title="逻辑代数" data-language-autonym="吴语" data-language-local-name="čínština (wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数 – čínština" lang="zh" hreflang="zh" data-title="逻辑代数" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E4%BB%A3%E6%95%B8" title="邏輯代數 – kantončina" lang="yue" hreflang="yue" data-title="邏輯代數" data-language-autonym="粵語" data-language-local-name="kantončina" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Vzhľad</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skryť</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">z Wikipédie, slobodnej encyklopédie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sk" dir="ltr"><p><b>Boolova algebra</b> je <a href="/wiki/Algebra" class="mw-disambig" title="Algebra">algebrická</a> štruktúra, ktorá modeluje vlastnosti množinových a logických operácií. Je nazvaná podľa írskeho matematika <a href="/wiki/George_Boole" title="George Boole">George Boolea</a>. </p><p>Boolova algebra je abstraktný formálny systém obsahujúci množinu prvkov (a, b, c, ...), nad ktorou sú definované dve binárne operácie symbolizované pomocou znakov <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" 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 \lor }"></span> 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 \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" 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 \land }"></span>. Boolova algebra je komplementárny a distributívny zväz pomenovaný podľa <a href="/wiki/George_Boole" title="George Boole">Georgea Boolea</a> (1815 – 1864). Boolova algebra je zvláštnym prípadom <a href="/wiki/%C5%A0trukt%C3%BAra" class="mw-disambig" title="Štruktúra">štruktúry</a> zvanej <a href="/wiki/Zv%C3%A4z_(matematika)" title="Zväz (matematika)">zväz</a>. </p><p>Boolova algebra má interpretácie v rôznych vedných disciplínach, napríklad: </p> <table class="wikitable"> <tbody><tr> <th>Boolova algebra</th> <th><a href="/wiki/Mno%C5%BEinov%C3%A1_algebra" title="Množinová algebra">množinová algebra</a></th> <th><a href="/wiki/V%C3%BDrokov%C3%A1_algebra" title="Výroková algebra">výroková algebra</a> (logika) </th></tr> <tr> <td>a, b, c, ... prvky<br /><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" 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 \lor }"></span> (alebo)</td> <td>A, B, C, ... podmnožiny<br />množiny I<br />U (zjednotenie)</td> <td>p, q, r, ... výroky<br />množiny U<br />∨ (disjunkcia) </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" 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 \land }"></span> (a)</td> <td>∩ (prienik)</td> <td>∧ (konjunkcia) </td></tr> <tr> <td>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 \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" 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 \lor }"></span> b = b <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" 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 \lor }"></span> a</td> <td>A U B = B U A</td> <td>p ∨ q = q ∨ p </td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definícia"><span id="Defin.C3.ADcia"></span>Definícia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=1" title="Upraviť sekciu: Definícia" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=1" title="Editovat zdrojový kód sekce Definícia"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Boolova algebra je definovaná ako distributívny komplementárny <a href="/wiki/Zv%C3%A4z_(matematika)" title="Zväz (matematika)">zväz</a>. </p><p>Boolova algebra je <a href="/wiki/Usporiadan%C3%A1_mno%C5%BEina" class="mw-redirect" title="Usporiadaná množina">šestica</a> (<i>A</i>, ∧, ∨, −, 0, 1), kde <i>A</i> je neprázdna <a href="/wiki/Mno%C5%BEina" title="Množina">množina</a>, 0 ∈ <i>A</i> je <a href="/w/index.php?title=Najmen%C5%A1%C3%AD_prvok&action=edit&redlink=1" class="new" title="Najmenší prvok (stránka neexistuje)">najmenší</a>, 1 ∈ <i>A</i> <a href="/w/index.php?title=Najv%C3%A4%C4%8D%C5%A1%C3%AD_prvok&action=edit&redlink=1" class="new" title="Najväčší prvok (stránka neexistuje)">najväčší prvok</a>, − je unárna operácia (<a href="/wiki/Komplement" class="mw-disambig" title="Komplement">komplement</a>) a ∧, ∨ sú binárne operácie (<a href="/wiki/Priese%C4%8Dn%C3%ADk" title="Priesečník">priesečník</a>, <a href="/wiki/Spojenie" class="mw-disambig" title="Spojenie">spojenie</a>) na <i>A</i>, spĺňajúce nasledujúce <a href="/wiki/Axi%C3%B3m" class="mw-redirect" title="Axióm">axiómy</a>. </p> <dl><dd><table cellpadding="5"> <tbody><tr> <td><a href="/wiki/Komutativita" class="mw-redirect" title="Komutativita">Komutativita</a>: </td> <td><span class="mwe-math-element"><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\lor y=y\lor x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∨</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lor y=y\lor x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ae0defb8c995a1ea0b001b20dd3bbaae177459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle x\lor y=y\lor x}"></span> </td> <td><span class="mwe-math-element"><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\land y=y\land x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∧</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\land y=y\land x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29630a3db3e9315226b64f64c7d151c4ad3ff09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle x\land y=y\land x}"></span> </td></tr> <tr> <td><a href="/w/index.php?title=Distributivita&action=edit&redlink=1" class="new" title="Distributivita (stránka neexistuje)">Distributivita</a>: </td> <td><span class="mwe-math-element"><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\lor (y\land z)=(x\lor y)\land (x\lor z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∧</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∨</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∨</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lor (y\land z)=(x\lor y)\land (x\lor z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a98ea87085569bb5a869cfab563218b22e5efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.916ex; height:2.843ex;" alt="{\displaystyle x\lor (y\land z)=(x\lor y)\land (x\lor z)}"></span> </td> <td><span class="mwe-math-element"><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\land (y\lor z)=(x\land y)\lor (x\land z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∨</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∧</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\land (y\lor z)=(x\land y)\lor (x\land z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d605239fc4493f0d535cca1c42c7e743493138b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.916ex; height:2.843ex;" alt="{\displaystyle x\land (y\lor z)=(x\land y)\lor (x\land z)}"></span> </td></tr> <tr> <td><a href="/wiki/Neutr%C3%A1lny_prvok" title="Neutrálny prvok">Neutralita</a> 0 a 1: </td> <td><span class="mwe-math-element"><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\lor 0=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lor 0=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4f0e08229faf04dd102d9aaef650b7d878e105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.503ex; height:2.176ex;" alt="{\displaystyle x\lor 0=x}"></span> </td> <td><span class="mwe-math-element"><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\land 1=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mn>1</mn> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\land 1=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a5bf445b2efef620ed6bb30df5ba69e8f88ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.503ex; height:2.176ex;" alt="{\displaystyle x\land 1=x}"></span> </td></tr> <tr> <td><a href="/w/index.php?title=Komplementarita_(matematika)&action=edit&redlink=1" class="new" title="Komplementarita (matematika) (stránka neexistuje)">Komplementarita</a>: </td> <td><span class="mwe-math-element"><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\lor -x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lor -x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b6c8a0bf944605e50ff3d130b7857a8d07eb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.311ex; height:2.343ex;" alt="{\displaystyle x\lor -x=1}"></span> </td> <td><span class="mwe-math-element"><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\land -x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\land -x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9470143b18ae12329357ba2280e510bd0a642bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.311ex; height:2.343ex;" alt="{\displaystyle x\land -x=0}"></span> </td></tr> <tr> <td>Nedegenerovanosť: </td> <td colspan="2"><span class="mwe-math-element"><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\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5add705e86314a6ce57c76d7493896b092661a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 0\neq 1}"></span> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=2" title="Upraviť sekciu: Vlastnosti" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=2" title="Editovat zdrojový kód sekce Vlastnosti"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pre Boolovu algebru <i>A</i> a každé <i>x</i>, <i>y</i>, <i>z</i> ∈ <i>A</i> platí: </p> <ul><li><a href="/wiki/Asociativita" class="mw-redirect" title="Asociativita">asociativita</a>: (<i>x</i> ∨ <i>y</i>) ∨ <i>z</i> = <i>x</i> ∨ (<i>y</i> ∨ <i>z</i>), (<i>x</i> ∧ <i>y</i>) ∧ <i>z</i> = <i>x</i> ∧ (<i>y</i> ∧ <i>z</i>)</li> <li>absorpcia: <i>x</i> ∨ (<i>x</i> ∧ <i>y</i>) = <i>x</i>, <i>x</i> ∧ (<i>x</i> ∨ <i>y</i>) = <i>x</i></li> <li>agresivita nuly: <i>x</i> ∧ 0 = 0</li> <li>agresivita jednotky: <i>x</i> ∨ 1 = 1</li> <li><a href="/w/index.php?title=Idempotencia&action=edit&redlink=1" class="new" title="Idempotencia (stránka neexistuje)">idempotencia</a>: <i>x</i> ∨ <i>x</i> = <i>x</i>, <i>x</i> ∧ <i>x</i> = <i>x</i></li> <li>absorpcia negácie: <i>x</i> ∨ (−<i>x</i> ∧ <i>y</i>) = <i>x</i> ∨ <i>y</i>, <i>x</i> ∧ (−<i>x</i> ∨ <i>y</i>) = <i>x</i> ∧ <i>y</i></li> <li>dvojitá negácia: −(−<i>x</i>) = <i>x</i></li> <li><a href="/wiki/De_Morganove_z%C3%A1kony" title="De Morganove zákony">De Morganove zákony</a>: −<i>x</i> ∧ −<i>y</i> = −(<i>x</i> ∨ <i>y</i>), −<i>x</i> ∨ −<i>y</i> = −(<i>x</i> ∧ <i>y</i>)</li> <li><i>0</i> a <i>1</i> sú vzájomne komplementárne: −0 = 1, −1 = 0</li></ul> <div class="mw-heading mw-heading2"><h2 id="Označovanie"><span id="Ozna.C4.8Dovanie"></span>Označovanie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=3" title="Upraviť sekciu: Označovanie" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=3" title="Editovat zdrojový kód sekce Označovanie"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Existujú najmenej tri najznámejšie tradície v označovaní v teórii Boolovej algebry. Vo vyššie použitej definícii sú použité symboly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor ,\land ,\lnot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor ,\land ,\lnot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baacf1b07121f70f63f47749810179ecd4310b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.719ex; height:2.343ex;" alt="{\displaystyle \lor ,\land ,\lnot }"></span>, ale bežne sú používané tiež <span class="mwe-math-element"><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 ,\cap ,\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> <mo>,</mo> <mo>∩</mo> <mo>,</mo> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup ,\cap ,\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8869c46adc5d28c97b1df95ce4980482516a347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.343ex;" alt="{\displaystyle \cup ,\cap ,\sim }"></span>, a na bežné použitie tiež <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +,\cdot ,-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +,\cdot ,-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdcb8bf76d5f5585286a4be4a28f9efbbf1dbd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.331ex; height:2.343ex;" alt="{\displaystyle +,\cdot ,-}"></span>. Symboly dvojargumentovýh operácií Boolovej algebry sú takmer vždy výberom jedného z páru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (+,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251907581f6b17b10e0298bac837c2997e367762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle (+,\cdot )}"></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 (\lor ,\land )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lor ,\land )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e917b7c2afc3b7426e81d4d8a17d0dd160a37c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.944ex; height:2.843ex;" alt="{\displaystyle (\lor ,\land )}"></span> alebo <span class="mwe-math-element"><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 ,\cap )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∪</mo> <mo>,</mo> <mo>∩</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cup ,\cap )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75390b703b6ff41a81c13688d9ad9108ffe3c2b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.944ex; height:2.843ex;" alt="{\displaystyle (\cup ,\cap )}"></span>. Označenie operácií jednoargumentovej algebry je menej, v dôsledku toho sa môžeme stretnúť ako so symbolmi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +,\cdot ,\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +,\cdot ,\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e4fdc01d4ff5ac059b8d833bbe9e55529910d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.331ex; height:2.343ex;" alt="{\displaystyle +,\cdot ,\sim }"></span> tak aj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor ,\land ,{}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor ,\land ,{}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3eb4c181a230d60525e07f344b6e2fbc145762c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle \lor ,\land ,{}^{\prime }}"></span>. </p><p>Symboly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge ,\vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge ,\vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95c04402ead9d1eca305bd9be599054fe841c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.135ex; height:2.343ex;" alt="{\displaystyle \wedge ,\vee }"></span> sa často používajú v súvislosti algebrickými teóriami. </p><p>Stretávame sa aj s použitím iných symbolov resp. ich kombinácií (na príklad  &  na miesto  <span class="mwe-math-element"><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>,  alebo  <span class="mwe-math-element"><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^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7749224d3755a26d61a125944c65aeaaec3a1fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.915ex; height:2.509ex;" alt="{\displaystyle a^{\prime }}"></span>  namiesto  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sim a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo>∼</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sim a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18518814e76973b1c22778d5a929fe86b1426f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.328ex; height:1.676ex;" alt="{\displaystyle \;\sim a}"></span>. V oblasti elektroniky a informačných technológií je často používaný ako OR, AND resp. NOT  na mieste  <span class="mwe-math-element"><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>, <span class="mwe-math-element"><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> resp. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Príklady"><span id="Pr.C3.ADklady"></span>Príklady</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=4" title="Upraviť sekciu: Príklady" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=4" title="Editovat zdrojový kód sekce Príklady"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Najjednoduchšia Boolova algebra obsahuje len jeden prvok, alebo 0 = 1 (tu nejde o spor, ale o dvojité označovanie jedného <a href="/wiki/Prvok_mno%C5%BEiny" title="Prvok množiny">prvku</a>). Všetky <a href="/w/index.php?title=Oper%C3%A1cia_(matematika)&action=edit&redlink=1" class="new" title="Operácia (matematika) (stránka neexistuje)">operácie</a> dávajú rovnaký výsledok (iné tu ani neexistujú), preto sa nazýva triviálne. Táto algebra samozrejme môže existovať jedine vtedy, keď sa vypustí Axiom nedegenerovanosti.</li> <li>Duálna algebra je algebra nad množinou <i>A</i>= (0, 1), kde operácie sú dané prirodzeným spôsobom.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Používané_Boolové_algebry"><span id="Pou.C5.BE.C3.ADvan.C3.A9_Boolov.C3.A9_algebry"></span>Používané Boolové algebry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=5" title="Upraviť sekciu: Používané Boolové algebry" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=5" title="Editovat zdrojový kód sekce Používané Boolové algebry"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Najvýznamnejšími príkladmi Boolových algebier sú algebry výrokov (alebo všeobecnejšie <a href="/w/index.php?title=Lindenbaumova_algebra&action=edit&redlink=1" class="new" title="Lindenbaumova algebra (stránka neexistuje)">Lindenbaumovej algebry</a> <a href="/wiki/Formula_(logika)" title="Formula (logika)">formulí</a>) a množinové algebry. </p> <ul><li>U algebier výrokov v dvojhodnotovej logike je <i>A</i>= (nepravda, pravda), 0 = nepravda, 1 = pravda, a operácie zodpovedajú <a href="/wiki/Disjunkcia" class="mw-disambig" title="Disjunkcia">disjunkcii</a>, <a href="/wiki/Konjunkcia_(logika)" title="Konjunkcia (logika)">konjunkcii</a> a <a href="/wiki/Neg%C3%A1cia" class="mw-disambig" title="Negácia">negácii</a>.</li></ul> <table border="0"> <tbody><tr> <td> <table class="wikitable"> <caption>Konjunkcia </caption> <tbody><tr> <td align="center" width="25"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" 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 \wedge }"></span> </td> <td align="center" width="25"><b>0</b> </td> <td align="center" width="25"><b>1</b> </td></tr> <tr> <td align="center" width="25"><b>0</b> </td> <td align="center" width="25">0 </td> <td align="center" width="25">0 </td></tr> <tr> <td align="center" width="25"><b>1</b> </td> <td align="center" width="25">0 </td> <td align="center" width="25">1 </td></tr></tbody></table> </td> <td>  </td> <td> <table class="wikitable"> <caption>Disjunkcia </caption> <tbody><tr> <td align="center" width="25"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" 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 \lor }"></span> </td> <td align="center" width="25"><b>0</b> </td> <td align="center" width="25"><b>1</b> </td></tr> <tr> <td align="center" width="25"><b>0</b> </td> <td align="center" width="25">0 </td> <td align="center" width="25">1 </td></tr> <tr> <td align="center" width="25"><b>1</b> </td> <td align="center" width="25">1 </td> <td align="center" width="25">1 </td></tr></tbody></table> </td> <td>  </td> <td> <table class="wikitable"> <caption>Negácia </caption> <tbody><tr> <td align="center" width="25">  </td> <td align="center" width="25"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span> </td></tr> <tr> <td align="center" width="25"><b>0</b> </td> <td align="center" width="25">1 </td></tr> <tr> <td align="center" width="25"><b>1</b> </td> <td align="center" width="25">0 </td></tr></tbody></table> </td></tr></tbody></table> <ul><li><a href="/w/index.php?title=Lindenbaumova_algebra&action=edit&redlink=1" class="new" title="Lindenbaumova algebra (stránka neexistuje)">Lindenbaumovy algebry</a> sú definované nad množinou <i>A</i> všetkých tried ekvivalencie <a href="/wiki/Formula_(logika)" title="Formula (logika)">formulou</a> dotknutého <a href="/w/index.php?title=Jazyk_(logika)&action=edit&redlink=1" class="new" title="Jazyk (logika) (stránka neexistuje)">jazyka</a> a operácie sú rovnaké ako u algebier výrokov.</li> <li>U množinových algebier je algebra definovaná nad <a href="/wiki/Poten%C4%8Dn%C3%A1_mno%C5%BEina" title="Potenčná množina">potenčnou množinou</a> ľubovoľnej množiny <i>S</i>, tzn. <i>A</i>= 2 <sup><i>S</i></sup>, najmenším prvkom 0 je <a href="/wiki/Pr%C3%A1zdna_mno%C5%BEina" title="Prázdna množina">prázdna množina</a>, najväčším prvkom 1 je sama množina <i>S</i>a operácie zodpovedajú <a href="/wiki/Prienik" class="mw-redirect" title="Prienik">prieniku</a>, <a href="/wiki/Zjednotenie" class="mw-disambig" title="Zjednotenie">zjednoteniu</a> a <a href="/wiki/Doplnok_(mno%C5%BEiny)" title="Doplnok (množiny)">doplnku</a> do množiny <i>S</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Pozri_aj">Pozri aj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolova_algebra&veaction=edit&section=6" title="Upraviť sekciu: Pozri aj" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolova_algebra&action=edit&section=6" title="Editovat zdrojový kód sekce Pozri aj"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/De_Morganove_z%C3%A1kony" title="De Morganove zákony">De Morganove zákony</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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