CINXE.COM

<!DOCTYPE html> <html class="client-nojs" lang="fi" dir="ltr"> <head> <meta charset="UTF-8"> <title>Boolen algebra – Wikipedia</title> <script>(function(){var className="client-js";var cookie=document.cookie.match(/(?:^|; )fiwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"fi normal","wgMonthNames":["","tammikuu","helmikuu","maaliskuu","huhtikuu","toukokuu","kesäkuu","heinäkuu","elokuu","syyskuu","lokakuu","marraskuu","joulukuu"],"wgRequestId":"b942ad0c-e816-4507-af31-5ff88072e0b3","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Boolen_algebra","wgTitle":"Boolen algebra","wgCurRevisionId":22769795,"wgRevisionId":22769795,"wgArticleId":71597,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null, "wgUserGroups":["*"],"wgCategories":["Seulonnan keskeiset artikkelit","Matemaattinen logiikka","Algebralliset rakenteet"],"wgPageViewLanguage":"fi","wgPageContentLanguage":"fi","wgPageContentModel":"wikitext","wgRelevantPageName":"Boolen_algebra","wgRelevantArticleId":71597,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Boolen_logiikka","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":3}}},"wgStableRevisionId":22769795,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"fi","pageLanguageDir":"ltr","pageVariantFallbacks":"fi"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgInternalRedirectTargetUrl":"/wiki/Boolen_algebra", "wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":true,"wgVector2022LanguageInHeader":false,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q173183","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.hidePersonalSandboxEdits":"ready","ext.gadget.fiwiki_flaggedrevs_css_rcfix":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.styles.legacy":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles": "ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","codex-search-styles":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.legacy.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.publicarttablesort","ext.gadget.ViikonKilpailu","ext.gadget.WikiLovesMonunmets","ext.gadget.ProtectionIndicator","ext.gadget.frwiki_infobox_v3","ext.gadget.linkeddata","ext.gadget.perustiedotwikidatassa","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.compactlinks","ext.uls.interface","ext.cx.eventlogging.campaigns", "ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=fi&modules=codex-search-styles%7Cext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.styles.legacy%7Cwikibase.client.init&only=styles&skin=vector"> <script async="" src="/w/load.php?lang=fi&modules=startup&only=scripts&raw=1&skin=vector"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=fi&modules=ext.gadget.fiwiki_flaggedrevs_css_rcfix%2ChidePersonalSandboxEdits&only=styles&skin=vector"> <link rel="stylesheet" href="/w/load.php?lang=fi&modules=site.styles&only=styles&skin=vector"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Boolen algebra – Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//fi.m.wikipedia.org/wiki/Boolen_algebra"> <link rel="alternate" type="application/x-wiki" title="Muokkaa" href="/w/index.php?title=Boolen_algebra&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (fi)"> <link rel="EditURI" type="application/rsd+xml" href="//fi.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://fi.wikipedia.org/wiki/Boolen_algebra"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fi"> <link rel="alternate" type="application/atom+xml" title="Wikipedia-Atom-syöte" href="/w/index.php?title=Toiminnot:Tuoreet_muutokset&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Boolen_algebra rootpage-Boolen_algebra skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Boolen algebra</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Wikipediasta</div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Ohjattu sivulta <a href="/w/index.php?title=Boolen_logiikka&redirect=no" class="mw-redirect" title="Boolen logiikka">Boolen logiikka</a>)</span></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Siirry navigaatioon</a> <a class="mw-jump-link" href="#searchInput">Siirry hakuun</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fi" dir="ltr"><p><b>Boolen algebra</b> on <a href="/wiki/George_Boole" title="George Boole">George Boolen</a> mukaan nimensä saanut <a href="/wiki/Algebrallinen_struktuuri" class="mw-redirect" title="Algebrallinen struktuuri">algebrallinen struktuuri</a>, joka toimii loogisen <a href="/wiki/Lausekalkyyli" class="mw-redirect" title="Lausekalkyyli">lausekalkyylin</a> ja <a href="/wiki/Joukko-oppi" title="Joukko-oppi">joukko-opin</a> mallina. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="fi" dir="ltr"><h2 id="mw-toc-heading">Sisällys</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Määritelmä"><span class="tocnumber">1</span> <span class="toctext">Määritelmä</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Boolen_algebra_ja_lausekalkyyli"><span class="tocnumber">2</span> <span class="toctext">Boolen algebra ja lausekalkyyli</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Boolen_algebran_läheinen_yhteys_joukko-oppiin"><span class="tocnumber">3</span> <span class="toctext">Boolen algebran läheinen yhteys joukko-oppiin</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Boolen_algebra_biteistä_koostuvana"><span class="tocnumber">3.1</span> <span class="toctext">Boolen algebra biteistä koostuvana</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Joukko-opillisen_Boolen_algebran_alkioina_käyttämät_osajoukot"><span class="tocnumber">3.2</span> <span class="toctext">Joukko-opillisen Boolen algebran alkioina käyttämät osajoukot</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Esimerkki_atomittomasta_Boolen_algebrasta"><span class="tocnumber">3.3</span> <span class="toctext">Esimerkki atomittomasta Boolen algebrasta</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Äärellisistä_Boolen_algebroista"><span class="tocnumber">3.4</span> <span class="toctext">Äärellisistä Boolen algebroista</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Joukko-opillisen_Boolen_algebran_yhteys_Sigma-algebraan"><span class="tocnumber">3.5</span> <span class="toctext">Joukko-opillisen Boolen algebran yhteys Sigma-algebraan</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Kuviollinen_esitys"><span class="tocnumber">4</span> <span class="toctext">Kuviollinen esitys</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Katso_myös"><span class="tocnumber">5</span> <span class="toctext">Katso myös</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Lähteet"><span class="tocnumber">6</span> <span class="toctext">Lähteet</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Aiheesta_muualla"><span class="tocnumber">7</span> <span class="toctext">Aiheesta muualla</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Määritelmä"><span id="M.C3.A4.C3.A4ritelm.C3.A4"></span>Määritelmä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=1" title="Muokkaa osiota Määritelmä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=1" title="Muokkaa osion lähdekoodia: Määritelmä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Boolen algebran muodostaa joukko, jossa on määritelty kaksi <a href="/wiki/Laskutoimitus" title="Laskutoimitus">laskutoimitusta</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> ja <span class="mwe-math-element"><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>, laskutoimitusten <a href="/wiki/Neutraalialkio" title="Neutraalialkio">neutraalialkiot</a> 0 ja 1 (joista käytetään joskus myös merkintöjä ⊥ ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span>) ja jossa jokaisella alkiolla on lisäksi <a href="/wiki/Komplementti" class="mw-disambig" title="Komplementti">komplementti</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 \neg a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e72f6b2a9120b875c42a17235dbf8d417e9abbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:1.676ex;" alt="{\displaystyle \neg a}"></span> siten, että ne toteuttavat seuraavat ehdot: </p> <dl><dd><dl><dd><table cellpadding="5"> <tbody><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 a\lor (b\lor c)=(a\lor b)\lor c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∨</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\lor (b\lor c)=(a\lor b)\lor c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29f59a522f4e46f01e318a3360327aa92424a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.516ex; height:2.843ex;" alt="{\displaystyle a\lor (b\lor c)=(a\lor b)\lor c}"></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 a\land (b\land c)=(a\land b)\land c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\land (b\land c)=(a\land b)\land c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80f3236791cddb20844f0f2c4ae2d17bfcc4c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.516ex; height:2.843ex;" alt="{\displaystyle a\land (b\land c)=(a\land b)\land c}"></span> </td> <td><a href="/wiki/Liit%C3%A4nt%C3%A4laki" class="mw-redirect" title="Liitäntälaki">liitäntälait</a> </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 a\lor b=b\lor 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\lor b=b\lor a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c667fd125343e1040d3e9720cc9b11b57aa72fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.718ex; height:2.176ex;" alt="{\displaystyle a\lor b=b\lor a}"></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 a\land b=b\land 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\land b=b\land a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe1def17ab431529915da41ae07e25ada77fa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.718ex; height:2.176ex;" alt="{\displaystyle a\land b=b\land a}"></span> </td> <td><a href="/wiki/Vaihdantalaki" class="mw-redirect" title="Vaihdantalaki">vaihdantalait</a> </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 a\lor (a\land b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\lor (a\land b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6d72148fbb065632557beb01401df218091d07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.76ex; height:2.843ex;" alt="{\displaystyle a\lor (a\land b)=a}"></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 a\land (a\lor b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\land (a\lor b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aabf1bc4aa530b4243da83d1676e9c6fbb83b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.76ex; height:2.843ex;" alt="{\displaystyle a\land (a\lor b)=a}"></span> </td> <td>absorptiolait </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 a\lor (b\land c)=(a\lor b)\land (a\lor c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\lor (b\land c)=(a\lor b)\land (a\lor c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9c372cc0d1eccdefd57aa503b28574a39aef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.138ex; height:2.843ex;" alt="{\displaystyle a\lor (b\land c)=(a\lor b)\land (a\lor c)}"></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 a\land (b\lor c)=(a\land b)\lor (a\land c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∨</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\land (b\lor c)=(a\land b)\lor (a\land c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff9ab68dfe58822c0855bb4b9584198c611a310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.138ex; height:2.843ex;" alt="{\displaystyle a\land (b\lor c)=(a\land b)\lor (a\land c)}"></span>   </td> <td><a href="/wiki/Osittelulaki" title="Osittelulaki">osittelulait</a> </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 a\lor {\neg }a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\lor {\neg }a=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51682b189b3627e3eb5de16cddad66bfaf3166bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.853ex; height:2.176ex;" alt="{\displaystyle a\lor {\neg }a=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 a\land {\neg }a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\land {\neg }a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043fe7cb925ea5383a17c99e69e2ab81a9529b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.853ex; height:2.176ex;" alt="{\displaystyle a\land {\neg }a=0}"></span> </td> <td> </td></tr></tbody></table></dd></dl></dd></dl> <p>Toisinaan määritelmän ehdoista jätetään pois absorptiolait, ja ne korvataan uusilla ehdoilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} a\lor 0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>a</mi> <mo>∨</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} a\lor 0=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e8e795dbf5c975a822d8ec0ceeb75b35f68d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.303ex; height:2.176ex;" alt="{\displaystyle \mathbf {} a\lor 0=a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} a\land 1=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>a</mi> <mo>∧</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} a\land 1=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6200982ed488fa31ea4659add7c111b0468b7cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.303ex; height:2.176ex;" alt="{\displaystyle \mathbf {} a\land 1=a}"></span> (Ehtojen kokonaismäärä pysyy siis kymmenessä.). Kummallakin tavalla kymmenen ehdon avulla esitetyt määritelmät ovat yhtäpitäviä, sillä voidaan osoittaa, että jos jokin struktuuri toteuttaa muut kahdeksan ehtoa ja lisäksi absorptiolait, se toteuttaa myös tässä esitetyt uudet ehdot, ja kääntäen, jos se toteuttaa muut kahdeksan ehtoa ja lisäksi nämä uudet ehdot se toteuttaa myös absorptiolait. </p><p>Toisinaan käytetään laskutoimitukselle <span class="mwe-math-element"><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\lor 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\lor b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a07a2a1dfb9a1620c981095b788e4113d2ae2ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\lor b}"></span> myös merkintää a + b ja laskutoimitukselle <span class="mwe-math-element"><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\land 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\land b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8a40f76f4fd75aa92b1a78e44fb7f76ab6ed70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\land b}"></span> merkintää ab tai a · b. Ne eivät kuitenkaan kaikilta osin noudata <a href="/wiki/Reaaliluku" title="Reaaliluku">reaalilukujen</a> laskutoimituksia. Edellä esitetyistä Boolen algebran laskusäännöistä vain liitäntä- ja vaihdantalait sekä jälkimmäinen osittelulaki pätevät myös reaaliluvuille. </p><p>Yksinkertaisin Boolen algebra käsittää vain yhden alkion. Toisinaan määritelmään kuitenkin lisätään ehto, jonka mukaan 0 ja 1 eivät saa olla sama alkio, mikä sulkee tämän tapauksen pois. </p> <div class="mw-heading mw-heading2"><h2 id="Boolen_algebra_ja_lausekalkyyli">Boolen algebra ja lausekalkyyli</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=2" title="Muokkaa osiota Boolen algebra ja lausekalkyyli" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=2" title="Muokkaa osion lähdekoodia: Boolen algebra ja lausekalkyyli"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Boolen algebraa voidaan käyttää matemaattisena esitystapana <a href="/wiki/Logiikka" title="Logiikka">loogiselle</a> <a href="/wiki/Lausekalkyyli" class="mw-redirect" title="Lausekalkyyli">lausekalkyylille</a>. Tällöin laskutoimitus <span class="mwe-math-element"><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> vastaa lauseiden <a href="/wiki/Konjunktio_(logiikka)" title="Konjunktio (logiikka)">konjunktiota</a> (JA, <a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>AND</i></span>), laskutoimitus <span class="mwe-math-element"><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> taas <a href="/wiki/Disjunktio" title="Disjunktio">disjunktiota</a> (TAI, <a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>OR</i></span>) ja komplementti <span class="mwe-math-element"><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><a href="/wiki/Negaatio" class="mw-redirect" title="Negaatio">negaatiota</a> (EI, <a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>NOT</i></span>). Alkio 0 tarkoittaa epätotta ja 1 totta lausetta. Mikäli perusjoukossa on muita alkioita, ne vastaavat yleensä lauseita, joiden totuusarvoa ei tunneta. Tällöin lauseet <i>a ja b</i>, <i>a tai b</i> sekä <i>ei a</i> ovat a:sta ja b:stä riippuen tosia tai epätosia niin kuin seuraavat ns. totuusarvotaulukot osoittavat: </p><p><b>JA</b>: Jos molemmat ehdot on tosia (eli arvo on 1), vastaus on tosi (1) </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><b>JA</b> </td> <td><b>0</b> </td> <td><b>1</b> </td></tr> <tr> <td><b>0</b> </td> <td>0 </td> <td>0 </td></tr> <tr> <td><b>1</b> </td> <td>0 </td> <td>1 </td></tr></tbody></table></dd></dl> <p><b>TAI</b>: Jos edes toinen ehdoista on tosi (eli arvo on 1), vastaus on tosi (1) </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><b>TAI</b> </td> <td><b>0</b> </td> <td><b>1</b> </td></tr> <tr> <td><b>0</b> </td> <td>0 </td> <td>1 </td></tr> <tr> <td><b>1</b> </td> <td>1 </td> <td>1 </td></tr></tbody></table></dd></dl> <p><b>EI</b> kääntää totuusarvon toiseksi: ykkösen nollaksi ja nollan ykköseksi. </p> <dl><dd><table class="wikitable"> <tbody><tr> <td> </td> <td><b>0</b> </td> <td><b>1</b> </td></tr> <tr> <td><b>EI</b> </td> <td>1 </td> <td>0 </td></tr></tbody></table></dd></dl> <dl><dd><table class="wikitable"> <tbody><tr> <td> </td> <td><b>X = 0</b> </td> <td><b>X = 1</b> </td> <td><b>X = a</b> </td></tr> <tr> <td><b>0 AND X</b> </td> <td>0 </td> <td>0 </td> <td>0 </td></tr> <tr> <td><b>1 AND X</b> </td> <td>0 </td> <td>1 </td> <td>a </td></tr> <tr> <td><b>0 OR X</b> </td> <td>0 </td> <td>1 </td> <td>a </td></tr> <tr> <td><b>1 OR X</b> </td> <td>1 </td> <td>1 </td> <td>1 </td></tr></tbody></table></dd></dl> <p>Voidaan osoittaa, että nämä loogiset operaatiot noudattavat kaikkia ylempänä esitettyjä Boolen algebran sääntöjä. </p><p>Disjunktio (TAI) merkitsee tässä ns. inklusiivista disjunktiota ("ja/tai"), joka on tosi, kun ainakin toinen lauseista a ja b on tosi. Tämän vuoksi sovellettaessa Boolen algebraa lausekalkyyliin ei operaatiolle 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 pidä käyttää merkintää a + b, koska sillä tarkoitetaan toisinaan ns. ekslusiivista disjunktiota, joka on tosi vain silloin, kun vain toinen sen yhdistämistä lauseista on tosi. </p><p>Esimerkiksi seuraavassa taulukossa analysoidaan, miten A OR B voidaan esittää toisessa muodossa: </p><p>A OR B = NOT( ( NOT(A) ) AND ( NOT(B) ) ) </p> <dl><dd><table class="wikitable"> <tbody><tr> <td> </td> <td>( A </td> <td>OR </td> <td>B ) = </td> <td>NOT( ( </td> <td>NOT( </td> <td>A) ) </td> <td>AND ( </td> <td>NOT( </td> <td>B) ) ) </td></tr> <tr> <td>A=0,B=0 </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <td>A=1,B=0 </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td></tr> <tr> <td>A=0,B=1 </td> <td>0 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>1 </td></tr> <tr> <td>A=1,B=1 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Boolen_algebran_läheinen_yhteys_joukko-oppiin"><span id="Boolen_algebran_l.C3.A4heinen_yhteys_joukko-oppiin"></span>Boolen algebran läheinen yhteys joukko-oppiin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=3" title="Muokkaa osiota Boolen algebran läheinen yhteys joukko-oppiin" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=3" title="Muokkaa osion lähdekoodia: Boolen algebran läheinen yhteys joukko-oppiin"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minkä tahansa <a href="/wiki/Perusjoukko" title="Perusjoukko">perusjoukon</a> osajoukoille voidaan määritellä joukko-opilliset operaatiot <a href="/wiki/Yhdiste_(matematiikka)" title="Yhdiste (matematiikka)">unioni</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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>, <a href="/wiki/Leikkaus_(matematiikka)" title="Leikkaus (matematiikka)">leikkaus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27b38cf9eac6060e67b61f66cd9beec5067f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cap B}"></span> ja <a href="/wiki/Komplementti_(joukko-oppi)" title="Komplementti (joukko-oppi)">komplementti</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cdb86a926c5b152a9d10526f527c094f2a61ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.687ex; height:2.343ex;" alt="{\displaystyle \mathbf {} A^{c}}"></span>. Nämä noudattavat myös Boolen algebran sääntöjä, kun unioni vastaa laskutoimitusta <span class="mwe-math-element"><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>, leikkaus laskutoimitusta <span class="mwe-math-element"><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> ja komplementti operaatiota <span class="mwe-math-element"><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>. Ykkösalkiona on tällöin perusjoukko, nolla-alkiona <a href="/wiki/Tyhj%C3%A4_joukko" title="Tyhjä joukko">tyhjä joukko</a>. Kääntäen voidaan osoittaa, että jokainen määritelmässä esitetyt "Boolen ehdot" toteuttava algebra on <a href="/wiki/Isomorfismi" title="Isomorfismi">isomorfinen</a> jonkin joukon osajoukkoalgebran kanssa eli se voidaan tulkita algebraksi, jonka alkioina ovat jonkin perusjoukon osajoukot ja laskutoimitukset on määritelty joukkojen unionin, leikkauksen ja komplementin perusteella. Tämä tulos seuraa <a href="/w/index.php?title=Stonen_esityslause&action=edit&redlink=1" class="new" title="Stonen esityslause (sivua ei ole)">Stonen esityslauseesta</a>. Tämä Boolen algebran joukko-opillinen esitys ei ole yksikäsitteinen, mutta kaikki tiettyyn Boolen algebraan liittyvät joukko-opilliset esitykset ovat isomorfisia keskenään ja kyseisen Boolen algebran kanssa. </p> <div class="mw-heading mw-heading3"><h3 id="Boolen_algebra_biteistä_koostuvana"><span id="Boolen_algebra_biteist.C3.A4_koostuvana"></span>Boolen algebra biteistä koostuvana</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=4" title="Muokkaa osiota Boolen algebra biteistä koostuvana" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=4" title="Muokkaa osion lähdekoodia: Boolen algebra biteistä koostuvana"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Boolen algebran ajatellaan usein koostuvan vain <a href="/wiki/Bitti" title="Bitti">biteistä</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f85d046cb5bf36f4d86fd5dfeba8887256d378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {} 0}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e368271392c4d18087c93c545a29f801a7a2910f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {} 1}"></span>, ja joukko-opillinen yhteys oikeuttaakin tämän tulkinnan siinä mielessä, että perusjoukon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span> jokaiseen pisteeseen voidaan ajatella liitetyn bitti-arvo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d1c919988c5ddd2591b3f1766e92d442fa2c12" 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 \mathbf {} B}"></span> sen mukaan kuuluuko kyseinen piste haluttuun joukkoon. Esimerkiksi jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E=\{a,b,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E=\{a,b,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/839b21f7737122c822fc8bdcd4a1b9d6dced20c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.501ex; height:2.843ex;" alt="{\displaystyle \mathbf {} E=\{a,b,c\}}"></span> tarkoittaa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} B(a)=1,B(b)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} B(a)=1,B(b)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aaac25fa66349caad03af9ab77a0c9dca39e689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.93ex; height:2.843ex;" alt="{\displaystyle \mathbf {} B(a)=1,B(b)=0}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} B(c)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} B(c)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b299cb037305edfa74f1d1b76a9c68720839b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.841ex; height:2.843ex;" alt="{\displaystyle \mathbf {} B(c)=1}"></span> joukkoa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \{a,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \{a,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b2f61ae616abf98311740c886f92db7db26cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.596ex; height:2.843ex;" alt="{\displaystyle \mathbf {} \{a,c\}}"></span>. Tästä seuraa se, että joukkojen joukko-opillisten operaatioiden voidaan tulkita tapahtuvan pisteittäin bitteihin sovellettavien lausekalkyylin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \lor ,\land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \lor ,\land }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffdf1d24737b80680363b2c8eec41608a23f5e5" 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 \mathbf {} \lor ,\land }"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3466b394e2ff53ca1d5af0bdad0aec0a31d6870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \mathbf {} \neg }"></span> avulla, jolloin esimerkiksi unionia vastaa "pistebiteittäin" sovellettava <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1e7c96940c4991ed37fb450fd96a0906826717" 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 \mathbf {} \lor }"></span>, sillä unioniin tullakseen riittää, että piste kuuluu edes toiseen joukoista, ja vastaavasti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1e7c96940c4991ed37fb450fd96a0906826717" 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 \mathbf {} \lor }"></span> antaa bittiarvon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e368271392c4d18087c93c545a29f801a7a2910f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {} 1}"></span>, jos edes toinen bitti on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e368271392c4d18087c93c545a29f801a7a2910f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {} 1}"></span>. Äärettömissä eli äärettömän monta alkiota omaavissa Boolen algebroissa "bittitulkinnassa" on kuitenkin se ongelma, että eri pisteisiin liitettäviä bittejä ei välttämättä voida valita toisistaan riippumattomasti, sillä riippumattomasti valittaessa alkioina käytettäviksi osajoukoiksi saataisiin kaikki perusjoukon osajoukot, mikä ei seuraavan alakappaleen mukaan ole aina mahdollista. Kuitenkin äärellisen monta alkiota omaavalle Boolen algebralle löytyy "Äärellisistä Boolen algebroista"-alakappaleen mukaisesti aina kaikki osajoukot käyttävä joukko-opillinen tulkinta, jolloin pisteisiin liitettävät bitit voidaan valita toisistaan riippumattomasti. </p> <div class="mw-heading mw-heading3"><h3 id="Joukko-opillisen_Boolen_algebran_alkioina_käyttämät_osajoukot"><span id="Joukko-opillisen_Boolen_algebran_alkioina_k.C3.A4ytt.C3.A4m.C3.A4t_osajoukot"></span>Joukko-opillisen Boolen algebran alkioina käyttämät osajoukot</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=5" title="Muokkaa osiota Joukko-opillisen Boolen algebran alkioina käyttämät osajoukot" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=5" title="Muokkaa osion lähdekoodia: Joukko-opillisen Boolen algebran alkioina käyttämät osajoukot"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jos joukko-opillisessa tulkinnassa mielivaltaisesti valitun perusjoukon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span> kaikki osajoukot otetaan alkioiksi, saadaan varmasti Boolen algebra, mutta on huomattava, että on olemassa Boolen algebroja, joiden yhdessäkään joukko-opillisessa esityksessä ei voida hyväksyä alkioiksi kaikkia käytetyn perusjoukon osajoukkoja, vaan algebran alkioina on vain osa niistä. Tämä seuraa siitä, että kaikkien osajoukkojen joukossa on myös perusjoukon yksittäisistä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90624ed41b0e92ba6447b167cebab1b35e66b51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \mathbf {} p}"></span>-pisteistä koostuvat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \{p\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \{p\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ca124f8ef40bba289ee4d70b6861e3b79a9414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.494ex; height:2.843ex;" alt="{\displaystyle \mathbf {} \{p\}}"></span>-joukot, jotka ovat selvästi atomeja eli sellaisia Boolen algebran alkioiksi hyväksyttyjä epätyhjiä osajoukkoja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c6ea5c45e09dc386fab0eb38449c8cf199eb43" 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 \mathbf {} A}"></span>, joilla kaikilla alkioiksi hyväksytyillä osajoukoilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caafa07fa12f498aa4095462b5d7e43d0fc8502c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle \mathbf {} P}"></span> pätee se, että <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A\cap P=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> <mo>∩</mo> <mi>P</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A\cap P=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/059df986f1050e00ac53e965de9e56c134a2ddd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.913ex; height:2.176ex;" alt="{\displaystyle \mathbf {} A\cap P=A}"></span> tai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A\cap P=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> <mo>∩</mo> <mi>P</mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A\cap P=\emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daaa6a129bf91cb349aecacdc4557f077a444758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.332ex; height:2.509ex;" alt="{\displaystyle \mathbf {} A\cap P=\emptyset }"></span>. Alkion atomi-ominaisuus säilyy isomorfismeissa, eli atomi-alkion vastine on myös atomi siinä Boolen algebrassa, joka on isomorfinen alkuperäisen Boolen algebran kanssa. On kuitenkin olemassa myös Boolen algebroja, joiden alkioista yksikään ei ole atomi, ja tällainen Boolen algebra ei siis voi olla isomorfisesti sama kuin sellainen Boolen algebra, jonka joukko-opillinen esitys ottaa alkioiksi kaikki osajoukot, sillä edellä todettiin, että kaikki osajoukot käyttävissä Boolen algebroissa on atomeja. </p> <div class="mw-heading mw-heading3"><h3 id="Esimerkki_atomittomasta_Boolen_algebrasta">Esimerkki atomittomasta Boolen algebrasta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=6" title="Muokkaa osiota Esimerkki atomittomasta Boolen algebrasta" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=6" title="Muokkaa osion lähdekoodia: Esimerkki atomittomasta Boolen algebrasta"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atomittomasta Boolen algebrasta saamme esimerkin valitsemalla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E=[0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E=[0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9821793c15da95a3bff138f15af769f69058678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.784ex; height:2.843ex;" alt="{\displaystyle \mathbf {} E=[0,1)}"></span> (Perusjoukkona <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span> on nyt siis puoliavoin reaalilukuväli.) ja ottamalla alkioiksi ne osajoukot, jotka saadaan valitsemalla ensin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd287e1308bd302d8149b1f372eceb73cc5903e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.873ex; height:2.843ex;" alt="{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}"></span> ja ottamalla sitten jokin unioni ("tyhjä unioni" antaa tyhjän joukon) muotoa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} [(m)/2^{n},(m+1)/2^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} [(m)/2^{n},(m+1)/2^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b9a85c55faa9b48b8676f530c17b735b3e1767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.374ex; height:2.843ex;" alt="{\displaystyle \mathbf {} [(m)/2^{n},(m+1)/2^{n})}"></span> olevista erillisistä puoliavoimista reaalilukuväleistä, missä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} m\in \{0,\cdots ,2^{n}-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>m</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} m\in \{0,\cdots ,2^{n}-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0f83a020337784a122f92c7e218ff410be3b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.93ex; height:2.843ex;" alt="{\displaystyle \mathbf {} m\in \{0,\cdots ,2^{n}-1\}}"></span>. Esimerkiksi joukko </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} {\Big [}[1/2^{4},2/2^{4})\cup [10/2^{4},11/2^{4})\cup [11/2^{4},12/2^{4}){\Big ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} {\Big [}[1/2^{4},2/2^{4})\cup [10/2^{4},11/2^{4})\cup [11/2^{4},12/2^{4}){\Big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286b60e95fddab349e85fc4ea0d0931d073afd97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.016ex; height:4.843ex;" alt="{\displaystyle \mathbf {} {\Big [}[1/2^{4},2/2^{4})\cup [10/2^{4},11/2^{4})\cup [11/2^{4},12/2^{4}){\Big ]}}"></span></center> <p>on tämän Boolen algebran alkio, joka on saatu unionina kolmesta annettua muotoa olevasta puoliavoimesta reaalilukuvälistä. Kyseessä on todella Boolen algebra, sillä tällaisiin joukkoihin sovelletut operaatiot antavat edelleen kyseistä muotoa olevia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span>:n osajoukkoja, mikä nähdään tarkastelemalla operaatiossa mukana olevia joukkoja suurimman niissä käytetyn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a727f8d81d6f2ed01efa3598517a15385a0537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle \mathbf {} n}"></span>-arvon mukaisina. Esimerkiksi </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} {\Big [}[2/2^{2},3/2^{2}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} {\Big [}[2/2^{2},3/2^{2}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93bdfb450dd777b0a0cb276f5edb80c381a82b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.812ex; height:4.843ex;" alt="{\displaystyle \mathbf {} {\Big [}[2/2^{2},3/2^{2}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}}"></span></center> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} ={\Big [}[4/2^{3},5/2^{3})\cup [5/2^{3},6/2^{3}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}={\Big [}[4/2^{3},5/2^{3}){\Big ]},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} ={\Big [}[4/2^{3},5/2^{3})\cup [5/2^{3},6/2^{3}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}={\Big [}[4/2^{3},5/2^{3}){\Big ]},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d439d093d72c85481296ad18d5e946476a1e6a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:93.771ex; height:4.843ex;" alt="{\displaystyle \mathbf {} ={\Big [}[4/2^{3},5/2^{3})\cup [5/2^{3},6/2^{3}){\Big ]}\cap {\Big [}[3/2^{3},4/2^{3})\cup [4/2^{3},5/2^{3})\cup [6/2^{3},7/2^{3}){\Big ]}={\Big [}[4/2^{3},5/2^{3}){\Big ]},}"></span></center> <p>missä leikkauksen ensimmäinen joukkokin on nyt tulkittu hienomman <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 1/2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 1/2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536d8095dfb0b291a52023640b4e298cc1967888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.542ex; height:3.176ex;" alt="{\displaystyle \mathbf {} 1/2^{3}}"></span>-jaon mukaisena. Yksikään tällainen epätyhjä joukko ei ole tämän Boolen algebran atomi, sillä jokaisessa joukossa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c6ea5c45e09dc386fab0eb38449c8cf199eb43" 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 \mathbf {} A}"></span> on käytetty jotain tiettyä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a727f8d81d6f2ed01efa3598517a15385a0537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle \mathbf {} n}"></span>-arvoa, jota voidaan aina kasvattaa yhdellä ja näin "hienontamalla" voidaan ottaa alkuperäisen joukon aito epätyhjä osajoukko, joka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caafa07fa12f498aa4095462b5d7e43d0fc8502c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle \mathbf {} P}"></span>:ksi ottamalla nähdään, että <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c6ea5c45e09dc386fab0eb38449c8cf199eb43" 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 \mathbf {} A}"></span> ei toteuta atomin määritelmää, sillä nyt (koska <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \emptyset \subset P\subset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi mathvariant="normal">∅</mi> <mo>⊂</mo> <mi>P</mi> <mo>⊂</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \emptyset \subset P\subset A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56bf7e3b2b03bdc2ac985dc9ef4bca65f3421c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.848ex; height:2.509ex;" alt="{\displaystyle \mathbf {} \emptyset \subset P\subset A}"></span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} A\cap P=P\neq \emptyset ,A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> <mo>∩</mo> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mo>≠</mo> <mi mathvariant="normal">∅</mi> <mo>,</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} A\cap P=P\neq \emptyset ,A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208098dc74a7b1e999f1223674fe2800d89b3b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.953ex; height:2.843ex;" alt="{\displaystyle \mathbf {} A\cap P=P\neq \emptyset ,A}"></span>. Esimerkiksi </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \emptyset \subset {\Big [}[203/2^{(7+1)},204/2^{(7+1)}){\Big ]}\subset {\Big [}[202/2^{(7+1)},203/2^{(7+1)})\cup [203/2^{(7+1)},204/2^{(7+1)}){\Big ]}={\Big [}[101/2^{7},102/2^{7}){\Big ]}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi mathvariant="normal">∅</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>203</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mn>204</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>202</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mn>203</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>203</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mn>204</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>101</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>,</mo> <mn>102</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \emptyset \subset {\Big [}[203/2^{(7+1)},204/2^{(7+1)}){\Big ]}\subset {\Big [}[202/2^{(7+1)},203/2^{(7+1)})\cup [203/2^{(7+1)},204/2^{(7+1)}){\Big ]}={\Big [}[101/2^{7},102/2^{7}){\Big ]}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ed1dab1394bee6bef7940d87d67e59f74b70b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:105.825ex; height:4.843ex;" alt="{\displaystyle \mathbf {} \emptyset \subset {\Big [}[203/2^{(7+1)},204/2^{(7+1)}){\Big ]}\subset {\Big [}[202/2^{(7+1)},203/2^{(7+1)})\cup [203/2^{(7+1)},204/2^{(7+1)}){\Big ]}={\Big [}[101/2^{7},102/2^{7}){\Big ]}.}"></span></center> <div class="mw-heading mw-heading3"><h3 id="Äärellisistä_Boolen_algebroista"><span id=".C3.84.C3.A4rellisist.C3.A4_Boolen_algebroista"></span>Äärellisistä Boolen algebroista</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=7" title="Muokkaa osiota Äärellisistä Boolen algebroista" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=7" title="Muokkaa osion lähdekoodia: Äärellisistä Boolen algebroista"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Boolen algebran sanotaan olevan äärellinen, jos sen alkioiden määrä on äärellinen. Äärellisille Boolen algebroille saadaan hyvin yksinkertainen joukko-opillinen esitysmuoto, sillä silloin voidaan perusjoukoksi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span> valita äärellinen joukko ja Boolen algebran alkioiksi kaikki <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span>:n osajoukot. Äärellisen Boolen algebran kohdalla löydetään siis perusjoukko, josta voidaan käyttää kaikki osajoukot, mikä on yksinkertaisin tilanne. Tästä selvästi seuraa myös, että äärellisen Boolen algebran alkioiden lukumäärä on välttämättä muotoa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faefbdc3fb87b2346b8664f16031e9ad373daf9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle \mathbf {} 2^{n}}"></span>, missä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd287e1308bd302d8149b1f372eceb73cc5903e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.873ex; height:2.843ex;" alt="{\displaystyle \mathbf {} n\in \{1,2,3,\cdots \}}"></span>. Yllä esimerkkinä annetussa atomittomassa Boolen algebrassa osajoukko-alkioiden määrä sen sijaan ei ole äärellinen, vaan se on <a href="/wiki/Numeroituvuus" class="mw-redirect" title="Numeroituvuus">numeroituvasti</a> <a href="/wiki/%C3%84%C3%A4rett%C3%B6myys" title="Äärettömyys">ääretön</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Joukko-opillisen_Boolen_algebran_yhteys_Sigma-algebraan">Joukko-opillisen Boolen algebran yhteys Sigma-algebraan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=8" title="Muokkaa osiota Joukko-opillisen Boolen algebran yhteys Sigma-algebraan" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=8" title="Muokkaa osion lähdekoodia: Joukko-opillisen Boolen algebran yhteys Sigma-algebraan"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Joukko-opillista Boolen algebraa vaativampi perusjoukon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span> osajoukkojen kokoelma on <a href="/wiki/Sigma-algebra" title="Sigma-algebra">sigma-algebra</a>. Joukko-opilliselta Boolen algebralta vaaditaan, että alkioksi hyväksytyn osajoukon komplementti on myös alkioksi hyväksytty ja kahden (Tätä kautta <a href="/wiki/Matemaattinen_induktio" title="Matemaattinen induktio">induktiivisesti</a> myös äärellisen monen.) alkioksi hyväksytyn osajoukon unioni on myös alkioksi hyväksytty, jolloin <a href="/wiki/De_Morganin_lait" title="De Morganin lait">De Morganin laeista</a> selvästi seuraa, että sama vaatimus toteutuu myös leikkauksen suhteen. Sigma-algebralta sen sijaan vaaditaan komplementti-ehto ja se, että numeroituvasti äärettömän monen (Boolen algebralla vain kahden) alkioksi hyväksytyn osajoukon unioni on myös alkioksi hyväksytty osajoukko, mikä on vaativampi ehto. Vastaavasti De Morganin laeista seuraa sigma-algebran kohdalla, että sigma-algebrassa myös numeroituvasti äärettömän monen alkioksi hyväksytyn osajoukon leikkaus on alkioksi hyväksytty osajoukko. Jokainen sigma-algebra on selvästi Boolen algebra, mutta käänteinen ei päde. Esimerkiksi yllä esitetty atomiton Boolen algebra ei ole sigma-algebra, sillä </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} {\Big [}[0/2^{1},1/2^{1}){\Big ]}\cap {\Big [}[0/2^{2},1/2^{2}){\Big ]}\cap {\Big [}[0/2^{3},1/2^{3}){\Big ]}\cap {\Big [}[0/2^{4},1/2^{4}){\Big ]}\cap \cdots =\{0\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>∩</mo> <mo>⋯</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} {\Big [}[0/2^{1},1/2^{1}){\Big ]}\cap {\Big [}[0/2^{2},1/2^{2}){\Big ]}\cap {\Big [}[0/2^{3},1/2^{3}){\Big ]}\cap {\Big [}[0/2^{4},1/2^{4}){\Big ]}\cap \cdots =\{0\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f23b4f89ff3301adbba5473a36e7baf668bfdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:75.74ex; height:4.843ex;" alt="{\displaystyle \mathbf {} {\Big [}[0/2^{1},1/2^{1}){\Big ]}\cap {\Big [}[0/2^{2},1/2^{2}){\Big ]}\cap {\Big [}[0/2^{3},1/2^{3}){\Big ]}\cap {\Big [}[0/2^{4},1/2^{4}){\Big ]}\cap \cdots =\{0\},}"></span></center> <p>mutta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6acfbbb7c82ed9ec6ba75ddda49847e9f3cd1404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \mathbf {} \{0\}}"></span> koostuu vain yhdestä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E=[0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E=[0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9821793c15da95a3bff138f15af769f69058678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.784ex; height:2.843ex;" alt="{\displaystyle \mathbf {} E=[0,1)}"></span>-perusjoukon pisteestä (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f85d046cb5bf36f4d86fd5dfeba8887256d378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {} 0}"></span>), eikä se siis selvästikään voi olla alkioksi hyväksytty <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {} E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d709d353f1df8d7c72ba7961d61c50623488f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathbf {} E}"></span>:n osajoukko, eli sigma-algebraa koskeva numeroituvasti äärettömän monen alkion leikkausta koskeva sääntö ei nyt toteudu, eli kyseinen Boolen algebra ei ole sigma-algebra. </p> <div class="mw-heading mw-heading2"><h2 id="Kuviollinen_esitys">Kuviollinen esitys</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=9" title="Muokkaa osiota Kuviollinen esitys" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=9" title="Muokkaa osion lähdekoodia: Kuviollinen esitys"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:Vennandornot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vennandornot.svg/500px-Vennandornot.svg.png" decoding="async" width="500" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vennandornot.svg/750px-Vennandornot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vennandornot.svg/1000px-Vennandornot.svg.png 2x" data-file-width="851" data-file-height="293" /></a><figcaption>Venn-diagrammit leikkaukselle, unionille ja komplementille.</figcaption></figure> <p><a href="/wiki/Venn-diagrammi" title="Venn-diagrammi">Venn-diagrammia</a> voidaan käyttää kuviolliseen esitykseen. </p> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=10" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=10" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tietokonearitmetiikka" title="Tietokonearitmetiikka">Tietokonearitmetiikka</a></li> <li><a href="/wiki/Looginen_portti" title="Looginen portti">Looginen portti</a></li> <li><a href="/wiki/Totuusfunktio" title="Totuusfunktio">Totuusfunktio</a></li> <li><a href="/wiki/Totuustaulu" title="Totuustaulu">Totuustaulu</a></li> <li><a href="/wiki/Toistorakenne" title="Toistorakenne">Toistorakenne</a></li> <li><a href="/wiki/Ikuinen_silmukka" title="Ikuinen silmukka">Ikuinen silmukka</a></li> <li><a href="/wiki/Valoportti" title="Valoportti">Valoportti</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=11" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=11" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"> <span class="kirjaviite" title="Kirjaviite">Thompson, Jan & Martinsson, Thomas: <i>Matematiikan käsikirja</i>, s. 47–49.  Helsinki:  Tammi, 1994.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-31-0471-0" title="Toiminnot:Kirjalähteet/951-31-0471-0">ISBN 951-31-0471-0</a> </span></span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="Aiheesta_muualla">Aiheesta muualla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boolen_algebra&veaction=edit&section=12" title="Muokkaa osiota Aiheesta muualla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Boolen_algebra&action=edit&section=12" title="Muokkaa osion lähdekoodia: Aiheesta muualla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/10px-Commons-logo.svg.png" decoding="async" width="10" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> Kuvia tai muita tiedostoja aiheesta <b><a href="https://commons.wikimedia.org/wiki/Category:Boolean_algebra" class="extiw" title="commons:Category:Boolean algebra">Boolen algebra</a></b> <a href="https://commons.wikimedia.org/wiki/Etusivu" class="extiw" title="commons:Etusivu">Wikimedia Commonsissa</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Noudettu kohteesta ”<a dir="ltr" href="https://fi.wikipedia.org/w/index.php?title=Boolen_algebra&oldid=22769795">https://fi.wikipedia.org/w/index.php?title=Boolen_algebra&oldid=22769795</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Toiminnot:Luokat" title="Toiminnot:Luokat">Luokat</a>: <ul><li><a href="/wiki/Luokka:Matemaattinen_logiikka" title="Luokka:Matemaattinen logiikka">Matemaattinen logiikka</a></li><li><a href="/wiki/Luokka:Algebralliset_rakenteet" title="Luokka:Algebralliset rakenteet">Algebralliset rakenteet</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Piilotettu luokka: <ul><li><a href="/wiki/Luokka:Seulonnan_keskeiset_artikkelit" title="Luokka:Seulonnan keskeiset artikkelit">Seulonnan keskeiset artikkelit</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigointivalikko</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Henkilökohtaiset työkalut</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="IP-osoitteesi käyttäjäsivu">Et ole kirjautunut</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Toiminnot:Oma_keskustelu" title="Keskustelu tämän IP-osoitteen muokkauksista [n]" accesskey="n"><span>Keskustelu</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Toiminnot:Omat_muokkaukset" title="Luettelo tästä IP-osoitteesta tehdyistä muokkauksista [y]" accesskey="y"><span>Muokkaukset</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Luo_tunnus&returnto=Boolen+algebra" title="On suositeltavaa luoda käyttäjätunnus ja kirjautua sisään. Se ei kuitenkaan ole pakollista."><span>Luo tunnus</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Kirjaudu_sis%C3%A4%C3%A4n&returnto=Boolen+algebra" title="On suositeltavaa kirjautua sisään. Se ei kuitenkaan ole pakollista. [o]" accesskey="o"><span>Kirjaudu sisään</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Nimiavaruudet</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/Boolen_algebra" title="Näytä sisältösivu [c]" accesskey="c"><span>Artikkeli</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/Keskustelu:Boolen_algebra" rel="discussion" title="Keskustele sisällöstä [t]" accesskey="t"><span>Keskustelu</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">suomi</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Näkymät</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/wiki/Boolen_algebra"><span>Lue</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=Boolen_algebra&veaction=edit" title="Muokkaa tätä sivua [v]" accesskey="v"><span>Muokkaa</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=Boolen_algebra&action=edit" title="Muokkaa tämän sivun lähdekoodia [e]" accesskey="e"><span>Muokkaa wikitekstiä</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=Boolen_algebra&action=history" title="Sivun aikaisemmat versiot [h]" accesskey="h"><span>Näytä historia</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Lisää valintoja" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Muut</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Haku</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Hae Wikipediasta" aria-label="Hae Wikipediasta" autocapitalize="sentences" title="Hae Wikipediasta [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Toiminnot:Haku"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Hae sivuilta tätä tekstiä" value="Hae"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Siirry sivulle, joka on tarkalleen tällä nimellä" value="Siirry"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/Wikipedia:Etusivu" title="Etusivu"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Valikko</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Etusivu" title="Siirry etusivulle [z]" accesskey="z"><span>Etusivu</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Tietoja"><span>Tietoja Wikipediasta</span></a></li><li id="n-allarticles" class="mw-list-item"><a href="/wiki/Wikipedia:Selaa_luokittain"><span>Kaikki sivut</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Toiminnot:Satunnainen_sivu" title="Avaa satunnainen sivu [x]" accesskey="x"><span>Satunnainen artikkeli</span></a></li> </ul> </div> </nav> <nav id="p-interaction" class="mw-portlet mw-portlet-interaction vector-menu-portal portal vector-menu" aria-labelledby="p-interaction-label" > <h3 id="p-interaction-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Osallistuminen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Ohje:Sis%C3%A4llys" title="Ohjeita"><span>Ohje</span></a></li><li id="n-Kahvihuone" class="mw-list-item"><a href="/wiki/Wikipedia:Kahvihuone"><span>Kahvihuone</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Wikipedia:Ajankohtaista" title="Taustatietoa tämänhetkisistä tapahtumista"><span>Ajankohtaista</span></a></li><li id="n-Tuoreet-odottavat-muutokset" class="mw-list-item"><a href="//fi.wikipedia.org/wiki/Toiminnot:Tuoreet_muutokset?damaging=&goodfaith=&hideliu=0&hideanons=0&userExpLevel=&hidemyself=0&hidebyothers=0&hidebots=1&hidehumans=0&hidepatrolled=1&hideunpatrolled=0&hideminor=0&hidemajor=0&hidepageedits=0&hidenewpages=0&hidecategorization=1&hideWikibase=1&hidelog=0&highlight=1&goodfaith__verylikelybad_color=c5&goodfaith__likelybad_color=c4&goodfaith__maybebad_color=c3&damaging__verylikelybad_color=c5&damaging__likelybad_color=c4&damaging__maybebad_color=c3"><span>Tuoreet odottavat muutokset</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Toiminnot:Tuoreet_muutokset" title="Luettelo tuoreista muutoksista [r]" accesskey="r"><span>Tuoreet muutokset</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_fi.wikipedia.org&uselang=fi" title="Tue meitä"><span>Lahjoitukset</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Työkalut</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Toiminnot:T%C3%A4nne_viittaavat_sivut/Boolen_algebra" title="Lista sivuista, jotka viittaavat tänne [j]" accesskey="j"><span>Tänne viittaavat sivut</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Toiminnot:Linkitetyt_muutokset/Boolen_algebra" rel="nofollow" title="Viimeisimmät muokkaukset sivuissa, joille viitataan tältä sivulta [k]" accesskey="k"><span>Linkitettyjen sivujen muutokset</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Toiminnot:Toimintosivut" title="Näytä toimintosivut [q]" accesskey="q"><span>Toimintosivut</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Boolen_algebra&oldid=22769795" title="Ikilinkki tämän sivun tähän versioon"><span>Ikilinkki</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Boolen_algebra&action=info" title="Enemmän tietoa tästä sivusta"><span>Sivun tiedot</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Viittaus&page=Boolen_algebra&id=22769795&wpFormIdentifier=titleform" title="Tietoa tämän sivun lainaamisesta"><span>Viitetiedot</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:UrlShortener&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FBoolen_algebra"><span>Lyhennä URL-osoite</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:QrCode&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FBoolen_algebra"><span>Lataa QR-koodi</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Tulosta/vie</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:DownloadAsPdf&page=Boolen_algebra&action=show-download-screen"><span>Lataa PDF-tiedostona</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Boolen_algebra&printable=yes" title="Tulostettava versio [p]" accesskey="p"><span>Tulostettava versio</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Muissa hankkeissa</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Boolean_algebra" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q173183" title="Linkki yhdistettyyn keskustietovaraston kohteeseen [g]" accesskey="g"><span>Wikidata-kohde</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Muilla kielillä</span> </h3> <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 — afrikaans" lang="af" hreflang="af" data-title="Boolse algebra" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" 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 — arabia" lang="ar" hreflang="ar" data-title="جبر بول" data-language-autonym="العربية" data-language-local-name="arabia" 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 — asturia" lang="ast" hreflang="ast" data-title="Álxebra de Boole" data-language-autonym="Asturianu" data-language-local-name="asturia" 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) — azeri" lang="az" hreflang="az" data-title="Bul cəbri (məntiqi)" data-language-autonym="Azərbaycanca" data-language-local-name="azeri" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_Boole" title="Aljabar Boole — indonesia" lang="id" hreflang="id" data-title="Aljabar Boole" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</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 — bengali" lang="bn" hreflang="bn" data-title="বুলিয়ান বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</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 — baškiiri" lang="ba" hreflang="ba" data-title="Логика алгебраһы" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Booleova_algebra" title="Booleova algebra — bosnia" lang="bs" hreflang="bs" data-title="Booleova algebra" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</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 — bulgaria" lang="bg" hreflang="bg" data-title="Булева алгебра" data-language-autonym="Български" data-language-local-name="bulgaria" 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/%C3%80lgebra_de_Boole" title="Àlgebra de Boole — katalaani" lang="ca" hreflang="ca" data-title="Àlgebra de Boole" data-language-autonym="Català" data-language-local-name="katalaani" 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%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 — tšuvassi" lang="cv" hreflang="cv" data-title="Каланăлăхсен алгебри" data-language-autonym="Чӑвашла" data-language-local-name="tšuvassi" 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/Booleova_algebra" title="Booleova algebra — tšekki" lang="cs" hreflang="cs" data-title="Booleova algebra" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra — tanska" lang="da" hreflang="da" data-title="Boolesk algebra" data-language-autonym="Dansk" data-language-local-name="tanska" 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 — saksa" lang="de" hreflang="de" data-title="Boolesche Algebra" data-language-autonym="Deutsch" data-language-local-name="saksa" 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/Boole%27i_algebra" title="Boole'i algebra — viro" lang="et" hreflang="et" data-title="Boole'i algebra" data-language-autonym="Eesti" data-language-local-name="viro" 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%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 — kreikka" lang="el" hreflang="el" data-title="Άλγεβρα Μπουλ" data-language-autonym="Ελληνικά" data-language-local-name="kreikka" 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 — englanti" lang="en" hreflang="en" data-title="Boolean algebra" data-language-autonym="English" data-language-local-name="englanti" 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/%C3%81lgebra_de_Boole" title="Álgebra de Boole — espanja" lang="es" hreflang="es" data-title="Álgebra de Boole" data-language-autonym="Español" data-language-local-name="espanja" 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/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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Booleren_aljebra" title="Booleren aljebra — baski" lang="eu" hreflang="eu" data-title="Booleren aljebra" data-language-autonym="Euskara" data-language-local-name="baski" 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 — persia" lang="fa" hreflang="fa" data-title="جبر بولی" data-language-autonym="فارسی" data-language-local-name="persia" 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/Alg%C3%A8bre_de_Boole_(logique)" title="Algèbre de Boole (logique) — ranska" lang="fr" hreflang="fr" data-title="Algèbre de Boole (logique)" data-language-autonym="Français" data-language-local-name="ranska" 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 — iiri" lang="ga" hreflang="ga" data-title="Ailgéabar Boole" data-language-autonym="Gaeilge" data-language-local-name="iiri" 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 — galicia" lang="gl" hreflang="gl" data-title="Álxebra de Boole" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Boolean_Logic" title="Boolean Logic — kikuju" lang="ki" hreflang="ki" data-title="Boolean Logic" data-language-autonym="Gĩkũyũ" data-language-local-name="kikuju" class="interlanguage-link-target"><span>Gĩkũyũ</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 — korea" lang="ko" hreflang="ko" data-title="불 논리" data-language-autonym="한국어" data-language-local-name="korea" 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/%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 — armenia" lang="hy" hreflang="hy" data-title="Բուլյան հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="armenia" 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%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) — hindi" lang="hi" hreflang="hi" data-title="बूलीय बीजगणित (तर्कशास्त्र)" 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/Booleova_algebra" title="Booleova algebra — kroatia" lang="hr" hreflang="hr" data-title="Booleova algebra" data-language-autonym="Hrvatski" data-language-local-name="kroatia" 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/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 — italia" lang="it" hreflang="it" data-title="Algebra di Boole" data-language-autonym="Italiano" data-language-local-name="italia" 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%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 — heprea" lang="he" hreflang="he" data-title="אלגברה בוליאנית" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</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 — kannada" lang="kn" hreflang="kn" data-title="ಬೂಲಿಯನ್ ಬೀಜಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</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 — kirgiisi" lang="ky" hreflang="ky" data-title="Айтылыштар алгебрасы" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target"><span>Кыргызча</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 — kurdi" lang="ku" hreflang="ku" data-title="Mantiqê Bûlî" data-language-autonym="Kurdî" data-language-local-name="kurdi" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_Booleana_(logica)" title="Algebra Booleana (logica) — latina" lang="la" hreflang="la" data-title="Algebra Booleana (logica)" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</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 — latvia" lang="lv" hreflang="lv" data-title="Būla algebra" data-language-autonym="Latviešu" data-language-local-name="latvia" class="interlanguage-link-target"><span>Latviešu</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 — liettua" lang="lt" hreflang="lt" data-title="Būlio algebra" data-language-autonym="Lietuvių" data-language-local-name="liettua" class="interlanguage-link-target"><span>Lietuvių</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) — unkari" lang="hu" hreflang="hu" data-title="Boole-algebra (informatika)" data-language-autonym="Magyar" data-language-local-name="unkari" 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%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 — makedonia" lang="mk" hreflang="mk" data-title="Булова алгебра" data-language-autonym="Македонски" data-language-local-name="makedonia" 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 — mirandeesi" lang="mwl" hreflang="mwl" data-title="Álgebra de Boole" data-language-autonym="Mirandés" data-language-local-name="mirandeesi" 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 — burma" lang="my" hreflang="my" data-title="ဘူလီယန်အက္ခရာသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burma" 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 — hollanti" lang="nl" hreflang="nl" data-title="Booleaanse algebra" data-language-autonym="Nederlands" data-language-local-name="hollanti" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Boolsk_algebra" title="Boolsk algebra — norjan bokmål" lang="nb" hreflang="nb" data-title="Boolsk algebra" data-language-autonym="Norsk bokmål" data-language-local-name="norjan 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/Boolsk_algebra" title="Boolsk algebra — norjan nynorsk" lang="nn" hreflang="nn" data-title="Boolsk algebra" data-language-autonym="Norsk nynorsk" data-language-local-name="norjan nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</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 — piemonte" lang="pms" hreflang="pms" data-title="Àlgebra ëd Boole" data-language-autonym="Piemontèis" data-language-local-name="piemonte" 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 — portugali" lang="pt" hreflang="pt" data-title="Álgebra booliana" data-language-autonym="Português" data-language-local-name="portugali" 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 — romania" lang="ro" hreflang="ro" data-title="Algebră booleană" data-language-autonym="Română" data-language-local-name="romania" 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 — venäjä" lang="ru" hreflang="ru" data-title="Алгебра логики" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Boolova_algebra" title="Boolova algebra — slovakki" lang="sk" hreflang="sk" data-title="Boolova algebra" data-language-autonym="Slovenčina" data-language-local-name="slovakki" 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/Booleova_algebra" title="Booleova algebra — sloveeni" lang="sl" hreflang="sl" data-title="Booleova algebra" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" 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 — serbia" lang="sr" hreflang="sr" data-title="Булова алгебра" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bulova_algebra" title="Bulova algebra — serbokroaatti" lang="sh" hreflang="sh" data-title="Bulova algebra" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaatti" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Boolesk_algebra" title="Boolesk algebra — ruotsi" lang="sv" hreflang="sv" data-title="Boolesk algebra" data-language-autonym="Svenska" data-language-local-name="ruotsi" 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/Alhebrang_Boolean" title="Alhebrang Boolean — tagalog" lang="tl" hreflang="tl" data-title="Alhebrang Boolean" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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%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 — tamili" lang="ta" hreflang="ta" data-title="பூலிய இயற்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="tamili" 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%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 — thai" lang="th" hreflang="th" data-title="พีชคณิตแบบบูล" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</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 — vietnam" lang="vi" hreflang="vi" data-title="Đại số Boole" data-language-autonym="Tiếng Việt" data-language-local-name="vietnam" class="interlanguage-link-target"><span>Tiếng Việt</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 — tadžikki" lang="tg" hreflang="tg" data-title="Алгебраи мантиқ" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžikki" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boole_cebiri" title="Boole cebiri — turkki" lang="tr" hreflang="tr" data-title="Boole cebiri" data-language-autonym="Türkçe" data-language-local-name="turkki" 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 — ukraina" lang="uk" hreflang="uk" data-title="Алгебра логіки" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</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 — wu-kiina" lang="wuu" hreflang="wuu" data-title="逻辑代数" data-language-autonym="吴语" data-language-local-name="wu-kiina" 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/%E9%82%8F%E8%BC%AF%E4%BB%A3%E6%95%B8" title="邏輯代數 — kantoninkiina" lang="yue" hreflang="yue" data-title="邏輯代數" data-language-autonym="粵語" data-language-local-name="kantoninkiina" 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/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数 — kiina" lang="zh" hreflang="zh" data-title="逻辑代数" data-language-autonym="中文" data-language-local-name="kiina" 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/Q173183#sitelinks-wikipedia" title="Muokkaa kieltenvälisiä linkkejä" class="wbc-editpage">Muokkaa linkkejä</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Sivua on viimeksi muutettu 12. marraskuuta 2024 kello 13.23.</li> <li id="footer-info-copyright">Teksti on saatavilla <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fi">Creative Commons Attribution/Share-Alike</a> -lisenssillä; lisäehtoja voi sisältyä. Katso <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/fi">käyttöehdot</a>.<br /> Wikipedia® on <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundationin</a> rekisteröimä tavaramerkki.<br /> <a href="/wiki/Wikipedia:Artikkelien_ongelmat" title="Wikipedia:Artikkelien ongelmat">Ongelma artikkelissa?</a></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Tietosuojakäytäntö</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Tietoja">Tietoja Wikipediasta</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Vastuuvapaus">Vastuuvapaus</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Käytössäännöstö</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Kehittäjät</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/fi.wikipedia.org">Tilastot</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Evästekäytäntö</a></li> <li id="footer-places-mobileview"><a href="//fi.m.wikipedia.org/w/index.php?title=Boolen_algebra&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobiilinäkymä</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> <script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("This page is using the deprecated ResourceLoader module \"codex-search-styles\".\n[1.43] Use a CodexModule with codexComponents to set your specific components used: https://www.mediawiki.org/wiki/Codex#Using_a_limited_subset_of_components");mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-fb7ms","wgBackendResponseTime":153,"wgPageParseReport":{"limitreport":{"cputime":"0.150","walltime":"0.310","ppvisitednodes":{"value":683,"limit":1000000},"postexpandincludesize":{"value":1725,"limit":2097152},"templateargumentsize":{"value":248,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":3410,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 91.871 1 -total"," 97.77% 89.826 4 Malline:Wikidata"," 56.33% 51.750 1 Malline:Commonscat-rivi"," 53.83% 49.454 1 Malline:Commons-rivi"," 35.96% 33.038 1 Malline:Viitteet"," 28.56% 26.239 1 Malline:Kirjaviite"," 5.47% 5.023 3 Malline:K-en"," 2.34% 2.150 3 Malline:K"]},"scribunto":{"limitreport-timeusage":{"value":"0.006","limit":"10.000"},"limitreport-memusage":{"value":632003,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-7c479b968-gld2b","timestamp":"20241118074631","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Boolen algebra","url":"https:\/\/fi.wikipedia.org\/wiki\/Boolen_algebra","sameAs":"http:\/\/www.wikidata.org\/entity\/Q173183","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q173183","author":{"@type":"Organization","name":"Wikimedia-hankkeiden muokkaajat"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-08-18T15:58:17Z","headline":"algebrallinen struktuuri, joka toimii loogisen lausekalkyylin ja joukko-opin mallina"}</script> </body> </html>