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class="vector-toc-numb">2.2</span> <span>Condensed truth tables for binary operators</span> </div> </a> <ul id="toc-Condensed_truth_tables_for_binary_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Truth_tables_in_digital_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Truth_tables_in_digital_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Truth tables in digital logic</span> </div> </a> <ul id="toc-Truth_tables_in_digital_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_of_truth_tables_in_digital_electronics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications_of_truth_tables_in_digital_electronics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Applications of truth tables in digital electronics</span> </div> </a> <ul id="toc-Applications_of_truth_tables_in_digital_electronics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Methods_of_writing_truth_tables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Methods_of_writing_truth_tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Methods of writing truth tables</span> </div> </a> <button aria-controls="toc-Methods_of_writing_truth_tables-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Methods of writing truth tables subsection</span> </button> <ul id="toc-Methods_of_writing_truth_tables-sublist" class="vector-toc-list"> <li id="toc-Alternating_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternating_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Alternating method</span> </div> </a> <ul id="toc-Alternating_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Combinatorial method</span> </div> </a> <ul id="toc-Combinatorial_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Size_of_truth_tables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Size_of_truth_tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Size of truth tables</span> </div> </a> <ul id="toc-Size_of_truth_tables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_Tables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Function_Tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Function Tables</span> </div> </a> <ul id="toc-Function_Tables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sentential_operator_truth_tables" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sentential_operator_truth_tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sentential operator truth tables</span> </div> </a> <button aria-controls="toc-Sentential_operator_truth_tables-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Sentential operator truth tables subsection</span> </button> <ul id="toc-Sentential_operator_truth_tables-sublist" class="vector-toc-list"> <li id="toc-Overview_table" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overview_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Overview table</span> </div> </a> <ul id="toc-Overview_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wittgenstein_table" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wittgenstein_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Wittgenstein table</span> </div> </a> <ul id="toc-Wittgenstein_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nullary_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nullary_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Nullary operations</span> </div> </a> <ul id="toc-Nullary_operations-sublist" class="vector-toc-list"> <li id="toc-Logical_true" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_true"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.1</span> <span>Logical true</span> </div> </a> <ul id="toc-Logical_true-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_false" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_false"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.2</span> <span>Logical false</span> </div> </a> <ul id="toc-Logical_false-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unary_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unary_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Unary operations</span> </div> </a> <ul id="toc-Unary_operations-sublist" class="vector-toc-list"> <li id="toc-Logical_identity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_identity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Logical identity</span> </div> </a> <ul id="toc-Logical_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_negation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_negation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.2</span> <span>Logical negation</span> </div> </a> <ul id="toc-Logical_negation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Binary_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Binary operations</span> </div> </a> <ul id="toc-Binary_operations-sublist" class="vector-toc-list"> <li id="toc-Logical_conjunction_(AND)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_conjunction_(AND)"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.1</span> <span>Logical conjunction (AND)</span> </div> </a> <ul id="toc-Logical_conjunction_(AND)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_disjunction_(OR)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_disjunction_(OR)"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.2</span> <span>Logical disjunction (OR)</span> </div> </a> <ul id="toc-Logical_disjunction_(OR)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_implication" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_implication"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.3</span> <span>Logical implication</span> </div> </a> <ul id="toc-Logical_implication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_equality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.4</span> <span>Logical equality</span> </div> </a> <ul id="toc-Logical_equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exclusive_disjunction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exclusive_disjunction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.5</span> <span>Exclusive disjunction</span> </div> </a> <ul id="toc-Exclusive_disjunction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_NAND" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_NAND"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.6</span> <span>Logical NAND</span> </div> </a> <ul id="toc-Logical_NAND-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_NOR" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_NOR"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.7</span> <span>Logical NOR</span> </div> </a> <ul id="toc-Logical_NOR-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Truth table</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 43 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-43" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">43 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Waarheidstabel" title="Waarheidstabel – Afrikaans" lang="af" hreflang="af" data-title="Waarheidstabel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8B%95%E1%8B%8D%E1%8A%90%E1%89%B3_%E1%88%A0%E1%8A%95%E1%8C%A0%E1%88%A8%E1%8B%A5" title="የዕውነታ ሠንጠረዥ – Amharic" lang="am" hreflang="am" data-title="የዕውነታ ሠንጠረዥ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%AF%D9%88%D9%84_%D8%A7%D9%84%D8%AD%D9%82%D9%8A%D9%82%D8%A9" title="جدول الحقيقة – Arabic" lang="ar" hreflang="ar" data-title="جدول الحقيقة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%A4%E0%A7%8D%E0%A6%AF%E0%A6%95_%E0%A6%B8%E0%A6%BE%E0%A6%B0%E0%A6%A3%E0%A6%BF" title="সত্যক সারণি – Bangla" lang="bn" hreflang="bn" data-title="সত্যক সারণি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Taula_de_veritat" title="Taula de veritat – Catalan" lang="ca" hreflang="ca" data-title="Taula de veritat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pravdivostn%C3%AD_tabulka" title="Pravdivostní tabulka – Czech" lang="cs" hreflang="cs" data-title="Pravdivostní tabulka" data-language-autonym="Čeština" data-language-local-name="Czech" 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/Sandhedstabel" title="Sandhedstabel – Danish" lang="da" hreflang="da" data-title="Sandhedstabel" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Wahrheitstabelle" title="Wahrheitstabelle – German" lang="de" hreflang="de" data-title="Wahrheitstabelle" data-language-autonym="Deutsch" data-language-local-name="German" 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/T%C3%B5ev%C3%A4%C3%A4rtustabel" title="Tõeväärtustabel – Estonian" lang="et" hreflang="et" data-title="Tõeväärtustabel" data-language-autonym="Eesti" data-language-local-name="Estonian" 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%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_%CE%B1%CE%BB%CE%B7%CE%B8%CE%B5%CE%AF%CE%B1%CF%82" title="Πίνακας αληθείας – Greek" lang="el" hreflang="el" data-title="Πίνακας αληθείας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tabla_de_verdad" title="Tabla de verdad – Spanish" lang="es" hreflang="es" data-title="Tabla de verdad" data-language-autonym="Español" data-language-local-name="Spanish" 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/Vertabelo" title="Vertabelo – Esperanto" lang="eo" hreflang="eo" data-title="Vertabelo" 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/Egia-taula" title="Egia-taula – Basque" lang="eu" hreflang="eu" data-title="Egia-taula" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%AF%D9%88%D9%84_%D8%A7%D8%B1%D8%B2%D8%B4" title="جدول ارزش – Persian" lang="fa" hreflang="fa" data-title="جدول ارزش" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Table_de_v%C3%A9rit%C3%A9" title="Table de vérité – French" lang="fr" hreflang="fr" data-title="Table de vérité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%84%EB%A6%AC%ED%91%9C" title="진리표 – Korean" lang="ko" hreflang="ko" data-title="진리표" data-language-autonym="한국어" data-language-local-name="Korean" 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%BB%D5%BD%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%A1%D5%B2%D5%B5%D5%B8%D6%82%D5%BD%D5%A1%D5%AF" title="Իսկության աղյուսակ – Armenian" lang="hy" hreflang="hy" data-title="Իսկության աղյուսակ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A4%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BE_%E0%A4%B8%E0%A4%BE%E0%A4%B0%E0%A4%A3%E0%A5%80" title="सत्यता सारणी – Hindi" lang="hi" hreflang="hi" data-title="सत्यता सारणी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tabel_kebenaran" title="Tabel kebenaran – Indonesian" lang="id" hreflang="id" data-title="Tabel kebenaran" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tabella_della_verit%C3%A0" title="Tabella della verità – Italian" lang="it" hreflang="it" data-title="Tabella della verità" data-language-autonym="Italiano" data-language-local-name="Italian" 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%98%D7%91%D7%9C%D7%AA_%D7%90%D7%9E%D7%AA" title="טבלת אמת – Hebrew" lang="he" hreflang="he" data-title="טבלת אמת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Ratio_propositionum" title="Ratio propositionum – Latin" lang="la" hreflang="la" data-title="Ratio propositionum" data-language-autonym="Latina" data-language-local-name="Latin" 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/Paties%C4%ABbas_tabula" title="Patiesības tabula – Latvian" lang="lv" hreflang="lv" data-title="Patiesības tabula" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D0%B0_%D0%BD%D0%B0_%D0%B2%D0%B8%D1%81%D1%82%D0%B8%D0%BD%D0%B8%D1%82%D0%BE%D1%81%D1%82" title="Таблица на вистинитост – Macedonian" lang="mk" hreflang="mk" data-title="Таблица на вистинитост" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jadual_kebenaran" title="Jadual kebenaran – Malay" lang="ms" hreflang="ms" data-title="Jadual kebenaran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Waarheidstabel" title="Waarheidstabel – Dutch" lang="nl" hreflang="nl" data-title="Waarheidstabel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9C%9F%E7%90%86%E5%80%A4%E8%A1%A8" title="真理値表 – Japanese" lang="ja" hreflang="ja" data-title="真理値表" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sannhetstabell" title="Sannhetstabell – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sannhetstabell" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tablica_prawdy" title="Tablica prawdy – Polish" lang="pl" hreflang="pl" data-title="Tablica prawdy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tabela-verdade" title="Tabela-verdade – Portuguese" lang="pt" hreflang="pt" data-title="Tabela-verdade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D0%B0_%D0%B8%D1%81%D1%82%D0%B8%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Таблица истинности – Russian" lang="ru" hreflang="ru" data-title="Таблица истинности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B6%AD%E0%B7%8F_%E0%B7%80%E0%B6%9C%E0%B7%94%E0%B7%80" title="සත්‍යතා වගුව – Sinhala" lang="si" hreflang="si" data-title="සත්‍යතා වගුව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Truth_table" title="Truth table – Simple English" lang="en-simple" hreflang="en-simple" data-title="Truth table" 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/Pravdivostn%C3%A1_tabu%C4%BEka" title="Pravdivostná tabuľka – Slovak" lang="sk" hreflang="sk" data-title="Pravdivostná tabuľka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D0%B5_%D0%B8%D1%81%D1%82%D0%B8%D0%BD%D0%B8%D1%82%D0%BE%D1%81%D1%82%D0%B8" title="Таблице истинитости – Serbian" lang="sr" hreflang="sr" data-title="Таблице истинитости" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Totuustaulu" title="Totuustaulu – Finnish" lang="fi" hreflang="fi" data-title="Totuustaulu" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sanningsv%C3%A4rdetabell" title="Sanningsvärdetabell – Swedish" lang="sv" hreflang="sv" data-title="Sanningsvärdetabell" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%B2%E0%B8%A3%E0%B8%B2%E0%B8%87%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%88%E0%B8%A3%E0%B8%B4%E0%B8%87" title="ตารางค่าความจริง – Thai" lang="th" hreflang="th" data-title="ตารางค่าความจริง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Fruluk_tablosu" title="Doğruluk tablosu – Turkish" lang="tr" hreflang="tr" data-title="Doğruluk tablosu" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D1%8F_%D1%96%D1%81%D1%82%D0%B8%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Таблиця істинності – Ukrainian" lang="uk" hreflang="uk" data-title="Таблиця істинності" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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/%E7%9C%9F%E5%80%BC%E8%A1%A8" title="真值表 – Wu" lang="wuu" hreflang="wuu" data-title="真值表" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8" title="真值表 – Cantonese" lang="yue" hreflang="yue" data-title="真值表" 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></th></tr><tr><td class="sidebar-content"> <table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.3em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Negation" title="Negation">NOT</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg 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 -A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf6db742ace65252b589963f7e7a07603ccb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {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 \sim A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bf96c247833282e773fae43602343150c1665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \sim A}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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/74954195333a8593163b93a9688695b8dc74da55" 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\land B}"></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 A\cdot 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\cdot B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}"></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 AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></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 A\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65ee34a8896390e0d1f193c137d9eb64815c1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.315ex; height:2.176ex;" alt="{\displaystyle A\&B}"></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 A\&\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3517dd859d7eaa8f3f1656c8125f99ada1470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.123ex; height:2.176ex;" alt="{\displaystyle A\&\&B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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{\overline {\land }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0167d56141342887a74d56a036e6fbbad7172b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\land }}B}"></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 A\uparrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↑</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\uparrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5723e9ef44d446f4410c273b056d7c7c8e6f2564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\uparrow B}"></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 A\mid 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\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></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 {\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cdot B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225f35bb78e90b9126458f1bc6bf1ed3f0724bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.301ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cdot B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" 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\lor B}"></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 A+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"></span>, <span class="mwe-math-element"><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\mid 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\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></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 A\parallel 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\parallel B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd0239f9b74f7ef1520ba4e30454b06e695289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle A\parallel B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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{\overline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591afbca3e984765b18abb189f4bb1b88116c400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\lor }}B}"></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 A\downarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↓</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\downarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5e77260e67880093dafe958880ea02f5026164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\downarrow B}"></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 {\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A+B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08840f8e2022f127fc459d801a8f8ce93f65f55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:3.176ex;" alt="{\displaystyle {\overline {A+B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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\ {\text{XNOR}}\ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>XNOR</mtext> </mrow> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ {\text{XNOR}}\ B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fbdc2cf9fad58dda134c0c9baa55a4712b3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.673ex; height:2.176ex;" alt="{\displaystyle A\ {\text{XNOR}}\ B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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\equiv 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\equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}"></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 A\Leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce08fffb4d36ba12921b8b3e06228887015b2b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftrightarrow B}"></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 A\leftrightharpoons B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇋</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightharpoons B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8639312e9e120cd65c98fc48a6d5256d57288c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightharpoons B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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{\underline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e1413692912954b90e99694e10c728e27a153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.06ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B}"></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 A\oplus 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\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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\not \equiv 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\not \equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339ed15cd58f7263c5eec8e5628168aa6006200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B}"></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 A\not \Leftrightarrow 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\not \Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfd97641b05a7f7fc0bcd02e83fa6532c62bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\not \Leftrightarrow B}"></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 A\nleftrightarrow 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\nleftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926467de6fd4709bdbc59c3168a21298cdf0d26c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\nleftrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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\Rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e560143d45c97e6387c7c3aa90e9d7745002228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow B}"></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 A\supset 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\supset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee952838d8b3e67045072a8f2b71e7fc0467dea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}"></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 A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\rightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><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\Leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇐</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8569d34530d97f701080546ca0f20c0defadf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftarrow B}"></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 A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></span>, <span class="mwe-math-element"><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\leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">←</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456da65c891c438fea04d7e40283b67d600fe92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftarrow B}"></span></td></tr></tbody></table></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Related concepts</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="line-height:1.3em;"><ul><li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li><li><a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a class="mw-selflink selflink">Truth table</a></li><li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li><li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="/wiki/Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Applications</th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="/wiki/Programming_language" title="Programming language">Programming languages</a></li><li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li><li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td> </tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;"> <span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Logical_connectives" title="Category:Logical connectives">Category</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Logical_connectives_sidebar" title="Template:Logical connectives sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Logical_connectives_sidebar&action=edit&redlink=1" class="new" title="Template talk:Logical connectives sidebar (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives_sidebar" title="Special:EditPage/Template:Logical connectives sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>truth table</b> is a <a href="/wiki/Mathematical_table" title="Mathematical table">mathematical table</a> used in <a href="/wiki/Logic" title="Logic">logic</a>—specifically in connection with <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>, <a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a>, and <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>—which sets out the functional values of logical <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> on each of their functional arguments, that is, for each <a href="/wiki/Valuation_(logic)" title="Valuation (logic)">combination of values taken by their logical variables</a>.<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> In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, <a href="/wiki/Validity_(logic)" title="Validity (logic)">logically valid</a>. </p><p>A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, <a href="#Exclusive_disjunction">A</a> <a href="/wiki/XOR" class="mw-redirect" title="XOR">XOR</a> <a href="#Exclusive_disjunction">B</a>). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values. </p><p>A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i> from A to F is a special <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a>, a subset of A×F, which simply means that <i>f</i> can be listed as a list of input-output pairs. Clearly, for the Boolean functions, the output belongs to a binary set, i.e. F = {0, 1}. For an n-ary Boolean function, the inputs come from a domain that is itself a <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian</a> product of binary sets corresponding to the input Boolean variables. For example for a binary function, <i>f</i>(A, B), the domain of <i>f</i> is A×B, which can be listed as: A×B = {(A = 0, B = 0), (A = 0, B = 1), (A = 1, B = 0), (A = 1, B = 1)}. Each element in the domain represents a combination of input values for the variables A and B. These combinations now can be combined with the output of the function corresponding to that combination, thus forming the set of input-output pairs as a special relation that is a subset of A×F. For a relation to be a function, the special requirement is that each element of the domain of the function must be mapped to one and only one member of the codomain. Thus, the function f itself can be listed as: <i>f</i> = {((0, 0), <i>f</i><sub>0</sub>), ((0, 1), <i>f</i><sub>1</sub>), ((1, 0), <i>f</i><sub>2</sub>), ((1, 1), <i>f</i><sub>3</sub>)}, where <i>f</i><sub>0</sub>, <i>f</i><sub>1</sub>, <i>f</i><sub>2</sub>, and <i>f</i><sub>3</sub> are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. Rather than a list (set) given above, the truth table then presents these input-output pairs in a tabular format, in which each row corresponds to a member of the domain paired with its corresponding output value, 0 or 1. Of course, for the Boolean functions, we do not have to list all the members of the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> with their <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">images</a> in the <a href="/wiki/Codomain" title="Codomain">codomain</a>; we can simply list the mappings that map the member to "1", because all the others will have to be mapped to "0" automatically (that leads us to the <a href="/wiki/Minterms" class="mw-redirect" title="Minterms">minterms</a> idea). </p><p><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> is generally credited with inventing and popularizing the truth table in his <i><a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></i>, which was completed in 1918 and published in 1921.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Such a system was also independently proposed in 1921 by <a href="/wiki/Emil_Leon_Post" title="Emil Leon Post">Emil Leon Post</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Irving_Anellis" title="Irving Anellis">Irving Anellis</a>'s research shows that <a href="/wiki/C.S._Peirce" class="mw-redirect" title="C.S. Peirce">C.S. Peirce</a> appears to be the earliest logician (in 1883) to devise a truth table matrix.<sup id="cite_ref-Peirce_4-0" class="reference"><a href="#cite_note-Peirce-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>From the summary of Anellis's paper:<sup id="cite_ref-Peirce_4-1" class="reference"><a href="#cite_note-Peirce-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p> In 1997, John Shosky discovered, on the <a href="/wiki/Verso" class="mw-redirect" title="Verso">verso</a> of a page of the typed transcript of <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the <i><a href="/wiki/American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a></i> in 1885 includes an example of an indirect truth table for the conditional. </p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Truth tables can be used to prove many other <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalences</a>. For example, consider the following truth table: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <caption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\Rightarrow q)\equiv (\lnot p\lor q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">⇒</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p\Rightarrow q)\equiv (\lnot p\lor q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae9420d9f7cbd5594a7560cf0d4effa2bd0b4bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.942ex; height:2.843ex;" alt="{\displaystyle (p\Rightarrow q)\equiv (\lnot p\lor q)}"></span> </caption> <tbody><tr style="background:paleturquoise"> <th style="width:12%"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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 p}"></span> </th> <th style="width:12%"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </th> <th style="width:12%"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7370d2cd179d1da7e80b853f12f741864a867405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \lnot p}"></span> </th> <th style="width:12%"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot p\lor q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot p\lor q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18664e27a52918e470335cc08ffb66b7da64e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle \lnot p\lor q}"></span> </th> <th style="width:12%"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Rightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">⇒</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\Rightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66a1178759137a460fdf9377cd24ee749dac8de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\Rightarrow q}"></span> </th></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr></tbody></table> <p>This demonstrates the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Rightarrow q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">⇒</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\Rightarrow q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66a1178759137a460fdf9377cd24ee749dac8de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\Rightarrow q}"></span> is <a href="/wiki/Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot p\lor q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo>∨</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot p\lor q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18664e27a52918e470335cc08ffb66b7da64e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle \lnot p\lor q}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Truth_table_for_most_commonly_used_logical_operators">Truth table for most commonly used logical operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=3" title="Edit section: Truth table for most commonly used logical operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here is a truth table that gives definitions of the 7 most commonly used out of the 16 possible truth functions of two Boolean variables P and Q: </p> <table class="wikitable" style="margin:1em auto 1em auto; text-align:center;"> <tbody><tr> <th><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">P</span></span></th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">Q</span></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\lor Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\lor Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2bc60d4b9ff5ec772fec5c2ef72a39536d4323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\lor Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\ {\underline {\lor }}\ Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\ {\underline {\lor }}\ Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37b28c7d42934e351e72dd9b0c01fd5f5fe1a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:6.298ex; height:3.176ex;" alt="{\displaystyle P\ {\underline {\lor }}\ Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\ {\underline {\land }}\ Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∧</mo> <mo>_</mo> </munder> </mrow> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\ {\underline {\land }}\ Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d103292b05279995b54258adf7f296150c22b7da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:6.298ex; height:3.176ex;" alt="{\displaystyle P\ {\underline {\land }}\ Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a57c9bc077d0b20e4f5ec006f5342cfbb18fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Leftarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇐</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda627141a82e9511bfb63ad9607fe0a32eecf36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Leftarrow Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe75af42226920bc628ac7bbd53c023928f346ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Leftrightarrow Q}"></span> </th></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T</td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">P</span></span></th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">Q</span></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\lor Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\lor Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2bc60d4b9ff5ec772fec5c2ef72a39536d4323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\lor Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\ {\underline {\lor }}\ Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\ {\underline {\lor }}\ Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37b28c7d42934e351e72dd9b0c01fd5f5fe1a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:6.298ex; height:3.176ex;" alt="{\displaystyle P\ {\underline {\lor }}\ Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\ {\underline {\land }}\ Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∧</mo> <mo>_</mo> </munder> </mrow> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\ {\underline {\land }}\ Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d103292b05279995b54258adf7f296150c22b7da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:6.298ex; height:3.176ex;" alt="{\displaystyle P\ {\underline {\land }}\ Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a57c9bc077d0b20e4f5ec006f5342cfbb18fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Leftarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇐</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda627141a82e9511bfb63ad9607fe0a32eecf36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Leftarrow Q}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe75af42226920bc628ac7bbd53c023928f346ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Leftrightarrow Q}"></span> </th></tr> <tr> <td></td> <td> </td> <td><a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a><br /> (conjunction) </td> <td><a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a> <br /> (disjunction) </td> <td><a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a> <br /> (exclusive or) </td> <td><a href="/wiki/Exclusive_nor" class="mw-redirect" title="Exclusive nor">XNOR</a> <br /> (exclusive nor) </td> <td><a href="/wiki/Logical_conditional" class="mw-redirect" title="Logical conditional">conditional <br /> "if-then"</a> </td> <td>conditional <br /> "if" </td> <td><a href="/wiki/If_and_only_if" title="If and only if">biconditional<br /> "if-and-only-if"</a> </td></tr> <tr> <td colspan="9"> <p><i>where</i> <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><span class="legend-color mw-no-invert" style="background-color:#96d6f6; color:black;"> T </span> <i>means</i> <b>true</b> <i>and</i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><span class="legend-color mw-no-invert" style="background-color:#f696d6; color:black;"> F </span> <i>means</i> <b>false</b> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Condensed_truth_tables_for_binary_operators">Condensed truth tables for binary operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=4" title="Edit section: Condensed truth tables for binary operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example, <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> uses this condensed truth table notation: </p> <table> <tbody><tr> <td style="width:80px;"> </td> <td> <table class="wikitable" style="margin:1em auto 1em auto; text-align:center;"> <tbody><tr> <th data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">∧ </th> <th>T </th> <th>F </th></tr> <tr> <th>T </th> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr> <tr> <th>F </th> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr></tbody></table> </td> <td style="width:80px;"> </td> <td> <table class="wikitable" style="margin:1em auto 1em auto; text-align:center;"> <tbody><tr> <th data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">∨ </th> <th>T </th> <th>F </th></tr> <tr> <th>T </th> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <th>F </th> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr></tbody></table> </td></tr></tbody></table> <p>This notation is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of the distribution of the values in the table which can assist the reader in grasping the rules more quickly. </p> <div class="mw-heading mw-heading3"><h3 id="Truth_tables_in_digital_logic">Truth tables in digital logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=5" title="Edit section: Truth tables in digital logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Truth tables are also used to specify the function of <a href="/wiki/Lookup_table#Hardware_LUTs" title="Lookup table">hardware look-up tables (LUTs)</a> in <a href="/wiki/Digital_circuit" class="mw-redirect" title="Digital circuit">digital logic circuitry</a>. For an n-input LUT, the truth table will have 2^<i>n</i> values (or rows in the above tabular format), completely specifying a Boolean function for the LUT. By representing each Boolean value as a <a href="/wiki/Bit" title="Bit">bit</a> in a <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary number</a>, truth table values can be efficiently encoded as <a href="/wiki/Integer" title="Integer">integer</a> values in <a href="/wiki/Electronic_design_automation" title="Electronic design automation">electronic design automation (EDA)</a> <a href="/wiki/Software" title="Software">software</a>. For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs. </p><p>When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit index <i>k</i> based on the input values of the LUT, in which case the LUT's output value is the <i>k</i>th bit of the integer. For example, to evaluate the output value of a LUT given an <a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">array</a> of <i>n</i> Boolean input values, the bit index of the truth table's output value can be computed as follows: if the <i>i</i>th input is true, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af98ff0a21a287cb51f7be55e5d5c06c2421f3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle V_{i}=1}"></span>, else let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f623dac5aad31a952bd5f4c00a1fbb6693c90de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle V_{i}=0}"></span>. Then the <i>k</i>th bit of the binary representation of the truth table is the LUT's output value, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n}\times 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n}\times 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2978880cd6b741d1dde4642ac7e181cd8c0ea6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.589ex; height:3.009ex;" alt="{\displaystyle k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n}\times 2^{n}}"></span>. </p><p>Truth tables are a simple and straightforward way to encode Boolean functions, however given the <a href="/wiki/Exponential_growth" title="Exponential growth">exponential growth</a> in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and <a href="/wiki/Binary_decision_diagram" title="Binary decision diagram">binary decision diagrams</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Applications_of_truth_tables_in_digital_electronics">Applications of truth tables in digital electronics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=6" title="Edit section: Applications of truth tables in digital electronics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a> or code. For example, a binary addition can be represented with the truth table: </p> <table class="wikitable"> <caption>Binary addition </caption> <tbody><tr> <th style="width:80px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> </th> <th style="width:80px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> </th> <th style="width:80px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> </th> <th style="width:80px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"></span> </th></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr> <tr> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #CEF; color:black; vertical-align: middle; text-align: center;" class="partial table-partial">T </td></tr> <tr> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td> <td style="background: #FCE; color:black; vertical-align: middle; text-align: center;" class="nonfree table-nonfree">F </td></tr></tbody></table> <p>where A is the first operand, B is the second operand, C is the carry digit, and R is the result. </p><p>This truth table is read left to right: </p> <ul><li>Value pair (A, B) equals value pair (C, R).</li> <li>Or for this example, A plus B equal result R, with the Carry C.</li></ul> <p>This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values. </p><p>With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation. </p><p>In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase. </p><p>For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2×2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3×3, or nine possible outputs. </p><p>The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a <a href="/wiki/Full_adder" class="mw-redirect" title="Full adder">full adder</a>'s logic: </p> <pre>A B C* | C R 0 0 0 | 0 0 0 1 0 | 0 1 1 0 0 | 0 1 1 1 0 | 1 0 0 0 1 | 0 1 0 1 1 | 1 0 1 0 1 | 1 0 1 1 1 | 1 1 Same as previous, but.. C* = Carry from previous adder </pre> <div class="mw-heading mw-heading2"><h2 id="Methods_of_writing_truth_tables">Methods of writing truth tables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=7" title="Edit section: Methods of writing truth tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regarding the <i>guide columns<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></i> to the left of a table, which represent <a href="/wiki/Propositional_variable" title="Propositional variable">propositional variables</a>, different authors have different recommendations about how to fill them in, although this is of no logical significance.<sup id="cite_ref-:13_6-0" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Alternating_method">Alternating method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=8" title="Edit section: Alternating method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lee Archie, a professor at <a href="/wiki/Lander_University" title="Lander University">Lander University</a>, recommends this procedure, which is commonly followed in published truth-tables: </p> <ol><li>Write out the number of variables (corresponding to the number of statements) in alphabetical order.</li> <li>The number of lines needed is 2<sup>n</sup> where n is the number of variables. (E. g., with three variables, 2<sup>3</sup> = 8).</li> <li>Start in the right-hand column and alternate <b>T</b>'s and <b>F</b>'s until you run out of lines.</li> <li>Then move left to the next column and alternate pairs of <b>T</b>'s and <b>F</b>'s until you run out of lines.</li> <li>Then continue to the next left-hand column and double the numbers of <b>T</b>'s and <b>F</b>'s until completed.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ol> <p>This method results in truth-tables such as the following table for "<i>P ⊃ (Q ∨ R ⊃ (R ⊃ ¬P))</i>", produced by <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a>:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th><i>P</i> </th> <th><i>Q</i> </th> <th><i>R</i> </th> <th><i>P</i> ⊃ (<i>Q</i> ∨ <i>R</i> ⊃ (<i>R</i> ⊃ ¬<i>P</i>)) </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">f</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">t </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_method">Combinatorial method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=9" title="Edit section: Combinatorial method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Colin_Howson" title="Colin Howson">Colin Howson</a>, on the other hand, believes that "it is a good practical rule" to do the following:</p><blockquote><p>to start with all Ts, then all the ways (three) two Ts can be combined with one F, then all the ways (three) one T can be combined with two Fs, and then finish with all Fs. If a compound is built up from n distinct sentence letters, its truth table will have 2<sup>n</sup> rows, since there are two ways of assigning T or F to the first letter, and for each of these there will be two ways of assigning T or F to the second, and for each of these there will be two ways of assigning T or F to the third, and so on, giving 2.2.2. …, n times, which is equal to 2<sup>n</sup>.<sup id="cite_ref-:13_6-1" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>This results in truth tables like this table "showing that (A→C)∧(B→C) and (A∨B)→C are <a href="/wiki/Truth_function" title="Truth function">truth-functionally</a> <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a>", modeled after a table produced by <a href="/wiki/Colin_Howson" title="Colin Howson">Howson</a>:<sup id="cite_ref-:13_6-2" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th>A </th> <th>B </th> <th>C </th> <th>(A → C) ∧ (B → C) </th> <th>(A ∨ B) → C </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Size_of_truth_tables">Size of truth tables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=10" title="Edit section: Size of truth tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If there are <i>n</i> input variables then there are 2<sup><i>n</i></sup> possible combinations of their truth values. A given function may produce true or false for each combination so the number of different functions of <i>n</i> variables is the <a href="/wiki/Double_exponential_function" title="Double exponential function">double exponential</a> 2<sup>2<sup><i>n</i></sup></sup>. </p> <table class="wikitable" style="text-align:right;"> <tbody><tr> <th><i>n</i></th> <th>2<sup><i>n</i></sup></th> <th colspan="2">2<sup>2<sup><i>n</i></sup></sup> </th></tr> <tr> <td>0</td> <td>1</td> <td style="border-right:0px solid transparent;">2</td> <td style="border-left:0px solid transparent;"> </td></tr> <tr> <td>1</td> <td>2</td> <td style="border-right:0px solid transparent;">4</td> <td style="border-left:0px solid transparent;"> </td></tr> <tr> <td>2</td> <td>4</td> <td style="border-right:0px solid transparent;">16</td> <td style="border-left:0px solid transparent;"> </td></tr> <tr> <td>3</td> <td>8</td> <td style="border-right:0px solid transparent;">256</td> <td style="border-left:0px solid transparent;"> </td></tr> <tr> <td>4</td> <td>16</td> <td style="border-right:0px solid transparent;">65,536</td> <td style="border-left:0px solid transparent;text-align:left;"> </td></tr> <tr> <td>5</td> <td>32</td> <td style="border-right:0px solid transparent;">4,294,967,296</td> <td style="border-left:0px solid transparent;text-align:left;">≈ 4.3<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000900000000000000♠"></span>9</span></sup> </td></tr> <tr> <td>6</td> <td>64</td> <td style="border-right:0px solid transparent;">18,446,744,073,709,551,616</td> <td style="border-left:0px solid transparent;text-align:left;">≈ 1.8<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001190000000000000♠"></span>19</span></sup> </td></tr> <tr> <td>7</td> <td>128</td> <td style="border-right:0px solid transparent;"><span class="nowrap"><span data-sort-value="7038340282366920938♠"></span>340,282,366,920,938,463,463,374,607,431,768,211,456</span></td> <td style="border-left:0px solid transparent;text-align:left;">≈ 3.4<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001380000000000000♠"></span>38</span></sup> </td></tr> <tr> <td>8</td> <td>256</td> <td style="border-right:0px solid transparent;"><span class="nowrap"><span data-sort-value="7077115792089237316♠"></span>115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936</span></td> <td style="border-left:0px solid transparent;text-align:left;">≈ 1.2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001770000000000000♠"></span>77</span></sup> </td></tr></tbody></table> <p>Truth tables for functions of three or more variables are rarely given. </p> <div class="mw-heading mw-heading2"><h2 id="Function_Tables">Function Tables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=11" title="Edit section: Function Tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It can be useful to have the output of a truth table expressed as a function of some variable values, instead of just a literal truth or false value. These may be called "function tables" to differentiate them from the more general "truth tables".<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, one value, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, may be used with an XOR gate to conditionally invert another value, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. In other words, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is false, the output is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, and when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is true, the output is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444e25985f32b5394ed8ed80b02211a1d6d4a145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle \lnot X}"></span>. The function table for this would look like: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G{\underline {\lor }}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G{\underline {\lor }}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d8bbef97b5ddaac7089bc472c7151ba259d755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.359ex; height:3.176ex;" alt="{\displaystyle G{\underline {\lor }}X}"></span> </th></tr> <tr> <td>F</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> </td></tr> <tr> <td>T</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 \lnot X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444e25985f32b5394ed8ed80b02211a1d6d4a145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle \lnot X}"></span> </td></tr></tbody></table> <p>Similarly, a 4-to-1 <a href="/wiki/Multiplexer" title="Multiplexer">multiplexer</a> with select imputs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span>, data inputs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, and output <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> (as displayed in the image) would have this function table: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Multiplexer_4-to-1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Multiplexer_4-to-1.svg/220px-Multiplexer_4-to-1.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Multiplexer_4-to-1.svg/330px-Multiplexer_4-to-1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Multiplexer_4-to-1.svg/440px-Multiplexer_4-to-1.svg.png 2x" data-file-width="250" data-file-height="175" /></a><figcaption>4-to-1 multiplexer</figcaption></figure> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> </th></tr> <tr> <td>F</td> <td>F</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> </td></tr> <tr> <td>F</td> <td>T</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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> </td></tr> <tr> <td>T</td> <td>F</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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> </td></tr> <tr> <td>T</td> <td>T</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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Sentential_operator_truth_tables">Sentential operator truth tables</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=12" title="Edit section: Sentential operator truth tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Overview_table">Overview table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=13" title="Edit section: Overview table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variables <i><b>p</b></i> and <i><b>q</b></i>:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:center;"> <tbody><tr> <th><i>p</i></th> <th><i>q</i> </th> <th style="background:black"> </th> <th><a href="/wiki/Contradiction" title="Contradiction">F</a><sup>0</sup></th> <th><a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a><sup>1</sup></th> <th><a href="/wiki/Converse_nonimplication" title="Converse nonimplication">↚</a><sup>2</sup></th> <th><a href="/wiki/Negation" title="Negation"><b>¬p</b></a><sup>3</sup></th> <th><a href="/wiki/Material_nonimplication" title="Material nonimplication">NIMPLY</a><sup>4</sup></th> <th><a href="/wiki/Negation" title="Negation"><b>¬q</b></a><sup>5</sup></th> <th><a href="/wiki/Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">XOR</a><sup>6</sup></th> <th><a href="/wiki/Logical_NAND" class="mw-redirect" title="Logical NAND">NAND</a><sup>7</sup></th> <th><a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a><sup>8</sup></th> <th><a href="/wiki/Logical_biconditional" title="Logical biconditional">XNOR</a><sup>9</sup></th> <th><a href="/wiki/Projection_function" class="mw-redirect" title="Projection function">q</a><sup>10</sup></th> <th><a href="/wiki/Material_conditional" title="Material conditional">IMPLY</a><sup>11</sup></th> <th><a href="/wiki/Projection_function" class="mw-redirect" title="Projection function">p</a><sup>12</sup></th> <th><a href="/wiki/Converse_implication" class="mw-redirect" title="Converse implication">←</a><sup>13</sup></th> <th><a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a><sup>14</sup></th> <th><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">T</a><sup>15</sup> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background:black"></td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background:black"></td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background:black"></td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background:black"></td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td colspan="19" style="background:black"> </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Commutative">Com</abbr> </td> <td style="background:black"></td> <td>✓</td> <td>✓</td> <td></td> <td></td> <td></td> <td></td> <td>✓</td> <td>✓</td> <td>✓</td> <td>✓</td> <td></td> <td></td> <td></td> <td></td> <td>✓</td> <td>✓ </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Associative">Assoc</abbr> </td> <td style="background:black"></td> <td>✓</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>✓</td> <td></td> <td>✓</td> <td>✓</td> <td>✓</td> <td></td> <td>✓</td> <td></td> <td>✓</td> <td>✓ </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Adjoint operator">Adj</abbr> </td> <td style="background:black"></td> <td>F<sup>0</sup></td> <td>NOR<sup>1</sup></td> <td>↛<sup>4</sup></td> <td>¬q<sup>5</sup></td> <td>↚<sup>2</sup></td> <td>¬p<sup>3</sup></td> <td>XOR<sup>6</sup></td> <td>NAND<sup>7</sup></td> <td>AND<sup>8</sup></td> <td>XNOR<sup>9</sup></td> <td>p<sup>12</sup></td> <td>←<sup>13</sup></td> <td>q<sup>10</sup></td> <td>→<sup>11</sup></td> <td>OR<sup>14</sup></td> <td>T<sup>15</sup> </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Negation">Neg</abbr> </td> <td style="background:black"></td> <td>T<sup>15</sup></td> <td>OR<sup>14</sup></td> <td>←<sup>13</sup></td> <td>p<sup>12</sup></td> <td>IMPLY<sup>11</sup></td> <td>q<sup>10</sup></td> <td>XNOR<sup>9</sup></td> <td>AND<sup>8</sup></td> <td>NAND<sup>7</sup></td> <td>XOR<sup>6</sup></td> <td>¬q<sup>5</sup></td> <td>NIMPLY<sup>4</sup></td> <td>¬p<sup>3</sup></td> <td>↚<sup>2</sup></td> <td>NOR<sup>1</sup></td> <td>F<sup>0</sup> </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Dual operator">Dual</abbr> </td> <td style="background:black"></td> <td>T<sup>15</sup></td> <td>NAND<sup>7</sup></td> <td>→<sup>11</sup></td> <td>¬p<sup>3</sup></td> <td>←<sup>13</sup></td> <td>¬q<sup>5</sup></td> <td>XNOR<sup>9</sup></td> <td>NOR<sup>1</sup></td> <td>OR<sup>14</sup></td> <td>XOR<sup>6</sup></td> <td>q<sup>10</sup></td> <td>↚<sup>2</sup></td> <td>p<sup>12</sup></td> <td>↛<sup>4</sup></td> <td>AND<sup>8</sup></td> <td>F<sup>0</sup> </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Left identities">L id</abbr> </td> <td style="background:black"></td> <td></td> <td></td> <td>F</td> <td></td> <td></td> <td></td> <td>F</td> <td></td> <td>T</td> <td>T</td> <td>T,F</td> <td>T</td> <td></td> <td></td> <td>F</td> <td> </td></tr> <tr> <td colspan="2" style="background: #;"><abbr title="Right identities">R id</abbr> </td> <td style="background:black"></td> <td></td> <td></td> <td></td> <td></td> <td>F</td> <td></td> <td>F</td> <td></td> <td>T</td> <td>T</td> <td></td> <td></td> <td>T,F</td> <td>T</td> <td>F</td> <td> </td></tr></tbody></table></dd></dl> <p>where </p> <dl><dd>T = true.</dd> <dd>F = false.</dd> <dd>The superscripts <sup>0</sup> to <sup>15</sup> is the number resulting from reading the four truth values as a <a href="/wiki/Binary_number" title="Binary number">binary number</a> with F = 0 and T = 1.</dd> <dd>The <b>Com</b> row indicates whether an operator, <b>op</b>, is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a> - <b>P op Q = Q op P</b>.</dd> <dd>The <b>Assoc</b> row indicates whether an operator, <b>op</b>, is <a href="/wiki/Associative_property" title="Associative property">associative</a> - <b>(P op Q) op R = P op (Q op R)</b>.</dd> <dd>The <b>Adj</b> row shows the operator <b>op2</b> such that <b>P op Q = Q op2 P</b>.</dd> <dd>The <b>Neg</b> row shows the operator <b>op2</b> such that <b>P op Q = ¬(P op2 Q)</b>.</dd> <dd>The <b>Dual</b> row shows the <a href="/wiki/Duality_principle_(Boolean_algebra)" class="mw-redirect" title="Duality principle (Boolean algebra)">dual operation</a> obtained by interchanging T with F, and AND with OR.</dd> <dd>The <b>L id</b> row shows the operator's <a href="/wiki/Left_identity" class="mw-redirect" title="Left identity">left identities</a> if it has any - values <b>I</b> such that <b>I op Q = Q</b>.</dd> <dd>The <b>R id</b> row shows the operator's <a href="/wiki/Right_identity" class="mw-redirect" title="Right identity">right identities</a> if it has any - values <b>I</b> such that <b>P op I = P</b>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wittgenstein_table">Wittgenstein table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=14" title="Edit section: Wittgenstein table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In proposition 5.101 of the <a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a>,<sup id="cite_ref-tlp5.101_11-0" class="reference"><a href="#cite_note-tlp5.101-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgenstein</a> listed the table above as follows: </p> <dl><dd><table class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:left;"> <tbody><tr> <th scope="col"> </th> <th scope="col">Truthvalues </th> <th scope="col"> </th> <th scope="col" colspan="2">Operator </th> <th scope="col">Operation name </th> <th scope="col">Tractatus<sup id="cite_ref-different_mapping_12-0" class="reference"><a href="#cite_note-different_mapping-12"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>0</td> <td>(F F F F)(p, q)</td> <td>⊥</td> <td><a href="/wiki/Falsum" class="mw-redirect" title="Falsum">false</a></td> <td><b>Opq</b></td> <td><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a></td> <td>p and not p; and q and not q </td></tr> <tr> <td>1</td> <td>(F F F T)(p, q)</td> <td>NOR</td> <td><b>p</b> ↓ <b>q</b></td> <td><b>Xpq</b></td> <td><a href="/wiki/Logical_NOR" title="Logical NOR">Logical NOR</a></td> <td>neither <i>p</i> nor <i>q</i> </td></tr> <tr> <td>2</td> <td>(F F T F)(p, q)</td> <td>↚</td> <td><b>p</b> ↚ <b>q</b></td> <td><b>Mpq</b></td> <td><a href="/wiki/Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</a></td> <td><i>q</i> and not <i>p</i> </td></tr> <tr> <td>3</td> <td>(F F T T)(p, q)</td> <td><b>¬p</b>, <b>~p</b></td> <td><b>¬p</b></td> <td><b>Np</b>, <b>Fpq</b></td> <td><a href="/wiki/Negation" title="Negation">Negation</a></td> <td>not <i>p</i> </td></tr> <tr> <td>4</td> <td>(F T F F)(p, q)</td> <td>↛</td> <td><b>p</b> ↛ <b>q</b> </td> <td><b>Lpq</b></td> <td><a href="/wiki/Material_nonimplication" title="Material nonimplication">Material nonimplication</a></td> <td><i>p</i> and not <i>q</i> </td></tr> <tr> <td>5</td> <td>(F T F T)(p, q)</td> <td><b>¬q</b>, <b>~q</b></td> <td><b>¬q</b></td> <td><b>Nq</b>, <b>Gpq</b></td> <td>Negation</td> <td>not <i>q</i> </td></tr> <tr> <td>6</td> <td>(F T T F)(p, q)</td> <td>XOR</td> <td><b>p</b> ⊕ <b>q</b></td> <td><b>Jpq</b></td> <td><a href="/wiki/Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">Exclusive disjunction</a></td> <td><i>p</i> or <i>q</i>, but not both </td></tr> <tr> <td>7</td> <td>(F T T T)(p, q)</td> <td>NAND</td> <td><b>p</b> ↑ <b>q</b></td> <td><b>Dpq</b></td> <td><a href="/wiki/Logical_NAND" class="mw-redirect" title="Logical NAND">Logical NAND</a></td> <td>not both <i>p</i> and <i>q</i> </td></tr> <tr> <td>8</td> <td>(T F F F)(p, q)</td> <td>AND</td> <td><b>p</b> ∧ <b>q</b></td> <td><b>Kpq</b></td> <td><a href="/wiki/Logical_conjunction" title="Logical conjunction">Logical conjunction</a></td> <td><i>p</i> and <i>q</i> </td></tr> <tr> <td>9</td> <td>(T F F T)(p, q)</td> <td>XNOR</td> <td><b>p</b> <a href="/wiki/If_and_only_if" title="If and only if">iff</a> <b>q</b></td> <td><b>Epq</b></td> <td><a href="/wiki/Logical_biconditional" title="Logical biconditional">Logical biconditional</a></td> <td>if <i>p</i> then <i>q</i>; and if <i>q</i> then <i>p</i> </td></tr> <tr> <td>10</td> <td>(T F T F)(p, q)</td> <td><b>q</b></td> <td><b>q</b></td> <td><b>Hpq</b></td> <td><a href="/wiki/Projection_function" class="mw-redirect" title="Projection function">Projection function</a></td> <td><i>q</i> </td></tr> <tr> <td>11</td> <td>(T F T T)(p, q)</td> <td><b>p</b> → <b>q</b></td> <td>if <b>p</b> then <b>q</b></td> <td><b>Cpq</b></td> <td><a href="/wiki/Material_conditional" title="Material conditional">Material implication</a></td> <td>if <i>p</i> then <i>q</i> </td></tr> <tr> <td>12</td> <td>(T T F F)(p, q)</td> <td><b>p</b></td> <td><b>p</b></td> <td><b>Ipq</b></td> <td>Projection function</td> <td><i>p</i> </td></tr> <tr> <td>13</td> <td>(T T F T)(p, q)</td> <td><b>p</b> ← <b>q</b></td> <td>if <b>q</b> then <b>p</b></td> <td><b>Bpq</b></td> <td><a href="/wiki/Converse_implication" class="mw-redirect" title="Converse implication">Converse implication</a></td> <td>if <i>q</i> then <i>p</i> </td></tr> <tr> <td>14</td> <td>(T T T F)(p, q)</td> <td>OR</td> <td><b>p</b> ∨ <b>q</b></td> <td><b>Apq</b></td> <td><a href="/wiki/Logical_disjunction" title="Logical disjunction">Logical disjunction</a></td> <td><i>p</i> or <i>q</i> </td></tr> <tr> <td>15</td> <td>(T T T T)(p, q)</td> <td>⊤</td> <td><a href="/wiki/Tee_(symbol)" title="Tee (symbol)">true</a></td> <td><b>Vpq</b></td> <td><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></td> <td>if p then p; and if q then q </td></tr></tbody></table></dd></dl> <p>The truth table represented by each row is obtained by appending the sequence given in <b>Truthvalues</b><sub>row</sub> to the table<sup id="cite_ref-different_mapping_12-1" class="reference"><a href="#cite_note-different_mapping-12"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:left;"> <tbody><tr> <th scope="row"><i>p</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <th scope="row"><i>q</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table></dd></dl> <p>For example, the table </p> <dl><dd><table class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:left;"> <tbody><tr> <th scope="row"><i>p</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <th scope="row"><i>q</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <th scope="row"><i>11</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table></dd></dl> <p>represents the truth table for <a href="/wiki/Material_conditional" title="Material conditional">Material implication</a>. Logical operators can also be visualized using <a href="/wiki/Venn_diagram#Overview" title="Venn diagram">Venn diagrams</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nullary_operations">Nullary operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=15" title="Edit section: Nullary operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 2 nullary operations: </p> <ul><li>Always true</li> <li>Never true, unary <i><a href="/wiki/Falsum" class="mw-redirect" title="Falsum">falsum</a></i></li></ul> <div class="mw-heading mw-heading4"><h4 id="Logical_true">Logical true</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=16" title="Edit section: Logical true"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The output value is always true, because this operator has zero operands and therefore no input values </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:80px"><i>p</i> </th> <th style="width:80px"><span class="texhtml"><i>T</i></span> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Logical_false">Logical false</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=17" title="Edit section: Logical false"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The output value is never true: that is, always false, because this operator has zero operands and therefore no input values </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:80px"><i>p</i> </th> <th style="width:80px"><span class="texhtml"><i>F</i></span> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Unary_operations">Unary operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=18" title="Edit section: Unary operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 2 unary operations: </p> <ul><li>Unary <i>identity</i></li> <li>Unary <i>negation</i></li></ul> <div class="mw-heading mw-heading4"><h4 id="Logical_identity">Logical identity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=19" title="Edit section: Logical identity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Identity_function" title="Identity function">Logical identity</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on one <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical value</a> p, for which the output value remains p. </p><p>The truth table for the logical identity operator is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:80px"><i>p</i> </th> <th style="width:80px"><span class="texhtml"><i>p</i></span> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Logical_negation">Logical negation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=20" title="Edit section: Logical negation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logical_negation" class="mw-redirect" title="Logical negation">Logical negation</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on one <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical value</a>, typically the value of a <a href="/wiki/Proposition" title="Proposition">proposition</a>, that produces a value of <i>true</i> if its operand is false and a value of <i>false</i> if its operand is true. </p><p>The truth table for <b>NOT p</b> (also written as <b>¬p</b>, <b>Np</b>, <b>Fpq</b>, or <b>~p</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:80px"><i>p</i> </th> <th style="width:80px"><span class="texhtml"><i>¬p</i></span> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Binary_operations">Binary operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=21" title="Edit section: Binary operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 16 possible <a href="/wiki/Truth_function" title="Truth function">truth functions</a> of two <a href="/wiki/Binary_variable" class="mw-redirect" title="Binary variable">binary variables</a>, each operator has its own name. </p> <div class="mw-heading mw-heading4"><h4 id="Logical_conjunction_(AND)"><span id="Logical_conjunction_.28AND.29"></span>Logical conjunction (AND)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=22" title="Edit section: Logical conjunction (AND)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logical_conjunction" title="Logical conjunction">Logical conjunction</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both of its operands are true. </p><p>The truth table for <b>p AND q</b> (also written as <b>p ∧ q</b>, <b>Kpq</b>, <b>p & q</b>, or <b>p</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span> <b>q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ∧ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table> <p>In ordinary language terms, if both <i>p</i> and <i>q</i> are true, then the conjunction <i>p</i> ∧ <i>q</i> is true. For all other assignments of logical values to <i>p</i> and to <i>q</i> the conjunction <i>p</i> ∧ <i>q</i> is false. </p><p>It can also be said that if <i>p</i>, then <i>p</i> ∧ <i>q</i> is <i>q</i>, otherwise <i>p</i> ∧ <i>q</i> is <i>p</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Logical_disjunction_(OR)"><span id="Logical_disjunction_.28OR.29"></span>Logical disjunction (OR)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=23" title="Edit section: Logical disjunction (OR)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logical_disjunction" title="Logical disjunction">Logical disjunction</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if at least one of its operands is true. </p><p>The truth table for <b>p OR q</b> (also written as <b>p ∨ q</b>, <b>Apq</b>, <b>p || q</b>, or <b>p + q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ∨ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table> <p>Stated in English, if <i>p</i>, then <i>p</i> ∨ <i>q</i> is <i>p</i>, otherwise <i>p</i> ∨ <i>q</i> is <i>q</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Logical_implication">Logical implication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=24" title="Edit section: Logical implication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Logical implication and the <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a> are both associated with an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, which produces a value of <i>false</i> if the first operand is true and the second operand is false, and a value of <i>true</i> otherwise. </p><p>The truth table associated with the logical implication <b>p implies q</b> (symbolized as <b>p ⇒ q</b>, or more rarely <b>Cpq</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ⇒ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p>The truth table associated with the material conditional <b>if p then q</b> (symbolized as <b>p → q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> → <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p><b>p ⇒ q</b> and <b>p → q</b> are equivalent to <b>¬p ∨ q</b>. </p> <div class="mw-heading mw-heading4"><h4 id="Logical_equality">Logical equality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=25" title="Edit section: Logical equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logical_equality" title="Logical equality">Logical equality</a> (also known as <a href="/wiki/Biconditional" class="mw-redirect" title="Biconditional">biconditional</a> or <a href="/wiki/Exclusive_nor" class="mw-redirect" title="Exclusive nor">exclusive nor</a>) is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both operands are false or both operands are true. </p><p>The truth table for <b>p XNOR q</b> (also written as <b>p ↔ q</b>, <b>Epq</b>, <b>p = q</b>, or <b>p ≡ q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ↔ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p>So p EQ q is true if p and q have the same <a href="/wiki/Truth_value" title="Truth value">truth value</a> (both true or both false), and false if they have different truth values. </p> <div class="mw-heading mw-heading4"><h4 id="Exclusive_disjunction">Exclusive disjunction</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=26" title="Edit section: Exclusive disjunction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">Exclusive disjunction</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if one but not both of its operands is true. </p><p>The truth table for <b>p XOR q</b> (also written as <b>Jpq</b>, or <b>p ⊕ q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><b>p</b> ⊕ <b>q</b> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table> <p>For two propositions, <b>XOR</b> can also be written as (p ∧ ¬q) ∨ (¬p ∧ q). </p> <div class="mw-heading mw-heading4"><h4 id="Logical_NAND">Logical NAND</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=27" title="Edit section: Logical NAND"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Logical_NAND" class="mw-redirect" title="Logical NAND">logical NAND</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>false</i> if both of its operands are true. In other words, it produces a value of <i>true</i> if at least one of its operands is false. </p><p>The truth table for <b>p NAND q</b> (also written as <b>p ↑ q</b>, <b>Dpq</b>, or <b>p | q</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ↑ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p>It is frequently useful to express a logical operation as a <a href="/wiki/Compound_operation_(computing)" class="mw-redirect" title="Compound operation (computing)">compound operation</a>, that is, as an operation that is built up or composed from other operations. Many such compositions are possible, depending on the operations that are taken as basic or "primitive" and the operations that are taken as composite or "derivative". </p><p>In the case of logical NAND, it is clearly expressible as a compound of NOT and AND. </p><p>The negation of a conjunction: ¬(<i>p</i> ∧ <i>q</i>), and the disjunction of negations: (¬<i>p</i>) ∨ (¬<i>q</i>) can be tabulated as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ∧ <i>q</i> </th> <th style="width:15%">¬(<i>p</i> ∧ <i>q</i>) </th> <th style="width:15%">¬<i>p</i> </th> <th style="width:15%">¬<i>q</i> </th> <th style="width:15%">(¬<i>p</i>) ∨ (¬<i>q</i>) </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Logical_NOR">Logical NOR</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=28" title="Edit section: Logical NOR"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Logical_NOR" title="Logical NOR">logical NOR</a> is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both of its operands are false. In other words, it produces a value of <i>false</i> if at least one of its operands is true. ↓ is also known as the <a href="/wiki/Peirce_arrow" class="mw-redirect" title="Peirce arrow">Peirce arrow</a> after its inventor, <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, and is a <a href="/wiki/Sole_sufficient_operator" class="mw-redirect" title="Sole sufficient operator">Sole sufficient operator</a>. </p><p>The truth table for <b>p NOR q</b> (also written as <b>p ↓ q</b>, or <b>Xpq</b>) is as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:15%"><i>p</i> </th> <th style="width:15%"><i>q</i> </th> <th style="width:15%"><i>p</i> ↓ <i>q</i> </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p>The negation of a disjunction ¬(<i>p</i> ∨ <i>q</i>), and the conjunction of negations (¬<i>p</i>) ∧ (¬<i>q</i>) can be tabulated as follows: </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <tbody><tr> <th style="width:10%"><i>p</i> </th> <th style="width:10%"><i>q</i> </th> <th style="width:10%"><i>p</i> ∨ <i>q</i> </th> <th style="width:10%">¬(<i>p</i> ∨ <i>q</i>) </th> <th style="width:10%">¬<i>p</i> </th> <th style="width:10%">¬<i>q</i> </th> <th style="width:10%">(¬<i>p</i>) ∧ (¬<i>q</i>) </th></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T </td></tr></tbody></table> <p>Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments <i>p</i> and <i>q</i>, produces the identical patterns of functional values for ¬(<i>p</i> ∧ <i>q</i>) as for (¬<i>p</i>) ∨ (¬<i>q</i>), and for ¬(<i>p</i> ∨ <i>q</i>) as for (¬<i>p</i>) ∧ (¬<i>q</i>). Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertain solely to their logical values. </p><p>This equivalence is one of <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=30" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Information about notation may be found in (<a href="#CITEREFBocheński1959">Bocheński 1959</a>), (<a href="#CITEREFEnderton2001">Enderton 2001</a>), and (<a href="#CITEREFQuine1982">Quine 1982</a>).</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">The operators here with equal left and right identities (XOR, AND, XNOR, and OR) are also <a href="/wiki/Monoid#Commutative_monoid" title="Monoid">commutative monoids</a> because they are also <a href="/wiki/Associative_property" title="Associative property">associative</a>. While this distinction may be irrelevant in a simple discussion of logic, it can be quite important in more advanced mathematics. For example, in <a href="/wiki/Category_theory" title="Category theory">category theory</a> an <a href="/wiki/Enriched_category" title="Enriched category">enriched category</a> is described as a base <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> enriched over a monoid, and any of these operators can be used for enrichment.</span> </li> <li id="cite_note-different_mapping-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-different_mapping_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-different_mapping_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Wittgenstein used a different mapping. In proposition 5.101 of the Tractatus one has to append <b>Truthvalues</b><sub>row</sub> to the table <dl><dd><table class="wikitable" style="margin:left margin:1em auto 1em auto; text-align:left;"> <tbody><tr> <th scope="row"><i>p</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr> <tr> <th scope="row"><i>q</i> </th> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F</td> <td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F </td></tr></tbody></table></dd></dl> <p>This explains why <b>Tractatus</b><sub>row</sub> in the table given here does not point to the same <b>Truthvalues</b><sub>row</sub> as in the Tractatus. </p> </span></li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=31" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFEnderton2001">Enderton 2001</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFvon_Wright1955" class="citation journal cs1"><a href="/wiki/Georg_Henrik_von_Wright" title="Georg Henrik von Wright">von Wright, Georg Henrik</a> (1955). "Ludwig Wittgenstein, A Biographical Sketch". <i>The Philosophical Review</i>. <b>64</b> (4): 527–545 (p. 532, note 9). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2182631">10.2307/2182631</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2182631">2182631</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Philosophical+Review&rft.atitle=Ludwig+Wittgenstein%2C+A+Biographical+Sketch&rft.volume=64&rft.issue=4&rft.pages=527-545+%28p.+532%2C+note+9%29&rft.date=1955&rft_id=info%3Adoi%2F10.2307%2F2182631&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2182631%23id-name%3DJSTOR&rft.aulast=von+Wright&rft.aufirst=Georg+Henrik&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPost1921" class="citation journal cs1"><a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Post, Emil</a> (July 1921). "Introduction to a general theory of elementary propositions". <i>American Journal of Mathematics</i>. <b>43</b> (3): 163–185. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2370324">10.2307/2370324</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuiuo.ark%3A%2F13960%2Ft9j450f7q">2027/uiuo.ark:/13960/t9j450f7q</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2370324">2370324</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=Introduction+to+a+general+theory+of+elementary+propositions&rft.volume=43&rft.issue=3&rft.pages=163-185&rft.date=1921-07&rft_id=info%3Ahdl%2F2027%2Fuiuo.ark%3A%2F13960%2Ft9j450f7q&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2370324%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2370324&rft.aulast=Post&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-Peirce-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Peirce_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Peirce_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnellis2012" class="citation journal cs1"><a href="/wiki/Irving_Anellis" title="Irving Anellis">Anellis, Irving H.</a> (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". <i>History and Philosophy of Logic</i>. <b>33</b>: 87–97. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01445340.2011.621702">10.1080/01445340.2011.621702</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170654885">170654885</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=History+and+Philosophy+of+Logic&rft.atitle=Peirce%27s+Truth-functional+Analysis+and+the+Origin+of+the+Truth+Table&rft.volume=33&rft.pages=87-97&rft.date=2012&rft_id=info%3Adoi%2F10.1080%2F01445340.2011.621702&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170654885%23id-name%3DS2CID&rft.aulast=Anellis&rft.aufirst=Irving+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://philosophy.lander.edu/logic/table.html">"How to Construct a Truth Table"</a>. <i>philosophy.lander.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-04-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=philosophy.lander.edu&rft.atitle=How+to+Construct+a+Truth+Table&rft_id=https%3A%2F%2Fphilosophy.lander.edu%2Flogic%2Ftable.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-:13-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:13_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:13_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:13_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowson1997" class="citation book cs1">Howson, Colin (1997). <i>Logic with trees: an introduction to symbolic logic</i>. London; New York: Routledge. p. 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-13342-5" title="Special:BookSources/978-0-415-13342-5"><bdi>978-0-415-13342-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+with+trees%3A+an+introduction+to+symbolic+logic&rft.place=London%3B+New+York&rft.pages=10&rft.pub=Routledge&rft.date=1997&rft.isbn=978-0-415-13342-5&rft.aulast=Howson&rft.aufirst=Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene2013" class="citation book cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4GzCAgAAQBAJ&pg=PA11"><i>Mathematical Logic</i></a>. Dover Books on Mathematics. Courier Corporation. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486317076" title="Special:BookSources/9780486317076"><bdi>9780486317076</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Logic&rft.series=Dover+Books+on+Mathematics&rft.pages=11&rft.pub=Courier+Corporation&rft.date=2013&rft.isbn=9780486317076&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4GzCAgAAQBAJ%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManoCiletti2018" class="citation book cs1">Mano, M. Morris; Ciletti, Michael (2018-07-13). <i>Digital Design, Global Edition</i> (6th ed.). Pearson Education, Limited. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781292231167" title="Special:BookSources/9781292231167"><bdi>9781292231167</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Design%2C+Global+Edition&rft.edition=6th&rft.pub=Pearson+Education%2C+Limited&rft.date=2018-07-13&rft.isbn=9781292231167&rft.aulast=Mano&rft.aufirst=M.+Morris&rft.au=Ciletti%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> <li id="cite_note-tlp5.101-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-tlp5.101_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWittgenstein1922" class="citation book cs1"><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgenstein, Ludwig</a> (1922). <a rel="nofollow" class="external text" href="https://www.gutenberg.org/files/5740/5740-pdf.pdf"><i>Tractatus Logico-Philosophicus</i></a> <span class="cs1-format">(PDF)</span>. Proposition 5.101.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tractatus+Logico-Philosophicus&rft.pages=Proposition+5.101&rft.date=1922&rft.aulast=Wittgenstein&rft.aufirst=Ludwig&rft_id=https%3A%2F%2Fwww.gutenberg.org%2Ffiles%2F5740%2F5740-pdf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=32" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBocheński1959" class="citation book cs1"><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński, Józef Maria</a> (1959). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YX8iCQAAQBAJ&pg=PP5"><i>A Précis of Mathematical Logic</i></a>. Translated by Bird, Otto. D. Reidel. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-0592-9">10.1007/978-94-017-0592-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-017-0592-9" title="Special:BookSources/978-94-017-0592-9"><bdi>978-94-017-0592-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Pr%C3%A9cis+of+Mathematical+Logic&rft.pub=D.+Reidel&rft.date=1959&rft_id=info%3Adoi%2F10.1007%2F978-94-017-0592-9&rft.isbn=978-94-017-0592-9&rft.aulast=Boche%C5%84ski&rft.aufirst=J%C3%B3zef+Maria&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYX8iCQAAQBAJ%26pg%3DPP5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton2001" class="citation book cs1"><a href="/wiki/Herbert_Enderton" title="Herbert Enderton">Enderton, H.</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dVncCl_EtUkC&pg=PR7"><i>A Mathematical Introduction to Logic</i></a> (2nd ed.). Harcourt Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-238452-0" title="Special:BookSources/0-12-238452-0"><bdi>0-12-238452-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.edition=2nd&rft.pub=Harcourt+Academic+Press&rft.date=2001&rft.isbn=0-12-238452-0&rft.aulast=Enderton&rft.aufirst=H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdVncCl_EtUkC%26pg%3DPR7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine1982" class="citation book cs1"><a href="/wiki/W.V._Quine" class="mw-redirect" title="W.V. Quine">Quine, W.V.</a> (1982). <i>Methods of Logic</i> (4th ed.). Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-57175-4" title="Special:BookSources/978-0-674-57175-4"><bdi>978-0-674-57175-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Logic&rft.edition=4th&rft.pub=Harvard+University+Press&rft.date=1982&rft.isbn=978-0-674-57175-4&rft.aulast=Quine&rft.aufirst=W.V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Truth_table&action=edit&section=33" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Truth_tables" class="extiw" title="commons:Category:Truth tables">Truth tables</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Truth_table">"Truth table"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Truth+table&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTruth_table&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html">Truth Tables, Tautologies, and Logical Equivalence</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnellis2011" class="citation arxiv cs1">Anellis, Irving H. (2011). "Peirce's Truth-functional Analysis and the Origin of Truth Tables". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1108.2429">1108.2429</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Peirce%27s+Truth-functional+Analysis+and+the+Origin+of+Truth+Tables&rft.date=2011&rft_id=info%3Aarxiv%2F1108.2429&rft.aulast=Anellis&rft.aufirst=Irving+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATruth+table" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.allaboutcircuits.com/vol_4/chpt_7/9.html">Converting truth tables into Boolean expressions</a></li></ul> <div class="navbox-styles"><link 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href="/wiki/Template_talk:Classical_logic" title="Template talk:Classical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_logic" title="Special:EditPage/Template:Classical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Connective</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a class="mw-selflink selflink">Truth tables</a></li> <li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Well-formed formula</a></li> <li><a href="/wiki/Idempotency_of_entailment" title="Idempotency of entailment">Idempotency of entailment</a></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">Problem of multiple generality</a></li> <li><a href="/wiki/Associative_property" title="Associative property">Associativity</a></li> <li><a href="/wiki/Distributive_property" title="Distributive property">Distribution</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logic.svg" class="mw-file-description"><img alt="Law of noncontradiction" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/75px-Logic.svg.png" decoding="async" width="75" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/113px-Logic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/150px-Logic.svg.png 2x" data-file-width="85" data-file-height="28" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical logics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Term_logic" title="Term logic">Term</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Principles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commutativity_of_conjunction" title="Commutativity of conjunction">Commutativity of conjunction</a></li> <li><a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">Excluded middle</a></li> <li><a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">Bivalence</a></li> <li><a href="/wiki/Law_of_noncontradiction" title="Law of noncontradiction">Noncontradiction</a></li> <li><a href="/wiki/Monotonicity_of_entailment" title="Monotonicity of entailment">Monotonicity of entailment</a></li> <li><a href="/wiki/Principle_of_explosion" title="Principle of explosion">Explosion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Rules</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)">Material implication</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></li> <li><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></li> <li><a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens">modus ponendo tollens</a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a></li> <li><a href="/wiki/Destructive_dilemma" title="Destructive dilemma">Destructive dilemma</a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism">Disjunctive syllogism</a></li> <li><a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">Hypothetical syllogism</a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)">Absorption</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Introduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation</a></li> <li><a href="/wiki/Double_negation_introduction" class="mw-redirect" title="Double negation introduction">Double negation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential</a></li> <li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal</a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction">Biconditional</a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction">Conjunction</a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction">Disjunction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Elimination</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Double_negation_elimination" class="mw-redirect" title="Double negation elimination">Double negation</a></li> <li><a href="/wiki/Existential_instantiation" title="Existential instantiation">Existential</a></li> <li><a href="/wiki/Universal_instantiation" title="Universal instantiation">Universal</a></li> <li><a href="/wiki/Biconditional_elimination" title="Biconditional elimination">Biconditional</a></li> <li><a href="/wiki/Conjunction_elimination" title="Conjunction elimination">Conjunction</a></li> <li><a href="/wiki/Disjunction_elimination" title="Disjunction elimination">Disjunction</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a></li> <li><a href="/wiki/George_Boole" title="George Boole">George Boole</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a></li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Hugh_MacColl" title="Hugh MacColl">Hugh MacColl</a></li> <li><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a></li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a></li> <li><a href="/wiki/Henry_M._Sheffer" title="Henry M. Sheffer">Henry M. Sheffer</a></li> <li><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a></li> <li><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a></li> <li><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li> <li><a href="/wiki/Function_and_Concept" title="Function and Concept">Function and Concept</a></li> <li><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></li> <li><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li> <li><a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a class="mw-selflink selflink">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> 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title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable 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style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of 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table","url":"https:\/\/en.wikipedia.org\/wiki\/Truth_table","sameAs":"http:\/\/www.wikidata.org\/entity\/Q219079","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q219079","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-07-26T20:53:23Z","dateModified":"2024-11-24T17:41:43Z","headline":"Mathematical table used in logic"}</script> </body> </html>