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id="toc-Eileamaid_ionnanachd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Co-iomlaideachd" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Co-iomlaideachd"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Co-iomlaideachd</span> </div> </a> <ul id="toc-Co-iomlaideachd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Roinnidheachd" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Roinnidheachd"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Roinnidheachd</span> </div> </a> <ul id="toc-Roinnidheachd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iom-thionndadh" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Iom-thionndadh"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.4</span> <span>Iom-thionndadh</span> </div> </a> <ul id="toc-Iom-thionndadh-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Co-thiomsachd" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Co-thiomsachd"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.5</span> <span>Co-thiomsachd</span> </div> </a> <ul id="toc-Co-thiomsachd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sgaoilidheachd" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sgaoilidheachd"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.6</span> <span>Sgaoilidheachd</span> </div> </a> <ul id="toc-Sgaoilidheachd-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Structairean_ailseabrach" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Structairean_ailseabrach"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Structairean ailseabrach</span> </div> </a> <button aria-controls="toc-Structairean_ailseabrach-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 Structairean ailseabrach subsection</span> </button> <ul id="toc-Structairean_ailseabrach-sublist" class="vector-toc-list"> <li id="toc-Magma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magma"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Magma</span> </div> </a> <ul id="toc-Magma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Leth-ghrùpa_agus_cuasai-ghrùpa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Leth-ghrùpa_agus_cuasai-ghrùpa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Leth-ghrùpa agus cuasai-ghrùpa</span> </div> </a> <ul id="toc-Leth-ghrùpa_agus_cuasai-ghrùpa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mònad_agus_lùb" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mònad_agus_lùb"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Mònad agus lùb</span> </div> </a> <ul id="toc-Mònad_agus_lùb-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grùpa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grùpa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Grùpa</span> </div> </a> <ul id="toc-Grùpa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fàinne" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fàinne"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Fàinne</span> </div> </a> <ul id="toc-Fàinne-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Raon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Raon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Raon</span> </div> </a> <ul id="toc-Raon-sublist" class="vector-toc-list"> </ul> </li> </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">Obrachadh càraideach</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 64 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-64" 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">64 cànan</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A%D8%A9" title="عملية ثنائية – Arabais" lang="ar" hreflang="ar" data-title="عملية ثنائية" data-language-autonym="العربية" data-language-local-name="Arabais" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Operaci%C3%B3n_binaria" title="Operación binaria – Astùrais" lang="ast" hreflang="ast" data-title="Operación binaria" data-language-autonym="Asturianu" data-language-local-name="Astùrais" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%8B%D1%8F" title="Бінарная аперацыя – Bealaruisis" lang="be" hreflang="be" data-title="Бінарная аперацыя" data-language-autonym="Беларуская" data-language-local-name="Bealaruisis" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарна операция – Bulgarais" lang="bg" hreflang="bg" data-title="Бинарна операция" data-language-autonym="Български" data-language-local-name="Bulgarais" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B6%E0%A6%A8_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="অপারেশন (গণিত) – Bangla" lang="bn" hreflang="bn" data-title="অপারেশন (গণিত)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – Bosnais" lang="bs" hreflang="bs" data-title="Binarna operacija" data-language-autonym="Bosanski" data-language-local-name="Bosnais" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Operaci%C3%B3_bin%C3%A0ria" title="Operació binària – Catalanais" lang="ca" hreflang="ca" data-title="Operació binària" data-language-autonym="Català" data-language-local-name="Catalanais" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%B1%D8%AF%D8%A7%D8%B1_(%D8%A8%DB%8C%D8%B1%DA%A9%D8%A7%D8%B1%DB%8C)" title="کردار (بیرکاری) – Cùrdais Mheadhanach" lang="ckb" hreflang="ckb" data-title="کردار (بیرکاری)" data-language-autonym="کوردی" data-language-local-name="Cùrdais Mheadhanach" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bin%C3%A1rn%C3%AD_operace" title="Binární operace – Seicis" lang="cs" hreflang="cs" data-title="Binární operace" data-language-autonym="Čeština" data-language-local-name="Seicis" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BB%C4%83_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8" title="Бинарлă операци – Chuvash" lang="cv" hreflang="cv" data-title="Бинарлă операци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bin%C3%A6r_operator" title="Binær operator – Danmhairgis" lang="da" hreflang="da" data-title="Binær operator" data-language-autonym="Dansk" data-language-local-name="Danmhairgis" 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/Zweistellige_Verkn%C3%BCpfung" title="Zweistellige Verknüpfung – Gearmailtis" lang="de" hreflang="de" data-title="Zweistellige Verknüpfung" data-language-autonym="Deutsch" data-language-local-name="Gearmailtis" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%B1%CE%B4%CE%B9%CE%BA%CE%AE_%CF%80%CF%81%CE%AC%CE%BE%CE%B7" title="Δυαδική πράξη – Greugais" lang="el" hreflang="el" data-title="Δυαδική πράξη" data-language-autonym="Ελληνικά" data-language-local-name="Greugais" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Binary_operation" title="Binary operation – Beurla" lang="en" hreflang="en" data-title="Binary operation" data-language-autonym="English" data-language-local-name="Beurla" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Duvalenta_operacio" title="Duvalenta operacio – Esperanto" lang="eo" hreflang="eo" data-title="Duvalenta operacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Operaci%C3%B3n_binaria" title="Operación binaria – Spàinntis" lang="es" hreflang="es" data-title="Operación binaria" data-language-autonym="Español" data-language-local-name="Spàinntis" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Binaarne_tehe" title="Binaarne tehe – Eastoinis" lang="et" hreflang="et" data-title="Binaarne tehe" data-language-autonym="Eesti" data-language-local-name="Eastoinis" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eragiketa_bitar" title="Eragiketa bitar – Basgais" lang="eu" hreflang="eu" data-title="Eragiketa bitar" data-language-autonym="Euskara" data-language-local-name="Basgais" 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%B9%D9%85%D9%84_%D8%AF%D9%88%D8%AA%D8%A7%DB%8C%DB%8C" title="عمل دوتایی – Peirsis" lang="fa" hreflang="fa" data-title="عمل دوتایی" data-language-autonym="فارسی" data-language-local-name="Peirsis" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bin%C3%A4%C3%A4rioperaatio" title="Binäärioperaatio – Fionnlannais" lang="fi" hreflang="fi" data-title="Binäärioperaatio" data-language-autonym="Suomi" data-language-local-name="Fionnlannais" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Op%C3%A9ration_binaire" title="Opération binaire – Fraingis" lang="fr" hreflang="fr" data-title="Opération binaire" data-language-autonym="Français" data-language-local-name="Fraingis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Operaci%C3%B3n_binaria" title="Operación binaria – Gailìsis" lang="gl" hreflang="gl" data-title="Operación binaria" data-language-autonym="Galego" data-language-local-name="Gailìsis" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%AA" title="פעולה בינארית – Eabhra" lang="he" hreflang="he" data-title="פעולה בינארית" data-language-autonym="עברית" data-language-local-name="Eabhra" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – Cròthaisis" lang="hr" hreflang="hr" data-title="Binarna operacija" data-language-autonym="Hrvatski" data-language-local-name="Cròthaisis" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Operation_binari" title="Operation binari – Interlingua" lang="ia" hreflang="ia" data-title="Operation binari" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Operasi_biner" title="Operasi biner – Innd-Innsis" lang="id" hreflang="id" data-title="Operasi biner" data-language-autonym="Bahasa Indonesia" data-language-local-name="Innd-Innsis" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/A%C3%B0ger%C3%B0_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Aðgerð (stærðfræði) – Innis Tìlis" lang="is" hreflang="is" data-title="Aðgerð (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Innis Tìlis" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operazione_binaria" title="Operazione binaria – Eadailtis" lang="it" hreflang="it" data-title="Operazione binaria" data-language-autonym="Italiano" data-language-local-name="Eadailtis" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E6%BC%94%E7%AE%97" title="二項演算 – Seapanais" lang="ja" hreflang="ja" data-title="二項演算" data-language-autonym="日本語" data-language-local-name="Seapanais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%ED%95%AD_%EC%97%B0%EC%82%B0" title="이항 연산 – Coirèanais" lang="ko" hreflang="ko" data-title="이항 연산" data-language-autonym="한국어" data-language-local-name="Coirèanais" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kiryar_(matemat%C3%AEk)" title="Kiryar (matematîk) – Cùrdais" lang="ku" hreflang="ku" data-title="Kiryar (matematîk)" data-language-autonym="Kurdî" data-language-local-name="Cùrdais" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bin%C4%81ra_oper%C4%81cija" title="Bināra operācija – Laitbheis" lang="lv" hreflang="lv" data-title="Bināra operācija" data-language-autonym="Latviešu" data-language-local-name="Laitbheis" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Operasi_biner" title="Operasi biner – Minangkabau" lang="min" hreflang="min" data-title="Operasi biner" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B5%8D%E0%B4%B5%E0%B4%AF%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%AF" title="ദ്വയാങ്കസംക്രിയ – Malayalam" lang="ml" hreflang="ml" data-title="ദ്വയാങ്കസംക്രിയ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Operasi_dedua" title="Operasi dedua – Malaidhis" lang="ms" hreflang="ms" data-title="Operasi dedua" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaidhis" 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/Binaire_operatie" title="Binaire operatie – Duitsis" lang="nl" hreflang="nl" data-title="Binaire operatie" data-language-autonym="Nederlands" data-language-local-name="Duitsis" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bin%C3%A6r_operasjon" title="Binær operasjon – Nynorsk na Nirribhidh" lang="nn" hreflang="nn" data-title="Binær operasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk na Nirribhidh" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bin%C3%A6r_operasjon" title="Binær operasjon – Bokmål na Nirribhidh" lang="nb" hreflang="nb" data-title="Binær operasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål na Nirribhidh" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/L%C3%A8i_de_composicion_int%C3%A8rna" title="Lèi de composicion intèrna – Ogsatanais" lang="oc" hreflang="oc" data-title="Lèi de composicion intèrna" data-language-autonym="Occitan" data-language-local-name="Ogsatanais" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dzia%C5%82anie_dwuargumentowe" title="Działanie dwuargumentowe – Pòlainnis" lang="pl" hreflang="pl" data-title="Działanie dwuargumentowe" data-language-autonym="Polski" data-language-local-name="Pòlainnis" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Operassion" title="Operassion – Piedmontese" lang="pms" hreflang="pms" data-title="Operassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Opera%C3%A7%C3%A3o_bin%C3%A1ria" title="Operação binária – Portagailis" lang="pt" hreflang="pt" data-title="Operação binária" data-language-autonym="Português" data-language-local-name="Portagailis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Opera%C8%9Bie_binar%C4%83" title="Operație binară – Romàinis" lang="ro" hreflang="ro" data-title="Operație binară" data-language-autonym="Română" data-language-local-name="Romàinis" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарная операция – Ruisis" lang="ru" hreflang="ru" data-title="Бинарная операция" data-language-autonym="Русский" data-language-local-name="Ruisis" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Opirazzioni_binaria" title="Opirazzioni binaria – Sisilis" lang="scn" hreflang="scn" data-title="Opirazzioni binaria" data-language-autonym="Sicilianu" data-language-local-name="Sisilis" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – Sèirb-Chròthaisis" lang="sh" hreflang="sh" data-title="Binarna operacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Sèirb-Chròthaisis" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binary_operation" title="Binary operation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binary operation" 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/Bin%C3%A1rna_oper%C3%A1cia" title="Binárna operácia – Slòbhacais" lang="sk" hreflang="sk" data-title="Binárna operácia" data-language-autonym="Slovenčina" data-language-local-name="Slòbhacais" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dvo%C4%8Dlena_operacija" title="Dvočlena operacija – Slòbhainis" lang="sl" hreflang="sl" data-title="Dvočlena operacija" data-language-autonym="Slovenščina" data-language-local-name="Slòbhainis" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Veprimi_binar" title="Veprimi binar – Albàinis" lang="sq" hreflang="sq" data-title="Veprimi binar" data-language-autonym="Shqip" data-language-local-name="Albàinis" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Бинарна операција – Sèirbis" lang="sr" hreflang="sr" data-title="Бинарна операција" data-language-autonym="Српски / srpski" data-language-local-name="Sèirbis" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bin%C3%A4r_operator" title="Binär operator – Suainis" lang="sv" hreflang="sv" data-title="Binär operator" data-language-autonym="Svenska" data-language-local-name="Suainis" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AF%86%E0%AE%AF%E0%AE%B2%E0%AE%BF" title="ஈருறுப்புச் செயலி – Taimilis" lang="ta" hreflang="ta" data-title="ஈருறுப்புச் செயலி" data-language-autonym="தமிழ்" data-language-local-name="Taimilis" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%94%E0%B8%B3%E0%B9%80%E0%B8%99%E0%B8%B4%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%97%E0%B8%A7%E0%B8%B4%E0%B8%A0%E0%B8%B2%E0%B8%84" title="การดำเนินการทวิภาค – Cànan nan Tàidh" lang="th" hreflang="th" data-title="การดำเนินการทวิภาค" data-language-autonym="ไทย" data-language-local-name="Cànan nan Tàidh" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Operasyong_tambalan" title="Operasyong tambalan – Tagalog" lang="tl" hreflang="tl" data-title="Operasyong tambalan" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0kili_i%C5%9Flem" title="İkili işlem – Turcais" lang="tr" hreflang="tr" data-title="İkili işlem" data-language-autonym="Türkçe" data-language-local-name="Turcais" 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%91%D1%96%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%96%D1%8F" title="Бінарна операція – Ucràinis" lang="uk" hreflang="uk" data-title="Бінарна операція" data-language-autonym="Українська" data-language-local-name="Ucràinis" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_to%C3%A1n_hai_ng%C3%B4i" title="Phép toán hai ngôi – Bhiet-Namais" lang="vi" hreflang="vi" data-title="Phép toán hai ngôi" data-language-autonym="Tiếng Việt" data-language-local-name="Bhiet-Namais" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E8%BF%90%E7%AE%97" title="二元运算 – Wu" lang="wuu" hreflang="wuu" data-title="二元运算" 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-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарн операция – Kalmyk" lang="xal" hreflang="xal" data-title="Бинарн операция" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%A9%D7%A2_%D7%90%D7%A4%D7%A2%D7%A8%D7%90%D7%A6%D7%99%D7%A2" title="בינארישע אפעראציע – Iùdhais" lang="yi" hreflang="yi" data-title="בינארישע אפעראציע" data-language-autonym="ייִדיש" data-language-local-name="Iùdhais" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E8%BF%90%E7%AE%97" title="二元运算 – Sìnis" lang="zh" hreflang="zh" data-title="二元运算" data-language-autonym="中文" data-language-local-name="Sìnis" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算 – Sìnis an Litreachais" lang="lzh" hreflang="lzh" data-title="二元運算" data-language-autonym="文言" data-language-local-name="Sìnis an Litreachais" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算 – Cantonais" lang="yue" hreflang="yue" data-title="二元運算" data-language-autonym="粵語" data-language-local-name="Cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q164307#sitelinks-wikipedia" title="Deasaich na ceanglaichean eadar-cànain" class="wbc-editpage">Deasaich na ceanglaichean</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ainm-spàsan"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> 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air eileamaidean dheth a tha <a href="/wiki/Ailseabra" title="Ailseabra">ailseabra</a> a' dèiligeadh. Tha i a' rannsachadh nam feartan coitcheann a thèid an tarraing bho na h-obrachaidhean seo gun sùil a thoirt air nàdar mionaideach an t-seata no nan obrachaidhean fhèin. Am measg nan obrachaidhean seo, tha <b>obrachaidhean càraideach</b> air leth cudromach. </p><p>'S e obrachadh eadar paidhir (no càraid) eileamaidean den t-seata a mhapas ri eileamaid den aon seata a th’ ann an <b>obrachadh càraideach</b>, ach feumaidh seo a bhith fìor do gach eileamaid den t-seata. Ann an comharran <a href="/wiki/Matamataig" title="Matamataig">matamataig</a>, tha an t-obrachadh * càraideach air an t-seata <b>A</b>, ma tha: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∀ <i>y</i> ∈ <b>A</b>, <i>x</i> * <i>y</i> ∈ <b>A</b></dd></dl></dd></dl> <p>Mar eisimpleir, 's e <b>obrachadh càraideach</b> a th’ ann an <a href="/wiki/Cur-ris" title="Cur-ris">cur-ris</a> air seata nan <a href="/w/index.php?title=%C3%80ireamh_n%C3%A0darra&action=edit&redlink=1" class="new" title="Àireamh nàdarra (chan eil duilleag ann fhathast)">àireamhan nàdarra</a> ℕ, oir 's e àireamh nàdarra a th' ann an suim dà àireimh nàdarra, ge b' e air bith na h-<a href="/wiki/%C3%80ireamh" title="Àireamh">àireamhan</a> a th' air an ròghnachadh. Ach chan e <b>obrachadh càraideach</b> a th' ann an <a href="/wiki/Toirt_air_falbh" title="Toirt air falbh">toirt air falbh</a>, oir tha e furasta gu leòr paidhir àireamhan a ròghnachadh nach map ri àireimh nàdarra fon obrachadh. </p> <dl><dd><dl><dd>m.e. 5 – 7 = –2 ∉ ℕ.</dd></dl></dd></dl> <p>Mur eil buil an obrachaidh na ball den aon seata, canaidh sinn gu bheil an t-obrachadh <i>fosgailte</i>. Ma tha a' bhuil na ball den t-seata, 's e obrachadh <i>dùinte</i> a th' ann. 'S ann <b>dùinte</b> a tha obrachadh càraideach. Tha toirt air falbh na h-obrachadh càraideach air seata ℤ nan <a href="/w/index.php?title=Sl%C3%A0n-%C3%A0ireamh&action=edit&redlink=1" class="new" title="Slàn-àireamh (chan eil duilleag ann fhathast)">slàn-àireamhan</a>, agus tha cur-ris, toirt air falbh agus <a href="/wiki/Iomadachadh" title="Iomadachadh">iomadachadh</a> nan obrachaidhean càraideach air na <a href="/w/index.php?title=F%C3%ACor-%C3%A0ireamh&action=edit&redlink=1" class="new" title="Fìor-àireamh (chan eil duilleag ann fhathast)">fìor-àireamhan</a> ℝ. Chan eil <a href="/w/index.php?title=Roinneadh&action=edit&redlink=1" class="new" title="Roinneadh (chan eil duilleag ann fhathast)">roinneadh</a> na obrachadh càraideach air ℝ oir chan fhaodar a roinn le neoni. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tuairisgeul_foirmeil">Tuairisgeul foirmeil</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=1" title="Deasaich earrann: Tuairisgeul foirmeil" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=1" title="Edit section's source code: Tuairisgeul foirmeil"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tha obrachadh càraideach air seata <b>A</b> na <a href="/w/index.php?title=Fuincsean&action=edit&redlink=1" class="new" title="Fuincsean (chan eil duilleag ann fhathast)">fuincsean</a> a <a href="/w/index.php?title=Mapadh&action=edit&redlink=1" class="new" title="Mapadh (chan eil duilleag ann fhathast)">mhapas</a> an <a href="/wiki/Toradh_Cartesach" title="Toradh Cartesach">toradh Cartesach</a> <b>A</b> × <b>A</b> ri <b>A</b>. </p> <dl><dd><dl><dd><i>f</i> : <b>A</b> × <b>A</b> → <b>A</b></dd></dl></dd></dl> <dl><dd><dl><dd>.i. ∀ (<i>x</i>, <i>y</i>) ∈ <b>A</b><sup>2</sup>, ∃ <i>z</i> ∈ <b>A</b> : <i>f</i>(<i>x</i>, <i>y</i>) = <i>z</i></dd></dl></dd></dl> <p>Mar as cumanta, tha comharra àraidh aig obrachadh càraideach (m.e. +, ×, ∩, 7c.) , mar eisimpleir cur-ris nan <a href="/w/index.php?title=Sl%C3%A0n-%C3%A0ireamh&action=edit&redlink=1" class="new" title="Slàn-àireamh (chan eil duilleag ann fhathast)">slàn-àireamhan</a>: </p> <dl><dd><dl><dd>+ : ℤ × ℤ → ℤ</dd> <dd>+ (2, 3) = 5</dd> <dd>+ (- 4, 11) = 7</dd></dl></dd></dl> <p>ged tha e nas cumanta an comharra seo a' sgrìobhadh eadar <a href="/wiki/%C3%80ireamh" title="Àireamh">àireamhan</a> na càraide: </p> <dl><dd><dl><dd>2 + 3 = 5</dd> <dd>- 4 + 11 = 7</dd></dl></dd></dl> <p>No, mas e * an comharra airson obrachaidh choitchinn: </p> <dl><dd><dl><dd><i>a</i> * <i>b</i> = <i>c</i></dd></dl></dd></dl> <p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Feartan_obrachaidh_càraidich"><span id="Feartan_obrachaidh_c.C3.A0raidich"></span>Feartan obrachaidh càraidich</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=2" title="Deasaich earrann: Feartan obrachaidh càraidich" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=2" title="Edit section's source code: Feartan obrachaidh càraidich"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Faodaidh feartan àraidh a bhith aig obrachadh càraideach. 'S ann mar as leanas na feartan as cudromach: </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Eileamaid_ionnanachd">Eileamaid ionnanachd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=3" title="Deasaich earrann: Eileamaid ionnanachd" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=3" title="Edit section's source code: Eileamaid ionnanachd"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha an làthair eileamaid <i>c</i> den t-seata: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∃ <i>c</i> ∈ <b>A</b> : <i>c</i> * <i>x</i> = <i>x</i></dd></dl></dd></dl> <p>'s e <b>eileamaid ionnanachd à clì</b> a th' ann an <i>c</i>. Ma tha: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∃ <i>d</i> ∈ <b>A</b> : <i>x</i> * <i>d</i> = <i>d</i></dd></dl></dd></dl> <p>'s e <b>eileamaid ionnanachd à deas</b> a th' ann an <i>d</i>. 'S e <b>eileamaid ionnanachd</b> a th' ann an <i>e</i> ma tha e ionnanachd à clì agus à deas: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∃ <i>e</i> ∈ <b>A</b> : <i>e</i> * <i>x</i> = <i>x</i> * <i>e</i> = <i>x</i>.</dd></dl></dd></dl> <p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Co-iomlaideachd">Co-iomlaideachd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=4" title="Deasaich earrann: Co-iomlaideachd" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=4" title="Edit section's source code: Co-iomlaideachd"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha paidhir eileamaidean ann <i>x</i> agus <i>y</i> far a bheil: </p> <dl><dd><dl><dd><i>x</i> * <i>y</i> = <i>y</i> * <i>x</i></dd></dl></dd></dl> <p>'s e <b>paidhir co-iomlaideach</b> a th' ann <i>x</i> agus <i>y</i>. Ma tha eileamaid <i>a</i> ann far a bheil, airson gach eileamaid <i>x</i> den t-seata: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∃ <i>a</i> ∈ <b>A</b> : <i>a</i> * <i>x</i> = <i>x</i> * <i>a</i></dd></dl></dd></dl> <p>'s e <b>eileamaid co-iomlaideach</b> a th' ann an <i>a</i>. Ma tha e fìor airson a h-uile paidhir eileamaidean den t-seata: </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>) ∈ <b>A</b><sup>2</sup>, <i>x</i> * <i>y</i> = <i>y</i> * <i>x</i></dd></dl></dd></dl> <p>'s e <b>obrachadh co-iomlaideach</b> a th' anns an *. </p><p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Roinnidheachd">Roinnidheachd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=5" title="Deasaich earrann: Roinnidheachd" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=5" title="Edit section's source code: Roinnidheachd"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha e fìor airson a h-uile paidhir eileamaid den t-seata: </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>) ∈ <b>A</b><sup>2</sup>, ∃ <i>c</i> ∈ <b>A</b> : <i>x</i> * <i>c</i> = <i>y</i></dd></dl></dd></dl> <p>'s e <b>so-roinnte à clì</b> a th’ anns an obrachadh *. Ma tha e fìor airson a h-uile paidhir eileamaid den t-seata: </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>) ∈ <b>A</b><sup>2</sup>, ∃ <i>d</i> ∈ <b>A</b> : <i>d</i> * <i>x</i> = <i>y</i></dd></dl></dd></dl> <p>'s e <b>so-roinnte à deas</b> a th' anns an obrachadh *. 'S e <b>obrachadh roinnidheach</b> a th' anns an * ma tha e so-roinnte à clì agus à deas. </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>) ∈ <b>A</b><sup>2</sup>, ∃ <i>e</i> ∈ <b>A</b> : <i>x</i> * <i>e</i> = <i>e</i> * <i>x</i> = <i>y</i>.</dd></dl></dd></dl> <p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Iom-thionndadh">Iom-thionndadh</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=6" title="Deasaich earrann: Iom-thionndadh" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=6" title="Edit section's source code: Iom-thionndadh"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha eileamaid ionnanachd <i>e</i> ann, agus: </p> <dl><dd><dl><dd>∀ <i>x</i> ∈ <b>A</b>, ∃ <i>y</i> ∈ <b>A</b> : <i>x</i> * <i>y</i> = <i>e</i></dd></dl></dd></dl> <p>'s e <b>eileamaidean iom-thionndaidh</b> a th' anns an <i>x</i> agus <i>y</i>. </p><p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Co-thiomsachd">Co-thiomsachd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=7" title="Deasaich earrann: Co-thiomsachd" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=7" title="Edit section's source code: Co-thiomsachd"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha <i>a</i>, <i>w</i>, <i>x</i>, <i>y</i> agus <i>z</i> nam buill den t-seata, agus ma tha <i>w</i> * <i>x</i> = <i>a</i> agus <i>a</i> * <i>y</i> = <i>z</i>. Faodar a sgrìobhadh: </p> <dl><dd><dl><dd>(<i>w</i> * <i>x</i>) * <i>y</i> = <i>z</i></dd></dl></dd></dl> <p>far a bheil na camagan ( ) a' ciallachadh gum feumar an t-obrachadh a-stigh fhuasgladh ron fhear a-muigh. Nise, ma tha e fìor gu bheil: </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <b>A</b><sup>3</sup>, (<i>x</i> * <i>y</i>) * <i>z</i> = <i>x</i> * (<i>y</i> * <i>z</i>)</dd></dl></dd></dl> <p>'s e <b>obrachadh co-thiomsach</b> a th' anns an *. </p><p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Sgaoilidheachd">Sgaoilidheachd</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=8" title="Deasaich earrann: Sgaoilidheachd" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=8" title="Edit section's source code: Sgaoilidheachd"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha dà obrachadh càraideach ann, ⊙ agus ⊕ can, agus ma tha e fìor gu bheil: </p> <dl><dd><dl><dd>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <b>A</b><sup>3</sup>, <i>x</i> ⊙ ( <i>y</i> ⊕ <i>z</i>) = (<i>x</i> ⊙ <i>y</i>) ⊕ (<i>x</i> ⊙ <i>z</i>)</dd></dl></dd></dl> <p>Canaidh sinn gu bheil an t-obrachadh ⊙ <b>sgaoileach</b> air an obrachadh ⊕. </p><p><br /> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Structairean_ailseabrach">Structairean ailseabrach</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=9" title="Deasaich earrann: Structairean ailseabrach" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=9" title="Edit section's source code: Structairean ailseabrach"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Magma">Magma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=10" title="Deasaich earrann: Magma" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=10" title="Edit section's source code: Magma"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>'S e <b>magma</b> an t-ainm air seata le obrachadh càraideach an lùib. Tha e cumanta seo a sgrìobhadh (<b>A</b>,*) far a bheil <b>A</b> an seata agus * na obrachadh càraideach. 'S e an structair as bunaitich ann an <a href="/wiki/Ailseabra" title="Ailseabra">ailseabra</a> a th' ann am <b>magma</b> far nach eil riaghailt no feart air a chur às leth an obrachaidh chàraidich aige. Ri linn seo, chan eil mòran ùidh air magma mar thà. </p> <div class="mw-heading mw-heading3"><h3 id="Leth-ghrùpa_agus_cuasai-ghrùpa"><span id="Leth-ghr.C3.B9pa_agus_cuasai-ghr.C3.B9pa"></span>Leth-ghrùpa agus cuasai-ghrùpa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=11" title="Deasaich earrann: Leth-ghrùpa agus cuasai-ghrùpa" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=11" title="Edit section's source code: Leth-ghrùpa agus cuasai-ghrùpa"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha obrachadh càraideach a' mhagma <i><a href="/w/index.php?title=Co-thiomsachd&action=edit&redlink=1" class="new" title="Co-thiomsachd (chan eil duilleag ann fhathast)">co-thiomsach</a></i>, 's e <b>leth-ghrùpa</b> a th' ann. Mur eil e <i>co-thiomsach</i> ach <i>so-roinnte</i>, 's e <b>quasi-ghrùpa</b> a th' ann. </p> <div class="mw-heading mw-heading3"><h3 id="Mònad_agus_lùb"><span id="M.C3.B2nad_agus_l.C3.B9b"></span>Mònad agus lùb</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=12" title="Deasaich earrann: Mònad agus lùb" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=12" title="Edit section's source code: Mònad agus lùb"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ma tha <i>eileamaid ionnanachd</i> aig leth-ghrùpa (.i. magma co-thiomsach le eileamaid ionnanachd), canaidh sinn <b>mònad</b> ris; ma tha <i>eileamaid ionnanachd</i> aig quasi-ghrùpa, canaidh sinn <b>lùb</b> ris. </p> <div class="mw-heading mw-heading3"><h3 id="Grùpa"><span id="Gr.C3.B9pa"></span>Grùpa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=13" title="Deasaich earrann: Grùpa" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=13" title="Edit section's source code: Grùpa"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Far a bheil <i>eileamaid iom-thionndaidh</i> aig gach eileamaid <i>mònaid</i>, 's e <b>grùpa</b> a th' ann. Agus 's e <b>grùpa</b> a th' ann cuideachd ma tha <i>lùb</i> <a href="/w/index.php?title=Co-thiomsachd&action=edit&redlink=1" class="new" title="Co-thiomsachd (chan eil duilleag ann fhathast)">co-thiomsach</a>. </p><p>Ma tha obrachadh càraideach a' ghrùpa <i><a href="/wiki/Co-iomlaideachd" title="Co-iomlaideachd">co-iomlaideach</a></i>, canaidh sinn gur e <b>grùpa aibèalach</b> a th' ann. </p> <div class="mw-heading mw-heading3"><h3 id="Fàinne"><span id="F.C3.A0inne"></span>Fàinne</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=14" title="Deasaich earrann: Fàinne" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=14" title="Edit section's source code: Fàinne"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>'S e <b>fàinne</b> a th' ann an structair le dà obrachadh càraideach: a' chiad fhear na <i>ghrùpa aibèalach</i> 's an dara fear na <i>mhonaid</i> agus e <i><a href="/w/index.php?title=Sgaoilidheachd&action=edit&redlink=1" class="new" title="Sgaoilidheachd (chan eil duilleag ann fhathast)">sgaoileach</a></i> air a' chiad fhear. Thathar a' smaoineachadh air dà obrachadh chàraideach mar gur e <a href="/wiki/Cur-ris" title="Cur-ris">cur-ris</a> agus <a href="/wiki/Iomadachadh" title="Iomadachadh">iomadachadh</a> a bhiodh ann. Tha e cumanta "cur-ris" a ràdh ri obrachadh a' ghrùpa aibèalaich agus "iomadachadh" ri obrachadh na monaid, ged sa chuis choitchinn nach biodh riaghailtean cur-ris no iomadachaidh nan àireamhan iomchaidh. Mas e fàinne (<b>F</b>, ⊕, ⊙): </p> <dl><dd><dl><dd><table> <tbody><tr> <td>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <b>F</b><sup>3</sup>, (<i>x</i> ⊕ <i>y</i>) ⊕ <i>z</i> = <i>x</i> ⊕ (<i>y</i> ⊕ <i>z</i>)</td> <td></td> <td>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <b>F</b><sup>3</sup>, (<i>x</i> ⊙ <i>y</i>) ⊙ <i>z</i> = <i>x</i> ⊙ (<i>y</i> ⊙ <i>z</i>) </td></tr> <tr> <td>∀ <i>x</i> ∈ <b>F</b>, ∃ <i>e</i> ∈ <b>F</b> : <i>e</i> ⊕ <i>x</i> = <i>x</i> ⊕ <i>e</i> = <i>x</i></td> <td></td> <td>∀ <i>x</i> ∈ <b>F</b>, ∃ <i>f</i> ∈ <b>F</b> : <i>f</i> ⊙ <i>x</i> = <i>x</i> ⊙ <i>f</i> = <i>x</i> </td></tr> <tr> <td>∀ <i>x</i> ∈ <b>F</b>, ∃ <i>y</i> ∈ <b>F</b> : <i>x</i> ⊕ <i>y</i> = <i>e</i></td> <td></td> <td>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <b>F</b><sup>3</sup>, <i>x</i> ⊙ ( <i>y</i> ⊕ <i>z</i>) = (<i>x</i> ⊙ <i>y</i>) ⊕ (<i>x</i> ⊙ <i>z</i>) </td></tr> <tr> <td>∀ (<i>x</i>, <i>y</i>) ∈ <b>F</b><sup>2</sup>, <i>x</i> ⊕ <i>y</i> = <i>y</i> ⊕ <i>x</i> </td></tr></tbody></table></dd></dl></dd></dl> <p>Ma tha an t-obrachadh ⊙ <i><a href="/wiki/Co-iomlaideachd" title="Co-iomlaideachd">co-iomlaideach</a></i> cuideachd (.i. ∀ (<i>x</i>, <i>y</i>) ∈ <b>F</b><sup>2</sup>, <i>x</i> ⊙ <i>y</i> = <i>y</i> ⊙ <i>x</i>), 's e <b>fàinne cho-iomlaideach</b> a chanar rithe. </p><p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Raon">Raon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&veaction=edit&section=15" title="Deasaich earrann: Raon" class="mw-editsection-visualeditor"><span>deasaich</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Obrachadh_c%C3%A0raideach&action=edit&section=15" title="Edit section's source code: Raon"><span>deasaich an tùs</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mas i fàinne cho-iomlaideach a th’ innte ach 's e grùpa a th' ann an dara obrachadh càraideach cuideachd, 's e <b>raon</b> a th' againn rithe. Mas e raon (<i><b>R</b></i>, ⊕, ⊙): </p> <dl><dd><dl><dd><table> <tbody><tr> <td>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <i><b>R</b></i><sup>3</sup>, (<i>x</i> ⊕ <i>y</i>) ⊕ <i>z</i> = <i>x</i> ⊕ (<i>y</i> ⊕ <i>z</i>)</td> <td></td> <td>∀ (<i>x</i>, <i>y</i>, <i>z</i>) ∈ <i><b>R</b></i><sup>3</sup>, (<i>x</i> ⊙ <i>y</i>) ⊙ <i>z</i> = <i>x</i> ⊙ (<i>y</i> ⊙ <i>z</i>) </td></tr> <tr> <td>∀ <i>x</i> ∈ <i><b>R</b></i>, ∃ <i>e</i> ∈ <i><b>R</b></i> : <i>e</i> ⊕ <i>x</i> = <i>x</i> ⊕ <i>e</i> = <i>x</i></td> <td></td> <td>∀ <i>x</i> ∈ <i><b>R</b></i>, ∃ <i>f</i> ∈ <i><b>R</b></i> : <i>f</i> ⊙ <i>x</i> = <i>x</i> ⊙ <i>f</i> = <i>x</i> </td></tr> <tr> <td>∀ <i>x</i> ∈ <i><b>R</b></i>, ∃ <i>y</i> ∈ <i><b>R</b></i> : <i>x</i> ⊕ <i>y</i> = <i>e</i></td> <td></td> <td>∀ <i>x</i> ∈ <i><b>R</b></i>, ∃ <i>y</i> ∈ <i><b>R</b></i> : <i>x</i> ⊙ <i>y</i> = <i>f</i> </td></tr> <tr> <td>∀ (<i>x</i>, <i>y</i>) ∈ <i><b>R</b></i><sup>2</sup>, <i>x</i> ⊕ <i>y</i> = <i>y</i> ⊕ <i>x</i></td> <td></td> <td>∀ 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