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id="toc-Relation-preserving_isomorphism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Category_theoretic_view" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Category_theoretic_view"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Category theoretic view</span> </div> </a> <button aria-controls="toc-Category_theoretic_view-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 Category theoretic view subsection</span> </button> <ul id="toc-Category_theoretic_view-sublist" class="vector-toc-list"> <li id="toc-Isomorphism_vs._bijective_morphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isomorphism_vs._bijective_morphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Isomorphism vs. bijective morphism</span> </div> </a> <ul id="toc-Isomorphism_vs._bijective_morphism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Isomorphism_class" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isomorphism_class"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Isomorphism class</span> </div> </a> <button aria-controls="toc-Isomorphism_class-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 Isomorphism class subsection</span> </button> <ul id="toc-Isomorphism_class-sublist" class="vector-toc-list"> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_equality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relation to equality</span> </div> </a> <ul id="toc-Relation_to_equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <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">Isomorphism</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 59 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-59" 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">59 languages</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%AA%D9%85%D8%A7%D9%83%D9%84" title="تماكل – Arabic" lang="ar" hreflang="ar" data-title="تماكل" data-language-autonym="العربية" data-language-local-name="Arabic" 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/Isomorfismu" title="Isomorfismu – Asturian" lang="ast" hreflang="ast" data-title="Isomorfismu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0zomorfluq" title="İzomorfluq – Azerbaijani" lang="az" hreflang="az" data-title="İzomorfluq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – Bashkir" lang="ba" hreflang="ba" data-title="Изоморфизм" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%B7%D0%B0%D0%BC%D0%B0%D1%80%D1%84%D1%96%D0%B7%D0%BC" title="Ізамарфізм – Belarusian" lang="be" hreflang="be" data-title="Ізамарфізм" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D1%8A%D0%BC" title="Изоморфизъм – Bulgarian" lang="bg" hreflang="bg" data-title="Изоморфизъм" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – Bosnian" lang="bs" hreflang="bs" data-title="Izomorfizam" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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/Isomorfisme" title="Isomorfisme – Catalan" lang="ca" hreflang="ca" data-title="Isomorfisme" 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/Izomorfismus" title="Izomorfismus – Czech" lang="cs" hreflang="cs" data-title="Izomorfismus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Isomorffedd" title="Isomorffedd – Welsh" lang="cy" hreflang="cy" data-title="Isomorffedd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Isomorfi" title="Isomorfi – Danish" lang="da" hreflang="da" data-title="Isomorfi" 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/Isomorphismus" title="Isomorphismus – German" lang="de" hreflang="de" data-title="Isomorphismus" 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/Isomorfism" title="Isomorfism – Estonian" lang="et" hreflang="et" data-title="Isomorfism" 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%99%CF%83%CE%BF%CE%BC%CE%BF%CF%81%CF%86%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Ισομορφισμός – Greek" lang="el" hreflang="el" data-title="Ισομορφισμός" 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/Isomorfismo" title="Isomorfismo – Spanish" lang="es" hreflang="es" data-title="Isomorfismo" 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/Izomorfio" title="Izomorfio – Esperanto" lang="eo" hreflang="eo" data-title="Izomorfio" 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/Isomorfismo" title="Isomorfismo – Basque" lang="eu" hreflang="eu" data-title="Isomorfismo" 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/%DB%8C%DA%A9%D8%B1%DB%8C%D8%AE%D8%AA%DB%8C" title="یکریختی – Persian" lang="fa" hreflang="fa" data-title="یکریختی" 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/Isomorphisme" title="Isomorphisme – French" lang="fr" hreflang="fr" data-title="Isomorphisme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Iseamorfacht" title="Iseamorfacht – Irish" lang="ga" hreflang="ga" data-title="Iseamorfacht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – Galician" lang="gl" hreflang="gl" data-title="Isomorfismo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8F%99%ED%98%95_%EC%82%AC%EC%83%81" title="동형 사상 – Korean" lang="ko" hreflang="ko" data-title="동형 사상" 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%A6%D5%B8%D5%B4%D5%B8%D6%80%D6%86%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Իզոմորֆություն (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Իզոմորֆություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – Croatian" lang="hr" hreflang="hr" data-title="Izomorfizam" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – Indonesian" lang="id" hreflang="id" data-title="Isomorfisme" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Isomorphismo" title="Isomorphismo – Interlingua" lang="ia" hreflang="ia" data-title="Isomorphismo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Isomorfismo" title="Isomorfismo – Italian" lang="it" hreflang="it" data-title="Isomorfismo" 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%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם – Hebrew" lang="he" hreflang="he" data-title="איזומורפיזם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC_(%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изоморфизм (Математика) – Kazakh" lang="kk" hreflang="kk" data-title="Изоморфизм (Математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – Kyrgyz" lang="ky" hreflang="ky" data-title="Изоморфизм" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Isomorphismus" title="Isomorphismus – Latin" lang="la" hreflang="la" data-title="Isomorphismus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Izomorfizmas" title="Izomorfizmas – Lithuanian" lang="lt" hreflang="lt" data-title="Izomorfizmas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Izomorfia" title="Izomorfia – Hungarian" lang="hu" hreflang="hu" data-title="Izomorfia" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84" title="Изоморф – Mongolian" lang="mn" hreflang="mn" data-title="Изоморф" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Isomorfisme" title="Isomorfisme – Dutch" lang="nl" hreflang="nl" data-title="Isomorfisme" 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/%E5%90%8C%E5%9E%8B%E5%86%99%E5%83%8F" title="同型写像 – Japanese" lang="ja" hreflang="ja" data-title="同型写像" 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/Isomorfisme" title="Isomorfisme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Isomorfisme" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Isomorfi" title="Isomorfi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Isomorfi" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Izomorfizm_(matematika)" title="Izomorfizm (matematika) – Uzbek" lang="uz" hreflang="uz" data-title="Izomorfizm (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%87%E0%A8%B8%E0%A9%8B%E0%A8%AE%E0%A9%8C%E0%A8%B0%E0%A8%AB%E0%A8%BF%E0%A8%9C%E0%A8%BC%E0%A8%AE" title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਆਇਸੋਮੌਰਫਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Isomorfism" title="Isomorfism – Piedmontese" lang="pms" hreflang="pms" data-title="Isomorfism" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Izomorfizm" title="Izomorfizm – Polish" lang="pl" hreflang="pl" data-title="Izomorfizm" 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/Isomorfismo" title="Isomorfismo – Portuguese" lang="pt" hreflang="pt" data-title="Isomorfismo" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Izomorfism" title="Izomorfism – Romanian" lang="ro" hreflang="ro" data-title="Izomorfism" data-language-autonym="Română" data-language-local-name="Romanian" 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%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%BC" title="Изоморфизм – Russian" lang="ru" hreflang="ru" data-title="Изоморфизм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Isomorphism" title="Isomorphism – Simple English" lang="en-simple" hreflang="en-simple" data-title="Isomorphism" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Izomorfizem" title="Izomorfizem – Slovenian" lang="sl" hreflang="sl" data-title="Izomorfizem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%B8%D0%B7%D0%B0%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изоморфизам (математика) – Serbian" lang="sr" hreflang="sr" data-title="Изоморфизам (математика)" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Izomorfizam" title="Izomorfizam – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Izomorfizam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a 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class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Inversible mapping (mathematics)</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about mathematics. For other uses, see <a href="/wiki/Isomorphism_(disambiguation)" class="mw-disambig" title="Isomorphism (disambiguation)">Isomorphism (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Isomorphism" title="Special:EditPage/Isomorphism">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Isomorphism%22">"Isomorphism"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Isomorphism%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Isomorphism%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Isomorphism%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Isomorphism%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Isomorphism%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">September 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:One5Root.svg" class="mw-file-description"><img alt="Fifth roots of unity" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/One5Root.svg/200px-One5Root.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/One5Root.svg/300px-One5Root.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/One5Root.svg/400px-One5Root.svg.png 2x" data-file-width="480" data-file-height="480" /></a></span></div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Regular_polygon_5_annotated.svg" class="mw-file-description"><img alt="Rotations of a pentagon" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Regular_polygon_5_annotated.svg/200px-Regular_polygon_5_annotated.svg.png" decoding="async" width="200" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Regular_polygon_5_annotated.svg/300px-Regular_polygon_5_annotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Regular_polygon_5_annotated.svg/400px-Regular_polygon_5_annotated.svg.png 2x" data-file-width="503" data-file-height="509" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of fifth <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.</div></div></div></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>isomorphism</b> is a structure-preserving <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> (a <a href="/wiki/Morphism" title="Morphism">morphism</a>) between two <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a> of the same type that can be reversed by an <a href="/wiki/Inverse_function" title="Inverse function">inverse mapping</a>. Two mathematical structures are <b>isomorphic</b> if an isomorphism exists between them. The word is derived from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i><a href="https://en.wiktionary.org/wiki/%E1%BC%B4%CF%83%CE%BF%CF%82" class="extiw" title="wikt:ἴσος">ἴσος</a><i> (isos)</i> 'equal' and <i> </i><a href="https://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AE" class="extiw" title="wikt:μορφή">μορφή</a><i> (morphe)</i> 'form, shape'. </p><p>The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are <em>the same <a href="/wiki/Up_to" title="Up to">up to</a> an isomorphism</em>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2021)">citation needed</span></a></i>]</sup> </p><p>An <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> is an isomorphism from a structure to itself. An isomorphism between two structures is a <b>canonical isomorphism</b> (a <a href="/wiki/Canonical_map" title="Canonical map">canonical map</a> that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a <a href="/wiki/Universal_property" title="Universal property">universal property</a>), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every <a href="/wiki/Prime_number" title="Prime number">prime number</a> <span class="texhtml mvar" style="font-style:italic;">p</span>, all <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> with <span class="texhtml mvar" style="font-style:italic;">p</span> elements are canonically isomorphic, with a unique isomorphism. The <a href="/wiki/Isomorphism_theorems" title="Isomorphism theorems">isomorphism theorems</a> provide canonical isomorphisms that are not unique. </p><p>The term <em>isomorphism</em> is mainly used for <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>. In this case, mappings are called <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>, and a homomorphism is an isomorphism <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a>. </p><p>In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example: </p> <ul><li>An <a href="/wiki/Isometry" title="Isometry">isometry</a> is an isomorphism of <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>.</li> <li>A <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> is an isomorphism of <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>.</li> <li>A <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> is an isomorphism of spaces equipped with a <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>, typically <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>.</li> <li>A <a href="/wiki/Symplectomorphism" title="Symplectomorphism">symplectomorphism</a> is an isomorphism of <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a>.</li> <li>A <a href="/wiki/Permutation" title="Permutation">permutation</a> is an automorphism of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>.</li> <li>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, isomorphisms and automorphisms are often called <a href="/wiki/Transformation_(function)" title="Transformation (function)">transformations</a>, for example <a href="/wiki/Rigid_transformation" title="Rigid transformation">rigid transformations</a>, <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>, <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformations</a>.</li></ul> <p><a href="/wiki/Category_theory" title="Category theory">Category theory</a>, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Logarithm_and_exponential">Logarithm and exponential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=2" title="Edit section: Logarithm and exponential"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{+}}"></span> be the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a> of <a href="/wiki/Positive_real_numbers" title="Positive real numbers">positive real numbers</a>, and 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> be the additive group of real numbers. </p><p>The <a href="/wiki/Logarithm_function" class="mw-redirect" title="Logarithm function">logarithm function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffef617958bbaa539fa183844ad1cacfaddf59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.39ex; height:2.843ex;" alt="{\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} }"></span> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(xy)=\log x+\log y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(xy)=\log x+\log y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90f980db27daf87d25fe2db8e2b544073921cc05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.408ex; height:2.843ex;" alt="{\displaystyle \log(xy)=\log x+\log y}"></span> for all <span class="mwe-math-element"><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,y\in \mathbb {R} ^{+},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} ^{+},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0745ca172382797ab2440077d3afdd617bf821f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.195ex; height:2.843ex;" alt="{\displaystyle x,y\in \mathbb {R} ^{+},}"></span> so it is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a>. The <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp :\mathbb {R} \to \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp :\mathbb {R} \to \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc8d9064ea0fb9ee61e51fa543537f9b4f6cdd39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.971ex; height:2.843ex;" alt="{\displaystyle \exp :\mathbb {R} \to \mathbb {R} ^{+}}"></span> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x+y)=(\exp x)(\exp y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(x+y)=(\exp x)(\exp y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f4c15f3bb32525bdd4e99a3deb3b2cb33a9b950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.768ex; height:2.843ex;" alt="{\displaystyle \exp(x+y)=(\exp x)(\exp y)}"></span> for all <span class="mwe-math-element"><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,y\in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b489ceaa39fa862e47e539193ade2453217380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.685ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} ,}"></span> so it too is a homomorphism. </p><p>The identities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \exp x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>exp</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \exp x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c797bda03ee72306125da548061676856363682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.056ex; height:2.509ex;" alt="{\displaystyle \log \exp x=x}"></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 \exp \log y=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \log y=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beff286ac6c896818ac2bc1d0c8a9361c5a94644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.708ex; height:2.509ex;" alt="{\displaystyle \exp \log y=y}"></span> show 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 \log }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></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 \exp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1185b570f67b4221307626254f64f9e619e769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }"></span> are <a href="/wiki/Inverse_function" title="Inverse function">inverses</a> of each other. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></span> is a homomorphism that has an inverse that is also a homomorphism, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></span> is an isomorphism of groups. </p><p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></span> function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a <a href="/wiki/Ruler" title="Ruler">ruler</a> and a <a href="/wiki/Table_of_logarithms" class="mw-redirect" title="Table of logarithms">table of logarithms</a>, or using a <a href="/wiki/Slide_rule" title="Slide rule">slide rule</a> with a logarithmic scale. </p> <div class="mw-heading mw-heading3"><h3 id="Integers_modulo_6">Integers modulo 6</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=3" title="Edit section: Integers modulo 6"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} _{6},+),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} _{6},+),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717201d3009257041c7fb833f62c618ef788efb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.903ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} _{6},+),}"></span> the integers from 0 to 5 with addition <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 6. Also consider the group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>+</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a6659fd97acfbf42ae76218b8dd1c50677da11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.735ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),}"></span> the ordered pairs where the <i>x</i> coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the <i>x</i>-coordinate is modulo 2 and addition in the <i>y</i>-coordinate is modulo 3. </p><p>These structures are isomorphic under addition, under the following scheme: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902d46e151cb52c69957ee0fa4cbdaac59979efa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:10.696ex; height:18.509ex;" alt="{\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}}"></span> or in general <span class="mwe-math-element"><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)\mapsto (3a+4b)\mod 6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mn>4</mn> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\mapsto (3a+4b)\mod 6.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe938c67d0a874afdb2e5dab1755adab712df2b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.958ex; height:2.843ex;" alt="{\displaystyle (a,b)\mapsto (3a+4b)\mod 6.}"></span> </p><p>For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1)+(1,0)=(0,1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1)+(1,0)=(0,1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80b44911407669e928f1c67fc5cd864bf24b5de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.09ex; height:2.843ex;" alt="{\displaystyle (1,1)+(1,0)=(0,1),}"></span> which translates in the other system as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+3=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+3=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94136030acf98cc212c60d755e392db44c5c2541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.073ex; height:2.343ex;" alt="{\displaystyle 1+3=4.}"></span> </p><p>Even though these two groups "look" different in that the sets contain different elements, they are indeed <b>isomorphic</b>: their structures are exactly the same. More generally, the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of two <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5474379674b9a5fd1b1336571cbeacbe81212d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.225ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{m}}"></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 \mathbb {Z} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b729c334a9863c47f0b7e3ad61342c2f0881bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{n}}"></span> is isomorphic 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 (\mathbb {Z} _{mn},+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} _{mn},+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e71169118c26d0b36d5d832f759d2405e665ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.863ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} _{mn},+)}"></span> if and only if <i>m</i> and <i>n</i> are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>, per the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation-preserving_isomorphism">Relation-preserving isomorphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=4" title="Edit section: Relation-preserving isomorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If one object consists of a set <i>X</i> with a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> R and the other object consists of a set <i>Y</i> with a binary relation S then an isomorphism from <i>X</i> to <i>Y</i> is a bijective function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> such that:<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> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">S</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if and only if </mtext> </mrow> <mspace width="1em" /> <mi mathvariant="normal">R</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec00dcd84a390ac745eda56007558aa66ab0dc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)}"></span> </p><p>S is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>, <a href="/wiki/Irreflexive_relation" class="mw-redirect" title="Irreflexive relation">irreflexive</a>, <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a>, <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>, <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">asymmetric</a>, <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, <a href="/wiki/Connected_relation" title="Connected relation">total</a>, <a href="/wiki/Homogeneous_relation#Properties" title="Homogeneous relation">trichotomous</a>, a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>, <a href="/wiki/Total_order" title="Total order">total order</a>, <a href="/wiki/Well-order" title="Well-order">well-order</a>, <a href="/wiki/Strict_weak_order" class="mw-redirect" title="Strict weak order">strict weak order</a>, <a href="/wiki/Strict_weak_order#Total_preorders" class="mw-redirect" title="Strict weak order">total preorder</a> (weak order), an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, or a relation with any other special properties, if and only if R is. </p><p>For example, R is an <a href="/wiki/Order_theory" title="Order theory">ordering</a> ≤ and S an ordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \sqsubseteq ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊑</mo> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \sqsubseteq ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36980b6e06ccb3a878c27f95a6bea594133e066d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.736ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \sqsubseteq ,}"></span> then an isomorphism from <i>X</i> to <i>Y</i> is a bijective function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>⊑</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if and only if </mtext> </mrow> <mspace width="1em" /> <mi>u</mi> <mo>≤</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6e4e0be373bdfcb7bc095253a9e74275ee972a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.277ex; height:2.843ex;" alt="{\displaystyle f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.}"></span> Such an isomorphism is called an <em><a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphism</a></em> or (less commonly) an <em>isotone isomorphism</em>. </p><p>If <span class="mwe-math-element"><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=Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83757ed7b7162654e88c32747556f2bf80fc57f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.499ex; height:2.509ex;" alt="{\displaystyle X=Y,}"></span> then this is a relation-preserving <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>. </p> <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=Isomorphism&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Algebra" title="Algebra">algebra</a>, isomorphisms are defined for all <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>. Some are more specifically studied; for example: </p> <ul><li><a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">Linear isomorphisms</a> between <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>; they are specified by <a href="/wiki/Invertible_matrices" class="mw-redirect" title="Invertible matrices">invertible matrices</a>.</li> <li><a href="/wiki/Group_isomorphism" title="Group isomorphism">Group isomorphisms</a> between <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>; the classification of <a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">isomorphism classes</a> of <a href="/wiki/Finite_group" title="Finite group">finite groups</a> is an open problem.</li> <li><a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">Ring isomorphism</a> between <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>.</li> <li>Field isomorphisms are the same as ring isomorphism between <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>; their study, and more specifically the study of <a href="/wiki/Field_automorphism" class="mw-redirect" title="Field automorphism">field automorphisms</a> is an important part of <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>.</li></ul> <p>Just as the <a href="/wiki/Automorphism" title="Automorphism">automorphisms</a> of an <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, the isomorphisms between two algebras sharing a common structure form a <a href="/wiki/Heap_(mathematics)" title="Heap (mathematics)">heap</a>. Letting a particular isomorphism identify the two structures turns this heap into a group. </p><p>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> is an isomorphism mapping hard <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> into easier <a href="/wiki/Algebra" title="Algebra">algebraic</a> equations. </p><p>In <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, an isomorphism between two graphs <i>G</i> and <i>H</i> is a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> map <i>f</i> from the vertices of <i>G</i> to the vertices of <i>H</i> that preserves the "edge structure" in the sense that there is an edge from <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a> <i>u</i> to vertex <i>v</i> in <i>G</i> if and only if there is an edge from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b282ccf3cf0f2b0b601e8629d61612d4951098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(u)}"></span> 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 f(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b2b785b7b6d9a22484d466da88d6328ed0b197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.215ex; height:2.843ex;" alt="{\displaystyle f(v)}"></span> in <i>H</i>. See <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">graph isomorphism</a>. </p><p>In mathematical analysis, an isomorphism between two <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> is a bijection preserving addition, scalar multiplication, and inner product. </p><p>In early theories of <a href="/wiki/Logical_atomism" title="Logical atomism">logical atomism</a>, the formal relationship between facts and true propositions was theorized by <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> and <a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> to be isomorphic. An example of this line of thinking can be found in Russell's <i><a href="/wiki/Introduction_to_Mathematical_Philosophy" title="Introduction to Mathematical Philosophy">Introduction to Mathematical Philosophy</a></i>. </p><p>In <a href="/wiki/Cybernetics" title="Cybernetics">cybernetics</a>, the <a href="/wiki/Good_regulator" title="Good regulator">good regulator</a> or Conant–Ashby theorem is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system. </p> <div class="mw-heading mw-heading2"><h2 id="Category_theoretic_view">Category theoretic view</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=6" title="Edit section: Category theoretic view"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, given a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> <i>C</i>, an isomorphism is a morphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:a\to b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">→</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:a\to b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5224bc5c18933c7ba94bf012e0f07c0f444b3f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.057ex; height:2.509ex;" alt="{\displaystyle f:a\to b}"></span> that has an inverse morphism <span class="mwe-math-element"><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:b\to a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>b</mi> <mo stretchy="false">→</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:b\to a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199fd0a26fbe3cf2449be01474d1465628c1cada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.541ex; height:2.509ex;" alt="{\displaystyle g:b\to a,}"></span> that 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 fg=1_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg=1_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6224e6e22e26369cff76f7e906579ed0056a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.593ex; height:2.509ex;" alt="{\displaystyle fg=1_{b}}"></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 gf=1_{a}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>f</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gf=1_{a}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff6dbaf6ad57ead7e92a5f0393988cfda730bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.404ex; height:2.509ex;" alt="{\displaystyle gf=1_{a}.}"></span> </p><p>Two categories <span class="texhtml mvar" style="font-style:italic;">C</span> and <span class="texhtml mvar" style="font-style:italic;">D</span> are <a href="/wiki/Isomorphism_of_categories" title="Isomorphism of categories">isomorphic</a> if there exist <a href="/wiki/Functor" title="Functor">functors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:C\to D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:C\to D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d082755579fdc0f7d9daca655472d59e8347242d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.983ex; height:2.176ex;" alt="{\displaystyle F:C\to D}"></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 G:D\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G:D\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e6e1f490befc894fb1c01e648a6df1a50eecc98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.069ex; height:2.176ex;" alt="{\displaystyle G:D\to C}"></span> which are mutually inverse to each other, that 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 FG=1_{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>G</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FG=1_{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344d63c61632f76427e259b4850ad526ac94890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.421ex; height:2.509ex;" alt="{\displaystyle FG=1_{D}}"></span> (the identity functor on <span class="texhtml mvar" style="font-style:italic;">D</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 GF=1_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>F</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GF=1_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d8801ff87bb21dfbe450b7f9b9ed179559850c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.31ex; height:2.509ex;" alt="{\displaystyle GF=1_{C}}"></span> (the identity functor on <span class="texhtml mvar" style="font-style:italic;">C</span>). </p> <div class="mw-heading mw-heading3"><h3 id="Isomorphism_vs._bijective_morphism">Isomorphism vs. bijective morphism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=7" title="Edit section: Isomorphism vs. bijective morphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Concrete_category" title="Concrete category">concrete category</a> (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the <a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a> or categories of algebraic objects (like the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>, the <a href="/wiki/Category_of_rings" title="Category of rings">category of rings</a>, and the <a href="/wiki/Category_of_modules" title="Category of modules">category of modules</a>), an isomorphism must be bijective on the <a href="/wiki/Underlying_set" class="mw-redirect" title="Underlying set">underlying sets</a>. In algebraic categories (specifically, categories of <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">varieties in the sense of universal algebra</a>), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces). </p> <div class="mw-heading mw-heading2"><h2 id="Isomorphism_class">Isomorphism class</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=8" title="Edit section: Isomorphism class"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since a composition of isomorphisms is an isomorphism, since the identity is an isomorphism and since the inverse of an isomorphism is an isomorphism, the relation that two mathematical objects are isomorphic is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. An <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> given by isomorphisms is commonly called an <b>isomorphism class</b>.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Examples of isomorphism classes are plentiful in mathematics. </p> <ul><li>Two sets are isomorphic if there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between them. The isomorphism class of a finite set can be identified with the non-negative integer representing the number of elements it contains.</li> <li>The isomorphism class of a <a href="/wiki/Finite-dimensional_vector_space" class="mw-redirect" title="Finite-dimensional vector space">finite-dimensional vector space</a> can be identified with the non-negative integer representing its dimension.</li> <li>The <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> enumerates the isomorphism classes of all <a href="/wiki/Finite_simple_groups" class="mw-redirect" title="Finite simple groups">finite simple groups</a>.</li> <li>The <a href="/wiki/Surface_(topology)#Classification_of_closed_surfaces" title="Surface (topology)">classification of closed surfaces</a> enumerates the isomorphism classes of all connected <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surfaces</a>.</li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinals</a> are essentially defined as isomorphism classes of well-ordered sets (though there are technical issues involved).</li></ul> <p>However, there are circumstances in which the isomorphism class of an object conceals vital information about it. </p> <ul><li>Given a <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a>, it is common that two substructures belong to the same isomorphism class. However, the way they are included in the whole structure can not be studied if they are identified. For example, in a finite-dimensional vector space, all <a href="/wiki/Linear_subspace" title="Linear subspace">subspaces</a> of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc.</li> <li>The <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebras</a> consisting of <a href="/wiki/Coquaternion" class="mw-redirect" title="Coquaternion">coquaternions</a> and 2 × 2 <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> are isomorphic as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>. Yet they appear in different contexts for application (plane mapping and kinematics) so the isomorphism is insufficient to merge the concepts.<sup class="noprint Inline-Template noprint Template-Opinion" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Neutral_point_of_view/FAQ#Assert_facts,_not_opinions" title="Wikipedia:Neutral point of view/FAQ"><span title="This statement may be opinion presented as fact. (February 2021)">opinion</span></a></i>]</sup></li> <li>In <a href="/wiki/Homotopy_theory" title="Homotopy theory">homotopy theory</a>, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a <a href="/wiki/Topological_space" title="Topological space">space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> at a point <span class="mwe-math-element"><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>, though technically denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(X,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d09ac1386cc00599b55d8fe4f273bda3c24a91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X,p)}"></span> to emphasize the dependence on the base point, is often written lazily as simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c502a8f1054224532cb495be2f6b5e65f660c2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X)}"></span> if <span class="mwe-math-element"><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> is <a href="/wiki/Connected_space#Path_connectedness" title="Connected space">path connected</a>. The reason for this is that the existence of a path between two points allows one to identify <a href="/wiki/Loop_(topology)" title="Loop (topology)">loops</a> at one with loops at the other; however, unless <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(X,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d09ac1386cc00599b55d8fe4f273bda3c24a91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X,p)}"></span> is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> this isomorphism is non-unique. Furthermore, the classification of <a href="/wiki/Covering_space" title="Covering space">covering spaces</a> makes strict reference to particular <a href="/wiki/Subgroup" title="Subgroup">subgroups</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(X,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d09ac1386cc00599b55d8fe4f273bda3c24a91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.372ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X,p)}"></span>, specifically distinguishing between isomorphic but <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugate</a> subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Relation_to_equality">Relation to equality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=10" title="Edit section: Relation to equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality (mathematics)</a> and <a href="/wiki/Coherent_isomorphism" class="mw-redirect" title="Coherent isomorphism">coherent isomorphism</a></div> <p>Although there are cases where isomorphic objects can be considered equal, one must distinguish <em><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a></em> and <em>isomorphism</em>.<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> Equality is when two objects are the same, and therefore everything that is true about one object is true about the other. On the other hand, isomorphisms are related to some structure, and two isomorphic objects share only the properties that are related to this structure. </p><p>For example, the sets <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> </mrow> <mo>}</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b250ba2c8cc24752de1dd746503ef1043d547d32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.347ex; height:3.343ex;" alt="{\displaystyle A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}}"></span> are <em>equal</em>; they are merely different representations—the first an <a href="/wiki/Intensional_definition" class="mw-redirect" title="Intensional definition">intensional</a> one (in <a href="/wiki/Set_builder_notation" class="mw-redirect" title="Set builder notation">set builder notation</a>), and the second <a href="/wiki/Extensional_definition" class="mw-redirect" title="Extensional definition">extensional</a> (by explicit enumeration)—of the same subset of the integers. By contrast, the sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,B,C\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A,B,C\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e76678c579efebb6722408baaacae933f842747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.666ex; height:2.843ex;" alt="{\displaystyle \{A,B,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 \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span> are not <em>equal</em> since they do not have the same elements. They are isomorphic as sets, but there are many choices (in fact 6) of an isomorphism between them: one isomorphism is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806e902d5f601760eac4ab9fe0ce65852e6e0616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.111ex; height:2.509ex;" alt="{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3,}"></span></dd></dl> <p>while another is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>3</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769e836ca36673f617cbc90142dbf4236a7c5bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.111ex; height:2.509ex;" alt="{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}"></span></dd></dl> <p>and no one isomorphism is intrinsically better than any other.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> On this view and in this sense, these two sets are not equal because one cannot consider them <em>identical</em>: one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism. </p><p>Also, <a href="/wiki/Integer" title="Integer">integers</a> and <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even numbers</a> are isomorphic as <a href="/wiki/Ordered_set" class="mw-redirect" title="Ordered set">ordered sets</a> and <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> (for addition), but cannot be considered equal sets, since one is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of the other. </p><p>On the other hand, when sets (or other <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>) are defined only by their properties, without considering the nature of their elements, one often considers them to be equal. This is generally the case with solutions of <a href="/wiki/Universal_properties" class="mw-redirect" title="Universal properties">universal properties</a>. </p><p>For example, the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> are usually defined as <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of pairs of integers, although nobody thinks of a rational number as a set (equivalence class). The universal property of the rational numbers is essentially that they form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> that contains the integers and does not contain any proper subfield. It results that given two fields with these properties, there is a unique field isomorphism between them. This allows identifying these two fields, since every property of one of them can be transferred to the other through the isomorphism. For example the <a href="/wiki/Real_number" title="Real number">real numbers</a> that are obtained by dividing two integers (inside the real numbers) form the smallest subfield of the real numbers. There is thus a unique isomorphism from the rational numbers (defined as equivalence classes of pairs) to the quotients of two real numbers that are integers. This allows identifying these two sorts of rational numbers. </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=Isomorphism&action=edit&section=11" 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 .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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relation</a></li> <li><a href="/wiki/Heap_(mathematics)" title="Heap (mathematics)">Heap (mathematics)</a></li> <li><a href="/wiki/Isometry" title="Isometry">Isometry</a></li> <li><a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">Isomorphism class</a></li> <li><a href="/wiki/Isomorphism_theorem" class="mw-redirect" title="Isomorphism theorem">Isomorphism theorem</a></li> <li><a href="/wiki/Universal_property" title="Universal property">Universal property</a></li> <li><a href="/wiki/Coherent_isomorphism" class="mw-redirect" title="Coherent isomorphism">Coherent isomorphism</a></li> <li><a href="/wiki/Balanced_category" title="Balanced category">Balanced category</a></li></ul> </div> <div 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=Isomorphism&action=edit&section=12" 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-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span> have a conventional order, namely the alphabetical order, and similarly 1, 2, 3 have the usual order of the integers. Viewed as ordered sets, there is only one isomorphism between them, namely <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> <mo stretchy="false">↦</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b21c1aa3b440c251fd56801306cc2cc93a40b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.111ex; height:2.509ex;" alt="{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}"></span></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=Isomorphism&action=edit&section=13" 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"><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="CITEREFVinberg,_Ėrnest_Borisovich2003" class="citation book cs1">Vinberg, Ėrnest Borisovich (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kd24d3mwaecC&pg=PA3"><i>A Course in Algebra</i></a>. American Mathematical Society. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821834138" title="Special:BookSources/9780821834138"><bdi>9780821834138</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+Course+in+Algebra&rft.pages=3&rft.pub=American+Mathematical+Society&rft.date=2003&rft.isbn=9780821834138&rft.au=Vinberg%2C+%C4%96rnest+Borisovich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkd24d3mwaecC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsomorphism" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAwodey,_Steve2006" class="citation book cs1"><a href="/wiki/Steve_Awodey" title="Steve Awodey">Awodey, Steve</a> (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IK_sIDI2TCwC&pg=PA11">"Isomorphisms"</a>. <i>Category theory</i>. Oxford University Press. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198568612" title="Special:BookSources/9780198568612"><bdi>9780198568612</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Isomorphisms&rft.btitle=Category+theory&rft.pages=11&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=9780198568612&rft.au=Awodey%2C+Steve&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIK_sIDI2TCwC%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsomorphism" 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"><a href="#CITEREFMazur2007">Mazur 2007</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isomorphism&action=edit&section=14" title="Edit section: Further reading"><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="CITEREFMazur2007" class="citation cs2"><a href="/wiki/Barry_Mazur" title="Barry Mazur">Mazur, Barry</a> (12 June 2007), <a rel="nofollow" class="external text" href="https://bpb-us-e1.wpmucdn.com/sites.harvard.edu/dist/a/189/files/2023/01/When-is-one-thing-equal-to-some-other-thing.pdf"><i>When is one thing equal to some other thing?</i></a> <span class="cs1-format">(PDF)</span></cite><span 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Isomorphism&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DIsomorphism&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsomorphism" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Isomorphism"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Isomorphism.html">"Isomorphism"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Isomorphism&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FIsomorphism.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsomorphism" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output 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