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title="此IP所修之議[n]" accesskey="n"><span>議</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站台"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目次" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至側邊欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隱藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-定義" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">一</span> <span>定義</span> </div> </a> <button aria-controls="toc-定義-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>切換定義子章節</span> </button> <ul id="toc-定義-sublist" class="vector-toc-list"> <li id="toc-最、極" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#最、極"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.一</span> <span>最、極</span> </div> </a> <ul id="toc-最、極-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-上界，下界" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#上界，下界"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.二</span> <span>上界，下界</span> </div> </a> <ul id="toc-上界，下界-sublist" class="vector-toc-list"> <li id="toc-完備" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#完備"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.二.一</span> <span>完備</span> </div> </a> <ul id="toc-完備-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-格" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#格"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.二.二</span> <span>格</span> </div> </a> <ul id="toc-格-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-全序" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#全序"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.三</span> <span>全序</span> </div> </a> <ul id="toc-全序-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-良序" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#良序"> <div class="vector-toc-text"> <span class="vector-toc-numb">一.四</span> <span>良序</span> </div> </a> <ul id="toc-良序-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例"> <div class="vector-toc-text"> <span class="vector-toc-numb">二</span> <span>例</span> </div> </a> <button aria-controls="toc-例-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>切換例子章節</span> </button> <ul id="toc-例-sublist" class="vector-toc-list"> <li id="toc-除盡" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#除盡"> <div class="vector-toc-text"> <span class="vector-toc-numb">二.一</span> <span>除盡</span> </div> </a> <ul id="toc-除盡-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-不大于" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#不大于"> <div class="vector-toc-text"> <span class="vector-toc-numb">二.二</span> <span>不大于</span> </div> </a> <ul id="toc-不大于-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-不小于" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#不小于"> <div class="vector-toc-text"> <span class="vector-toc-numb">二.三</span> <span>不小于</span> </div> </a> <ul id="toc-不小于-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-註" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註"> <div class="vector-toc-text"> <span class="vector-toc-numb">三</span> <span>註</span> </div> </a> <ul id="toc-註-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="目次" 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="切換目次" > <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">切換目次</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">偏序</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="前往別種語言的條目。有三四種語言可用" > <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-34" 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">三四種語言</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/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A9_%D9%85%D8%B1%D8%AA%D8%A8%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A7" title="مجموعة مرتبة جزئيا – 阿拉伯文" lang="ar" hreflang="ar" data-title="مجموعة مرتبة جزئيا" data-language-autonym="العربية" data-language-local-name="阿拉伯文" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%BA%D0%BE%D0%B2%D0%B0_%D1%9E%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%B0%D0%B5_%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Часткова ўпарадкаванае мноства – 白俄羅斯文" lang="be" hreflang="be" data-title="Часткова ўпарадкаванае мноства" data-language-autonym="Беларуская" data-language-local-name="白俄羅斯文" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjunt_parcialment_ordenat" title="Conjunt parcialment ordenat – 加泰蘭文" lang="ca" hreflang="ca" data-title="Conjunt parcialment ordenat" data-language-autonym="Català" data-language-local-name="加泰蘭文" 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/%C4%8C%C3%A1ste%C4%8Dn%C4%9B_uspo%C5%99%C3%A1dan%C3%A1_mno%C5%BEina" title="Částečně uspořádaná množina – 捷克文" lang="cs" hreflang="cs" data-title="Částečně uspořádaná množina" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Teilweise_geordnete_Menge" title="Teilweise geordnete Menge – 德文" lang="de" hreflang="de" data-title="Teilweise geordnete Menge" data-language-autonym="Deutsch" data-language-local-name="德文" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" title="Partially ordered set – 英文" lang="en" hreflang="en" data-title="Partially ordered set" data-language-autonym="English" data-language-local-name="英文" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Parta_ordo" title="Parta ordo – 世界文" lang="eo" hreflang="eo" data-title="Parta ordo" data-language-autonym="Esperanto" data-language-local-name="世界文" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjunto_parcialmente_ordenado" title="Conjunto parcialmente ordenado – 西班牙文" lang="es" hreflang="es" data-title="Conjunto parcialmente ordenado" data-language-autonym="Español" data-language-local-name="西班牙文" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Osaliselt_j%C3%A4rjestatud_hulk" title="Osaliselt järjestatud hulk – 愛沙尼亞文" lang="et" hreflang="et" data-title="Osaliselt järjestatud hulk" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87_%D8%AC%D8%B2%D8%A6%DB%8C%E2%80%8C%D9%85%D8%B1%D8%AA%D8%A8" title="مجموعه جزئی‌مرتب – 波斯文" lang="fa" hreflang="fa" data-title="مجموعه جزئی‌مرتب" data-language-autonym="فارسی" data-language-local-name="波斯文" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Osittain_j%C3%A4rjestetty_joukko" title="Osittain järjestetty joukko – 芬蘭文" lang="fi" hreflang="fi" data-title="Osittain järjestetty joukko" data-language-autonym="Suomi" data-language-local-name="芬蘭文" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ensemble_partiellement_ordonn%C3%A9" title="Ensemble partiellement ordonné – 法文" lang="fr" hreflang="fr" data-title="Ensemble partiellement ordonné" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Conxunto_parcialmente_ordenado" title="Conxunto parcialmente ordenado – 加利西亞文" lang="gl" hreflang="gl" data-title="Conxunto parcialmente ordenado" data-language-autonym="Galego" data-language-local-name="加利西亞文" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8_%D7%97%D7%9C%D7%A7%D7%99" title="סדר חלקי – 希伯來文" lang="he" hreflang="he" data-title="סדר חלקי" data-language-autonym="עברית" data-language-local-name="希伯來文" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Parcijalno_ure%C4%91en_skup" title="Parcijalno uređen skup – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Parcijalno uređen skup" data-language-autonym="Hrvatski" data-language-local-name="克羅埃西亞文" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%A9szbenrendezett_halmaz" title="Részbenrendezett halmaz – 匈牙利文" lang="hu" hreflang="hu" data-title="Részbenrendezett halmaz" data-language-autonym="Magyar" data-language-local-name="匈牙利文" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Himpunan_terurut_parsial" title="Himpunan terurut parsial – 印尼文" lang="id" hreflang="id" data-title="Himpunan terurut parsial" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutr%C3%B6%C3%B0unarvensl" title="Hlutröðunarvensl – 冰島文" lang="is" hreflang="is" data-title="Hlutröðunarvensl" data-language-autonym="Íslenska" data-language-local-name="冰島文" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja badge-Q70893996 mw-list-item" title=""><a href="https://ja.wikipedia.org/wiki/%E5%8D%8A%E9%A0%86%E5%BA%8F%E9%9B%86%E5%90%88" title="半順序集合 – 日文" lang="ja" hreflang="ja" data-title="半順序集合" data-language-autonym="日本語" data-language-local-name="日文" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합 – 韓文" lang="ko" hreflang="ko" data-title="부분 순서 집합" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Parti%C3%ABle_orde" title="Partiële orde – 荷蘭文" lang="nl" hreflang="nl" data-title="Partiële orde" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ordna_mengd" title="Ordna mengd – 耐諾斯克挪威文" lang="nn" hreflang="nn" data-title="Ordna mengd" data-language-autonym="Norsk nynorsk" data-language-local-name="耐諾斯克挪威文" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cz%C4%99%C5%9Bciowy_porz%C4%85dek" title="Częściowy porządek – 波蘭文" lang="pl" hreflang="pl" data-title="Częściowy porządek" data-language-autonym="Polski" data-language-local-name="波蘭文" 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/Conjunto_parcialmente_ordenado" title="Conjunto parcialmente ordenado – 葡萄牙文" lang="pt" hreflang="pt" data-title="Conjunto parcialmente ordenado" data-language-autonym="Português" data-language-local-name="葡萄牙文" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%BE_%D1%83%D0%BF%D0%BE%D1%80%D1%8F%D0%B4%D0%BE%D1%87%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Частично упорядоченное множество – 俄文" lang="ru" hreflang="ru" data-title="Частично упорядоченное множество" data-language-autonym="Русский" data-language-local-name="俄文" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/%C4%8Ciasto%C4%8Dne_usporiadan%C3%A1_mno%C5%BEina" title="Čiastočne usporiadaná množina – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Čiastočne usporiadaná množina" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Delno_urejena_mno%C5%BEica" title="Delno urejena množica – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Delno urejena množica" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Partialordnad_m%C3%A4ngd" title="Partialordnad mängd – 瑞典文" lang="sv" hreflang="sv" data-title="Partialordnad mängd" data-language-autonym="Svenska" data-language-local-name="瑞典文" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%A4%E0%AE%BF_%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3_%E0%AE%95%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="பகுதி வரிசையுள்ள கணம் – 坦米爾文" lang="ta" hreflang="ta" data-title="பகுதி வரிசையுள்ள கணம்" data-language-autonym="தமிழ்" data-language-local-name="坦米爾文" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C4%B1smi_s%C4%B1ral%C4%B1_k%C3%BCme" title="Kısmi sıralı küme – 土耳其文" lang="tr" hreflang="tr" data-title="Kısmi sıralı küme" data-language-autonym="Türkçe" data-language-local-name="土耳其文" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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class="mw-body-content"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">頁面版本狀態</h2><p class="cdx-dialog__header__subtitle">該頁面尚未檢查</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="關閉" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">此為底本，未經<a href="/wiki/%E8%AA%AA%E6%98%8E:%E6%A0%A1%E5%B0%8D" title="說明:校對">審校</a>。</div></div><div tabindex="0"></div></div><div class="mw-content-ltr mw-parser-output" lang="lzh" dir="ltr"><div style="border:1px solid #A3B1BF; padding:0.5em 2em; width:60%; float:left;"> 註︰蓋當今數學之事，誠難僅以文述，而無符號，故凡數學之文，咸有漢字、拉丁字相易之事，以合文言、數學，則無論文理之人，皆可明之也。</div> <div style="clear:left;"></div> <p><b>偏序</b>者，柣序也。大小，長短，前後，皆偏序也。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定義"><span id=".E5.AE.9A.E7.BE.A9"></span>定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=1" title="所纂章名：定義"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>偏序者，<a href="/wiki/%E9%97%9C%E4%BF%82_(%E6%95%B8%E5%AD%B8)" title="關係 (數學)">關係</a>也，且曰「不大於」，必以下是從。 </p> <ul><li>物不大於己。曰<b>反射</b>。</li> <li>甲不大於乙，乙不大於甲，則甲同乎乙。曰<b>反對稱</b>。</li> <li>甲不大於乙，乙不大於丙，則甲不大於丙。曰<b>傳遞</b>。</li></ul> <div class="mw-heading mw-heading3"><h3 id="最、極"><span id=".E6.9C.80.E3.80.81.E6.A5.B5"></span>最、極</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=2" title="所纂章名：最、極"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>凡一界之内，小於無物者，曰<b>極大</b>；大於無物者，曰<b>極小</b>。 </p><p>物皆小於其者，曰<b>最大</b>；物皆大於其者，曰<b>最小</b>。最大必極大，最小必極小，反之則不然。 </p> <div class="mw-heading mw-heading3"><h3 id="上界，下界"><span id=".E4.B8.8A.E7.95.8C.EF.BC.8C.E4.B8.8B.E7.95.8C"></span>上界，下界</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=3" title="所纂章名：上界，下界"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>取子集，大於此集之所有，曰<b>上界</b>；小於此集之所有，曰<b>下界</b>。 </p><p>若上界有最小者，曰<b>最小上界</b>；若下界有最大者，曰<b>最大下界</b>。 </p> <div class="mw-heading mw-heading4"><h4 id="完備"><span id=".E5.AE.8C.E5.82.99"></span>完備</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=4" title="所纂章名：完備"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>完備</b>者，凡子集有上界者必有最小上界也，同乎凡子集有下界者必有最大下界耳。 </p> <div class="mw-heading mw-heading4"><h4 id="格"><span id=".E6.A0.BC"></span>格</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=5" title="所纂章名：格"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>格</b>者，凡二物必有最小上界及最大下界也。 </p> <div class="mw-heading mw-heading3"><h3 id="全序"><span id=".E5.85.A8.E5.BA.8F"></span>全序</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=6" title="所纂章名：全序"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>全序</b>者，凡二物，必有前者小於後，或後者小於前。全序必格也。 </p> <div class="mw-heading mw-heading3"><h3 id="良序"><span id=".E8.89.AF.E5.BA.8F"></span>良序</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=7" title="所纂章名：良序"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>良序</b>者，凡子集必有最小物也。良序必全序也。良序必完備也。噫，良序集果真井然有序耳，故得此名。 </p><p><a href="/wiki/%E8%89%AF%E5%BA%8F%E5%85%AC%E7%90%86" class="mw-redirect" title="良序公理">良序公理</a>曰：「集皆可良序也。」此與選擇公理等價耳。 </p> <div class="mw-heading mw-heading2"><h2 id="例"><span id=".E4.BE.8B"></span>例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=8" title="所纂章名：例"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="除盡"><span id=".E9.99.A4.E7.9B.A1"></span>除盡</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=9" title="所纂章名：除盡"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>以甲為自然數集，乙為甲<a href="/wiki/%E8%A3%9C%E9%9B%86" title="補集">去</a>一。以「除盡」為偏序，故有三不少於六，五不少於百。然三與百無可比耳，故「除盡」非全序耳。 </p><p>一為甲之最小。三、七為乙之極小，然乙無最小者也。甲乙皆無極大者。 </p><p>取甲之子集，其物之<a href="/w/index.php?title=%E5%85%AC%E5%9B%A0%E6%95%B8&action=edit&redlink=1" class="new" title="公因數（查無此頁）">公因數</a>乃下界，最大公因數為最大下界。 </p><p>取甲之二物，最大公因數為最大下界，最小<a href="/w/index.php?title=%E5%85%AC%E5%80%8D%E6%95%B8&action=edit&redlink=1" class="new" title="公倍數（查無此頁）">公倍數</a>為最小上界，故甲乃格也。取乙之二、三，並無最大下界<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>一<span class="cite-bracket">]</span></a></sup>，故乙非格也。 </p><p>不論甲乙，凡子集有下界者，其最大公因數亦存，故有最大下界。故甲乙皆完備也。 </p> <div class="mw-heading mw-heading3"><h3 id="不大于"><span id=".E4.B8.8D.E5.A4.A7.E4.BA.8E"></span>不大于</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=10" title="所纂章名：不大于"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>觀分數集，以「不大于」為偏序。凡二物必可比，故此乃全序耳。立方大于二者，成一子集，有下界零，然無最大下界<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>二<span class="cite-bracket">]</span></a></sup>，是以分數集非完備耳。 </p><p>觀實數集，以「不大于」為偏序，完備耳。然無最小者，故非良序。 </p><p>觀正實數集，以「不大于」為偏序，完備耳。立方大于二者，成一子集，二開立方為最大下界，然非子集之物也，故子集無最小者也，是以正實數集非良序。 </p><p>觀自然數集，以「不大于」為偏序，良序耳。 </p> <div class="mw-heading mw-heading3"><h3 id="不小于"><span id=".E4.B8.8D.E5.B0.8F.E4.BA.8E"></span>不小于</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=11" title="所纂章名：不小于"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>觀自然數集，以「不小于」為偏序。故三不大於二，五不大於三<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>三<span class="cite-bracket">]</span></a></sup>，故自然數無最小者也。故非良序耳。 </p> <div class="mw-heading mw-heading2"><h2 id="註"><span id=".E8.A8.BB"></span>註</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%81%8F%E5%BA%8F&action=edit&section=12" title="所纂章名：註"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">乙無一也。</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">二開立方非有理數也</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">以大作小，抽象之甚也，初學者常難之。</span> </li> </ol> <div class="NavFrame" style="clear:both"> <div class="NavHead"><a href="/wiki/%E9%9B%86%E8%AB%96" title="集論">集論</a></div> <div class="NavContent"> <p><a href="/wiki/%E9%9B%86" title="集">元素</a>｜ <a href="/wiki/%E9%9B%86" title="集">集</a>｜ <a href="/wiki/%E6%97%8F_(%E6%95%B8%E5%AD%B8)" title="族 (數學)">族</a>｜ <a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a>｜ <a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>｜ <a href="/wiki/%E4%BD%B5%E9%9B%86" title="併集">併集</a>｜ <a href="/wiki/%E8%A3%9C%E9%9B%86" title="補集">補集</a>｜ <a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a>｜ <a href="/wiki/%E6%9C%89%E5%BA%8F%E5%B0%8D" title="有序對">有序對</a>｜ <a href="/wiki/%E7%9B%B4%E7%A9%8D" title="直積">直積</a>｜ <a href="/wiki/%E9%97%9C%E4%BF%82_(%E6%95%B8%E5%AD%B8)" title="關係 (數學)">關係</a>｜ <a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a>｜ <a href="/wiki/%E7%AD%89%E5%83%B9" title="等價">等價</a>｜ <a class="mw-selflink selflink">偏序</a> </p> </div> <div 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