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id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <h1 id="firstHeading" class="firstHeading mw-first-heading">查看“︁二元关系”︁的源代码</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="这篇条目只存在于此语言中。添加其他语言的条目" > <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-0" 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">添加语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span></div> </div> 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id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="vector-tab-noicon mw-list-item"><a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB"><span>阅读</span></a></li><li id="ca-edit" class="selected vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB&action=edit" title="编辑该页面"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="vector-more-collapsible-item mw-list-item"><a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB"><span>阅读</span></a></li><li id="ca-more-edit" class="selected vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fw%2Findex.php%3Ftitle%3D%25E4%25BA%258C%25E5%2585%2583%25E5%2585%25B3%25E7%25B3%25BB%26action%3Dedit"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fw%2Findex.php%3Ftitle%3D%25E4%25BA%258C%25E5%2585%2583%25E5%2585%25B3%25E7%25B3%25BB%26action%3Dedit"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q130901" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> </div> <div id="contentSub"><div id="mw-content-subtitle">←<a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" title="二元关系">二元关系</a></div></div> <div id="mw-content-text" class="mw-body-content"><p>根据以下的原因，您并无权限去编辑该页面： </p> <ul class="permissions-errors"><li class="mw-permissionerror-blockedtext"> <table id="mw-blocked-text" style="border:1px solid var(--border-color-notice,rgba(0,0,0,0.4)); border-radius: 3px; width:100%; margin:0 auto 6px auto; padding: 0;"> <tbody><tr> <td> <div style="width:100%; text-align:center; margin: 1.3em 0 1.3em 0;"><div style="font-size:1.6em; color:var(--color-error,rgba(221,51,51,1));"><b>您現在使用的IP地址目前不能編輯維基百科</b></div> <div style="text-align: center;"><div style="display: inline-block; padding: 0 2em; border-radius: 3px;"><b>您可以瀏覽頁面</b>，但不能編輯或建立頁面</div></div> </div> <div style="margin:auto; width:70%;"> <div style="text-align:justify;"> <p>目前，IP地址區段「<u>8.222.0.0/16</u>」已被‪Jimmy-abot‬<b><a href="/wiki/Wikipedia:BLOCK" class="mw-redirect" title="Wikipedia:BLOCK">封禁</a></b>，而您所使用的IP地址正位於被封禁的範圍內。該IP地址區段被封禁的 </p> <div style="margin:1em; background: var(--background-color-neutral,#eaecf0); padding:0.6em 2em; border:0px var(--border-color-notice,#666) solid; border-radius: 3px;" id="mw-blocked-text-reason"> <div style="color: var(--color-progressive--active,rgba(42,75,141,1))">封禁理由：</div> <div style="color: var(--color-error,rgba(221,51,51,1)); margin:0.5em 1em"> <table class="user-block" style="border: 1px solid var(--border-color-base,#AAA); padding: 0.5em; width: 100%;"> <tbody><tr> <td> <figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/File:Banned_proxys.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Banned_proxys.svg/50px-Banned_proxys.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Banned_proxys.svg/75px-Banned_proxys.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Banned_proxys.svg/100px-Banned_proxys.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption></figcaption></figure>您正在使用的<a href="/wiki/IP%E4%BD%8D%E5%9D%80" class="mw-redirect" title="IP位址">IP地址</a>被判定用于代理工具，因此这一IP已经被封禁。 <p>为防止破坏，<b><a href="https://meta.wikimedia.org/wiki/No_open_proxies/zh" class="extiw" title="m:No open proxies/zh">维基媒体基金会禁止使用部分代理工具来编辑维基百科</a></b>。 </p> </td></tr> <tr> <td style="border-top: 1px solid #AAA; padding-top: 0.5em;"> <p><b>为什么？</b> </p><p>您当前访问维基百科使用的IP被我们判定为代理（包括<a href="/wiki/%E7%AA%81%E7%A0%B4%E7%BD%91%E7%BB%9C%E5%AE%A1%E6%9F%A5" title="突破网络审查">突破网络审查</a>）工具所使用的IP。因代理工具可以隐藏用户的真实IP，使破坏者可以以人工或程序对维基百科进行破坏，故禁止使用公共代理工具进行编辑。 </p><p><b>怎么办？</b> </p> <ul><li>直接访问维基百科，不经过代理服务。</li> <li>如果您在中国大陆而必须使用代理服务器，或是有特殊需要，请您 <span class="plainlinks"><a href="/wiki/Wikipedia:Unblock-zh/Blocked_proxy" class="mw-redirect" title="Wikipedia:Unblock-zh/Blocked proxy"><span class="mw-ui-button mw-ui-progressive" role="button">使用网站</span></a></span> 或者 <span class="plainlinks"><a href="/wiki/Wikipedia:IPBEMAIL" class="mw-redirect" title="Wikipedia:IPBEMAIL"><span class="mw-ui-button mw-ui-progressive" role="button">使用电邮</span></a></span> 申请IP封禁豁免。带有此权限的用户可以使用被封禁的代理编辑维基百科。</li> <li>如果您相信自己并未运行或使用代理，那很可能是因为上一个从<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%9C%8D%E5%8A%A1%E4%BE%9B%E5%BA%94%E5%95%86" title="互联网服务供应商">互联网服务供应商</a>分配到此IP地址的用户运行了代理。那么请在您的<a href="/wiki/Help:%E8%AE%A8%E8%AE%BA%E9%A1%B5" class="mw-redirect" title="Help:讨论页">讨论页</a>使用{{<a href="/wiki/Template:Unblock" title="Template:Unblock">unblock</a>}}模板申请解除封禁。极少数情况下，您的网络设备或服务提供商可能未进行正确配置，或已被恶意软件（如病毒）入侵。</li> <li>您如果无法自行解决问题，可至<a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93" title="Wikipedia:IRC聊天频道">IRC聊天频道</a>寻求其他编者的帮助。</li></ul> </td></tr></tbody></table> </div> <div style="color: var(--color-progressive--active,rgba(42,75,141,1))">這次封禁的結束時間是：<b>2025年7月23日 (三) 10:52 (<a href="/wiki/UTC" class="mw-redirect" title="UTC">UTC</a>)</b></div> </div> <p>如您对此封禁有疑问或异议，您可以： </p> <div style="margin:1em; background: var(--background-color-neutral,#eaecf0); color: var(--color-progressive--active,rgba(42,75,141,1)); padding:0.6em 2em; border:0px var(--border-color-notice,#666 solid); border-radius: 3px;"> <div style="margin-bottom: 0.4em;"><b>网站申诉</b>：前往<a rel="nofollow" class="external text" href="https://unblock-zh.org/new/zhwp">-{R|https://unblock-zh.org/new/zhwp}-</a>，如果您没有账户，请选择“无法创建账户”。如果您有账户，请选择“已有账户，但无法编辑”。在页面下方的“IP地址和IPBE”一节，请您选择“检测并提供我的IP地址信息，同时申请IPBE权限”。如果您使用代理/翻墙工具，请确保您开启全局模式，以保证我们获得您的准确IP信息。管理员将手工处理您的请求。</div> <div style="margin-bottom: 0.4em;"><b>邮件申诉</b>：将您的用户名和申诉理由写入电子邮件，发送至 <b><samp>unblock-zh<span class="skin-invert" typeof="mw:File"><span title="@"><img alt="@" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Nospam_at.svg/12px-Nospam_at.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Nospam_at.svg/18px-Nospam_at.svg.png 1.5x, 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class="extiw" title="m:Special:MyLanguage/Global blocks">在所有维基媒体基金会维基受到封禁</a>的地址段。</b> <p>封禁执行者是<a href="/wiki/User:Jon_Kolbert" title="User:Jon Kolbert">‪Jon Kolbert‬</a>。封禁原因是<i><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/NOP" class="extiw" title="m:Special:MyLanguage/NOP">Open proxy/Webhost</a>: See the <a href="https://meta.wikimedia.org/wiki/WM:OP/H" class="extiw" title="m:WM:OP/H">help page</a> if you are affected </i>。 </p> <ul><li>封禁开始时间：2023年8月27日 (日) 15:12</li> <li>封禁期满时间：2028年8月27日 (日) 15:12</li></ul> <p>您当前的IP地址是8.222.208.146。封禁的地址段是8.222.128.0/17。 </p> 请在进行咨询时提供上面所有详情。如果您认为您受到错误封禁，您可以在<a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/No_open_proxies" class="extiw" title="m:Special:MyLanguage/No open proxies">禁止公开代理</a>全域方针里找到更多信息和说明。您也可以<a href="https://meta.wikimedia.org/wiki/Steward_requests/Global" class="extiw" title="m:Steward requests/Global">到元维基发布复查申请</a>讨论该封禁。您还可以给<a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">监管员</a><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a>队列（<kbd>stewards@wikimedia.org</kbd>）发送提供上面所有详情的电子邮件。</li></ul><hr /> <p>您可以查看和复制此页面的源代码。 </p><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="zh" dir="ltr" name="wpTextbox1">[[数学]]上，'''二元关系'''（{{lang-en|Binary relation}}，或简称'''关系'''）用於讨论两种物件的连系。诸如[[算术]]中的「大於」及「等於」、[[几何学]]中的「相似」或[[集合论]]中的「为……之元素」、「为……之子集」。 == 定义 == 设<math>A,B</math>为集合，<math>A\times B</math>的任何子集称作<math>A</math>到<math>B</math>的二元关系，特别是当<math>A=B</math>时，称作<math>A</math>上的二元关系,一般记作<math>R</math>。若<math>R \subseteq A \times B</math>, <math>R</math>是从<math>A</math>到<math>B</math>的二元关系；若<math>R \subseteq A \times A</math>,那么<math>R</math>是<math> A</math>上的二元关系或是以[[一阶逻辑#量詞的簡寫|正式的邏輯符號]]表述為 :<math>(\forall r \in R)(\exists x)(\exists y)[\,r = (x,\,y)\,]</math> 例一：有四件物件 {''球''，''糖''，''车''，''枪''} 及四个人 {''甲''，''乙''，''丙，丁''} 。若甲擁有球、乙擁有糖、丙一無所有但丁擁有车，则「擁有」的二元关系可以寫為 :<math>R</math> = {(''球''，''甲''), (''糖''，''乙''), (''车''，''丁'')} 其中二元[[有序对]]的第一项是被擁有的物件，第二项是擁有者。例二：[[实数系|實數系]] <math>\mathbb{R}</math> 上的「大於關係」可定義為 :<math>> \,:= \{\, (a,\, b) \in {\mathbb{R}}^2 \,|\, (\exists r \in \mathbb{R})[\,(a = b + r) \wedge (r \neq 0)\,] \} </math> 由於習慣上 <math>(a,\,b) \in\, ></math> 通常都是寫為 <math>a > b</math> ，更一般來說，不引起混淆的話會把 <math>(x,\,y)\in R</math> 簡寫成 <math>xRy</math> 。 == 集合的關係 == [[集合 (数学)|集合]]<math>X</math>与集合<math>Y</math>上的二元关系則定義為 <math>R=(X, Y, G(R)) </math> ，当中 <math>G(R)\subseteq X \times Y</math> ( 請參見[[笛卡儿积]] ) ，称为 <math>R</math> 的'''图'''。若 <math>(x, y) \in G(R)</math> 则称 <math>x</math> 与 <math>y</math> 有关系 <math>R</math> ，并记作 <math>xRy</math> 或 <math>R(x, y)</math> 。但经常地我们把关系与其图等价起来，即若 <math>R \subseteq X \times Y</math> 则 <math>R</math> 是一个关系。话虽如此，我们很多時候索性把集合間的關係 <math>R</math> 定义为 <math>G(R)</math> 而 “有序对 <math> (x,y) \in G(R) </math> ” 即是 “ <math>(x,y) \in R</math> ”。 == 特殊的二元关系 == 设<math>A</math>是一个集合，则 # 空集<math>\varnothing</math>称作<math>A</math>上的'''空关系''' # <math>E_{A} = A \times A</math>称作<math>A</math>上的'''全域关系'''（'''完全關係'''） # <math>I_{A} = \{(x, x)|x \in A\}</math>称作<math>A</math>上的'''恒等关系''' == 关系矩阵 == 设<math>X = \{x_1, x_2, \ldots, x_n\}</math>及<math>Y = \{y_1, y_2, \ldots, y_m\}</math>，<math>R</math>是<math>X</math> 和<math> Y</math>上的关系，令 :<math> r_{ij} = \begin{cases} 1 & (x_i, y_j) \in R \\ 0 & (x_i, y_j) \notin R \end{cases} </math> 则[[0,1矩阵]] :<math> (r_{ij}) = \begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1m} \\ r_{21} & r_{22} & \cdots & r_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ r_{n1} & r_{n2} & \cdots & r_{nm} \end{bmatrix} </math> 称为<math>R</math>的'''关系矩阵'''，记作<math>M_{R}</math>。 == 关系图 == 设<math>A = \{x_1, x_2, \ldots, x_n\}</math>，<math>R</math>是<math>A</math>上的关系，令[[图 (数学)|图]]<math>G=(V, E)</math>，其中[[顶点 (图论)|顶点]]集合<math>V = A</math>，边集合为<math>E</math>，且对于任意的<math>x_i, x_j \in V</math>，满足<math>(x_i, x_j) \in E</math>当且仅当<math>(x_i, x_j) \in R</math>。则称图<math>G</math>是关系<math>R</math>的'''关系图'''，记作<math>G_R</math>。 == 运算 == 关系的基本运算有以下几种： * 设<math>R</math>为二元关系，<math>R</math>中所有[[有序对]]的第一元素构成的集合称为<math>R</math>的'''定义域'''，记作<math>\mbox{dom}(R)</math>。形式化表示为 :<math> \mbox{dom}(R) = \{\, x\,|\,(\exists y)[\, (x, y) \in R\,]\, \} </math> * 设<math>R</math>为二元关系，<math>R</math>中所有[[有序对]]的第二元素构成的集合称为<math>R</math>的'''值域'''，记作<math>\mbox{ran}(R)</math>。形式化表示为 :<math> \mbox{ran}(R) = \{\,y\,|\,(\exists x)[\,(x, y) \in R \,]\, \} </math> * 设<math>R</math>为二元关系，<math>R</math>的[[定义域]]和[[值域]]的并集称作<math>R</math>的'''域'''，记作<math>\mbox{fld}(R)</math>，形式化表示为 :<math> \mbox{fld}(R) = \mbox{dom}(R)\cup\mbox{ran}(R) </math> * 设<math>R</math>为二元关系，<math>R</math>的'''逆关系'''，简称<math>R</math>的'''逆'''，记作<math>R^{-1}</math>，其中 :<math> R^{-1} = \{\,p\,|\,(\exists x)(\exists y)[\,(x, y) \in R \wedge p = (y, x)\,]\,\} </math> * 设<math>F, G</math>为二元关系，<math>G</math>與<math>F</math>的'''[[合成關係]]'''记作<math>F \circ G</math>，其中 :<math> F \circ G = \{ (x, y) | \exists t,~(x, t) \in F \wedge (t, y) \in G \} </math> * 设<math>R</math>为二元关系，<math>A</math>是一个集合。<math>R</math>在<math>A</math>上的'''限制'''记作<math>R \upharpoonright A</math>，其中 :<math> R \upharpoonright A = \{ (x, y) | (x, y) \in R \wedge x \in A\} </math> * 设<math>R</math>为二元关系，<math>A</math>是一个集合。<math>A</math>在<math>R</math>下的'''像'''记作<math>R[A]</math>，其中 :<math> R[A] = \mbox{ran}(R \upharpoonright A) </math> * 设<math>R</math>为<math>A</math>上的二元关系，在右复合的基础上可以定义关系的'''幂运算'''： :<math>R^0 = I_A \ </math> <br><math>R^{n+1} = R^n \circ R \ </math> == 性质 == 关系的性质主要有以下五种： * [[自反关系|自反性]]：<math>\forall x \in A,~(x, x) \in R</math> :在集合X上的关系R，如对任意<math>x \in X </math>，有<math>(x,x) \in R </math>，则称R是自反的。 * 非自反性（自反性的否定的強型式）：<math>\forall x \in A,~~(x, x) \notin R</math> :在集合X上的关系R，如对任意<math>x \in X</math>，有<math>(x,x) \notin R</math>，则称R是非自反的。 * [[对称关系|对称性]]：<math>\forall x, y \in A,~(x, y) \in R \implies (y, x) \in R</math> :在集合X上的关系R，如果有<math>(x,y) \in R</math>且<math>x \neq y</math>必有<math>(y,x) \in R</math>，则称R是对称的。 * [[反对称关系|反对称性]]（不是對稱性的否定）：<math>\forall x, y \in A,~((x, y) \in R \wedge (y, x) \in R) \implies x = y</math> * [[非对称关系|非對稱性]]（對稱性的否定的強型式）：<math>\forall x, y \in A,~(x, y) \in R \implies (y, x) \notin R</math> :非對稱性是滿足非自反性的反對稱性。 * [[传递关系|传递性]]：<math>\forall x, y, z \in A, ~((x, y) \in R \wedge (y, z) \in R) \implies (x, z) \in R</math> 设<math>R</math>为集合<math>A</math>上的关系，下面给出<math>R</math>的五种性质成立的充要条件： # <math>R</math>在<math>A</math>上自反，当且仅当<math>I_A \subseteq R</math> # <math>R</math>在<math>A</math>上非自反，当且仅当<math>R \cap I_{A} = \emptyset</math> # <math>R</math>在<math>A</math>上对称，当且仅当<math>R = R^{-1} \ </math> # <math>R</math>在<math>A</math>上反对称，当且仅当<math>R \cap R^{-1} \subseteq I_{A}</math> # <math>R</math>在<math>A</math>上非對稱，当且仅当<math>R \cap R^{-1} = \emptyset</math> # <math>R</math>在<math>A</math>上传递，当且仅当<math>R \circ R \subseteq R</math> == 闭包 == 设<math>R</math>是非空集合<math>A</math>上的关系，<math>R</math>的自反（对称或传递）'''闭包'''是<math>A</math>上的关系<math>R'</math>，满足 # <math>R'</math>是自反的（对称的或传递的） # <math>R \subseteq R'</math> # 对<math>A</math>上任何包含<math>R</math>的自反（对称或传递）关系<math>R''</math>有<math>R' \subseteq R''</math> 一般将<math>R</math>的自反闭包记作<math>r(R)</math>，对称闭包记作<math>s(R)</math>，[[传递闭包]]记作<math>t(R)</math>。下列三个定理给出了构造闭包的方法： # <math>r(R) = R \cup R^0</math> # <math>s(R) = R \cup R^{-1}</math> # <math>t(R) = R \cup R^2 \cup R^3 \cup \cdots</math> 对于有限集合<math>A</math>上的关系<math>R</math>，存在一个正整数<math>r</math>，使得 :<math>t(R) = R \cup R^2 \cup \cdots \cup R^r</math> 求传递闭包是图论中一个非常重要的问题，例如给定了一个城市的交通地图，可利用求传递闭包的方法获知任意两个地点之间是否有路相连通。可以直接利用关系矩阵相乘来求传递闭包，但那样做复杂度比较高；好一点的办法是在计算矩阵相乘的时候用[[分治法]]降低时间复杂度；但最好的方法是利用基于[[动态规划]]的[[Floyd-Warshall算法]]来求传递闭包。 == 参见 == * [[有序对]] * [[二元集合]] * [[笛卡儿积]] * [[偏序关系]] * [[等价关系]] * [[相容关系]] [[Category:数学关系]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p>本页<a 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