{"id":3095,"date":"2025-06-07T10:02:47","date_gmt":"2025-06-07T10:02:47","guid":{"rendered":"https:\/\/diznr.com\/?p=3095"},"modified":"2025-06-07T10:02:47","modified_gmt":"2025-06-07T10:02:47","slug":"part-01-discrete-mathematics-for-gate-partial-order-relations-and-its-representation-matrix","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/part-01-discrete-mathematics-for-gate-partial-order-relations-and-its-representation-matrix\/","title":{"rendered":"Part 01-Discrete mathematics for gate-Partial Order Relations and it&#8217;s matrix representation"},"content":{"rendered":"<p>Part 01-Discrete mathematics for gate-Partial Order Relations and it&#8217;s matrix representation<\/p>\n<p>[fvplayer id=&#8221;250&#8243;]<\/p>\n<h3 class=\"\" data-start=\"0\" data-end=\"93\"><strong data-start=\"4\" data-end=\"61\">Partial Order Relations and Its Matrix Representation<\/strong> (Discrete Mathematics for GATE)<\/h3>\n<p class=\"\" data-start=\"95\" data-end=\"269\"><strong data-start=\"95\" data-end=\"141\">1. Introduction to Partial Order Relations<\/strong><br data-start=\"141\" data-end=\"144\" \/>A <strong data-start=\"146\" data-end=\"180\">partial order relation (poset)<\/strong> is a binary relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> on a set <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> that satisfies the following properties:<\/p>\n<ul data-start=\"273\" data-end=\"463\">\n<li class=\"\" data-start=\"273\" data-end=\"325\">\n<p class=\"\" data-start=\"275\" data-end=\"325\"><strong data-start=\"275\" data-end=\"290\">Reflexivity<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">aRaaRa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> for all <span class=\"katex\"><span class=\"katex-mathml\">a\u2208Sa \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"329\" data-end=\"395\">\n<p class=\"\" data-start=\"331\" data-end=\"395\"><strong data-start=\"331\" data-end=\"347\">Antisymmetry<\/strong>: If <span class=\"katex\"><span class=\"katex-mathml\">aRbaRb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bRabRa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">a=ba = b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"399\" data-end=\"463\">\n<p class=\"\" data-start=\"401\" data-end=\"463\"><strong data-start=\"401\" data-end=\"417\">Transitivity<\/strong>: If <span class=\"katex\"><span class=\"katex-mathml\">aRbaRb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bRcbRc<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">aRcaRc<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"465\" data-end=\"569\">A set <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> along with a partial order relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is called a <strong data-start=\"535\" data-end=\"568\">partially ordered set (poset)<\/strong>.<\/p>\n<p class=\"\" data-start=\"576\" data-end=\"767\"><strong data-start=\"576\" data-end=\"631\">2. Matrix Representation of Partial Order Relations<\/strong><br data-start=\"631\" data-end=\"634\" \/>The <strong data-start=\"638\" data-end=\"657\">relation matrix<\/strong> of a partial order relation on a finite set is a <strong data-start=\"707\" data-end=\"724\">square matrix<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">MRM_R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">R<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> of size <span class=\"katex\"><span class=\"katex-mathml\">n\u00d7nn \\times n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span>, where:<\/p>\n<ul data-start=\"771\" data-end=\"842\">\n<li class=\"\" data-start=\"771\" data-end=\"808\">\n<p class=\"\" data-start=\"773\" data-end=\"808\"><span class=\"katex\"><span class=\"katex-mathml\">Mij=1M_{ij} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> if <span class=\"katex\"><span class=\"katex-mathml\">aiRaja_i R a_j<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">j<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"812\" data-end=\"842\">\n<p class=\"\" data-start=\"814\" data-end=\"842\"><span class=\"katex\"><span class=\"katex-mathml\">Mij=0M_{ij} = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span> otherwise<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"844\" data-end=\"949\"><strong data-start=\"844\" data-end=\"856\">Example:<\/strong><br data-start=\"856\" data-end=\"859\" \/>Let <span class=\"katex\"><span class=\"katex-mathml\">S={1,2,3}S = \\{1, 2, 3\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> with a relation <span class=\"katex\"><span class=\"katex-mathml\">R={(1,1),(2,2),(3,3),(1,2),(2,3)}R = \\{(1,1), (2,2), (3,3), (1,2), (2,3)\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span>.<\/p>\n<p class=\"\" data-start=\"951\" data-end=\"988\">The <strong data-start=\"955\" data-end=\"984\">relation matrix <span class=\"katex\"><span class=\"katex-mathml\">MRM_R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">R<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> is:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">MR=[110011001]M_R = \\begin{bmatrix} 1 &amp; 1 &amp; 0 \\\\ 0 &amp; 1 &amp; 1 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">R<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">100<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">110<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"col-align-c\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">011<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"1068\" data-end=\"1259\">\n<li class=\"\" data-start=\"1068\" data-end=\"1124\">\n<p class=\"\" data-start=\"1070\" data-end=\"1124\">The <strong data-start=\"1074\" data-end=\"1101\">diagonal elements are 1<\/strong>, ensuring reflexivity.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1125\" data-end=\"1181\">\n<p class=\"\" data-start=\"1127\" data-end=\"1181\">The matrix is <strong data-start=\"1141\" data-end=\"1158\">not symmetric<\/strong>, proving antisymmetry.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1182\" data-end=\"1259\">\n<p class=\"\" data-start=\"1184\" data-end=\"1259\">The <strong data-start=\"1188\" data-end=\"1210\">transitive closure<\/strong> can be found using the <strong data-start=\"1234\" data-end=\"1258\">Warshall\u2019s Algorithm<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"1266\" data-end=\"1371\"><strong data-start=\"1266\" data-end=\"1301\">3. Hasse Diagram Representation<\/strong><br data-start=\"1301\" data-end=\"1304\" \/>A <strong data-start=\"1306\" data-end=\"1323\">Hasse diagram<\/strong> is a graphical representation of a poset where:<\/p>\n<ul data-start=\"1375\" data-end=\"1510\">\n<li class=\"\" data-start=\"1375\" data-end=\"1409\">\n<p class=\"\" data-start=\"1377\" data-end=\"1409\"><strong data-start=\"1377\" data-end=\"1390\">Reflexive<\/strong> edges are omitted.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1413\" data-end=\"1463\">\n<p class=\"\" data-start=\"1415\" data-end=\"1463\"><strong data-start=\"1415\" data-end=\"1435\">Transitive edges<\/strong> are removed for simplicity.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1467\" data-end=\"1510\">\n<p class=\"\" data-start=\"1469\" data-end=\"1510\"><strong data-start=\"1469\" data-end=\"1481\">Elements<\/strong> are arranged hierarchically.<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"1512\" data-end=\"1560\">For the above example, the <strong data-start=\"1539\" data-end=\"1556\">Hasse Diagram<\/strong> is:<\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">3<br \/>\n\u2191<br \/>\n2<br \/>\n\u2191<br \/>\n1<br \/>\n<\/code><\/div>\n<\/div>\n<p class=\"\" data-start=\"1596\" data-end=\"1642\"><strong data-start=\"1596\" data-end=\"1642\">4. Applications of Partial Order Relations<\/strong><\/p>\n<ul data-start=\"1643\" data-end=\"1787\">\n<li class=\"\" data-start=\"1643\" data-end=\"1670\">\n<p class=\"\" data-start=\"1645\" data-end=\"1670\"><strong data-start=\"1645\" data-end=\"1668\">Scheduling Problems<\/strong><\/p>\n<\/li>\n<li class=\"\" data-start=\"1671\" data-end=\"1702\">\n<p class=\"\" data-start=\"1673\" data-end=\"1702\"><strong data-start=\"1673\" data-end=\"1700\">Hierarchical Structures<\/strong><\/p>\n<\/li>\n<li class=\"\" data-start=\"1703\" data-end=\"1748\">\n<p class=\"\" data-start=\"1705\" data-end=\"1748\"><strong data-start=\"1705\" data-end=\"1746\">Dependency Graphs in Computer Science<\/strong><\/p>\n<\/li>\n<li class=\"\" data-start=\"1749\" data-end=\"1787\">\n<p class=\"\" data-start=\"1751\" data-end=\"1787\"><strong data-start=\"1751\" data-end=\"1785\">Data Organization in Databases<\/strong><\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"1789\" data-end=\"1875\">Would you like a <strong data-start=\"1806\" data-end=\"1830\">detailed explanation<\/strong> or <strong data-start=\"1834\" data-end=\"1857\">GATE-level problems<\/strong> on this topic?<\/p>\n<h3 data-start=\"1789\" data-end=\"1875\"><a href=\"https:\/\/niamt.ac.in\/WriteReadData\/Mathematics%20(Discrete%20Structure).pdf\" target=\"_blank\" rel=\"noopener\">Part 01-Discrete mathematics for gate-Partial Order Relations and it&#8217;s matrix representation<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/mrcet.com\/downloads\/digital_notes\/CSE\/II%20Year\/DISCRETE%20MATHEMATICS%20NOTES.pdf\" target=\"_blank\" rel=\"noopener\">DIGITAL NOTES ON Discrete Mathematics B.TECH II YEAR<\/a><\/h3>\n<h3><a href=\"https:\/\/bsh.gecgudlavalleru.ac.in\/images\/admin\/pdf\/1594720102_II%20-%20I%20-%20DMS.pdf\" target=\"_blank\" rel=\"noopener\">Discrete Mathematical Structures Unit-1<\/a><\/h3>\n<p data-start=\"0\" data-end=\"53\">Here is a detailed, student-friendly explanation for:<\/p>\n<hr data-start=\"55\" data-end=\"58\" \/>\n<h2 data-start=\"60\" data-end=\"111\">\ud83d\udcd8 <strong data-start=\"66\" data-end=\"109\">Part 01 \u2013 Discrete Mathematics for GATE<\/strong><\/h2>\n<h3 data-start=\"112\" data-end=\"185\">\ud83c\udfaf <strong data-start=\"119\" data-end=\"183\">Topic: Partial Order Relations and Its Matrix Representation<\/strong><\/h3>\n<p data-start=\"186\" data-end=\"324\"><strong data-start=\"186\" data-end=\"198\">Language<\/strong>: English (can also be provided in Hindi upon request)<br data-start=\"252\" data-end=\"255\" \/><strong data-start=\"255\" data-end=\"269\">Useful for<\/strong>: GATE, B.Tech, UGC-NET, BCA, MCA, Competitive CS Exams<\/p>\n<hr data-start=\"326\" data-end=\"329\" \/>\n<h2 data-start=\"331\" data-end=\"377\">\ud83d\udd37 <strong data-start=\"337\" data-end=\"377\">1. What is a Relation in Set Theory?<\/strong><\/h2>\n<p data-start=\"379\" data-end=\"494\">A <strong data-start=\"381\" data-end=\"393\">relation<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> from a set <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> to <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> is a subset of <span class=\"katex\"><span class=\"katex-mathml\">A\u00d7AA \\times A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>, i.e., a set of ordered pairs.<\/p>\n<p data-start=\"496\" data-end=\"593\">Example:<br data-start=\"504\" data-end=\"507\" \/>Let <span class=\"katex\"><span class=\"katex-mathml\">A={1,2,3}A = \\{1, 2, 3\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><br data-start=\"532\" data-end=\"535\" \/>A relation <span class=\"katex\"><span class=\"katex-mathml\">R={(1,1),(2,2),(3,3),(1,2),(2,3)}R = \\{(1,1), (2,2), (3,3), (1,2), (2,3)\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span><\/p>\n<hr data-start=\"595\" data-end=\"598\" \/>\n<h2 data-start=\"600\" data-end=\"646\">\ud83d\udd36 <strong data-start=\"606\" data-end=\"646\">2. What is a Partial Order Relation?<\/strong><\/h2>\n<p data-start=\"648\" data-end=\"752\">A <strong data-start=\"650\" data-end=\"676\">Partial Order Relation<\/strong> on a set <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> is a binary relation <span class=\"katex\"><span class=\"katex-mathml\">R\u2286A\u00d7AR \\subseteq A \\times A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mrel\">\u2286<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> that is:<\/p>\n<ol data-start=\"754\" data-end=\"970\">\n<li data-start=\"754\" data-end=\"808\">\n<p data-start=\"757\" data-end=\"808\"><strong data-start=\"757\" data-end=\"770\">Reflexive<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">(a,a)\u2208R(a, a) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> \u2200 <span class=\"katex\"><span class=\"katex-mathml\">a\u2208Aa \\in A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"809\" data-end=\"888\">\n<p data-start=\"812\" data-end=\"888\"><strong data-start=\"812\" data-end=\"829\">Antisymmetric<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">(a,b)\u2208R(a, b) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">(b,a)\u2208R(b, a) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> \u27f9 <span class=\"katex\"><span class=\"katex-mathml\">a=ba = b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"889\" data-end=\"970\">\n<p data-start=\"892\" data-end=\"970\"><strong data-start=\"892\" data-end=\"906\">Transitive<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">(a,b)\u2208R(a, b) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">(b,c)\u2208R(b, c) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> \u27f9 <span class=\"katex\"><span class=\"katex-mathml\">(a,c)\u2208R(a, c) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p data-start=\"972\" data-end=\"1145\">\ud83d\udcdd If a relation satisfies these three properties, it is called a <strong data-start=\"1038\" data-end=\"1055\">partial order<\/strong>, and the set <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> with relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is called a <strong data-start=\"1111\" data-end=\"1144\">partially ordered set (poset)<\/strong>.<\/p>\n<hr data-start=\"1147\" data-end=\"1150\" \/>\n<h2 data-start=\"1152\" data-end=\"1200\">\ud83d\udd37 <strong data-start=\"1158\" data-end=\"1200\">3. Matrix Representation of a Relation<\/strong><\/h2>\n<p data-start=\"1202\" data-end=\"1290\">To represent a relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> on set <span class=\"katex\"><span class=\"katex-mathml\">A={a1,a2,&#8230;,an}A = \\{a_1, a_2, &#8230;, a_n\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">&#8230;<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> as a <strong data-start=\"1279\" data-end=\"1289\">matrix<\/strong>:<\/p>\n<ul data-start=\"1292\" data-end=\"1405\">\n<li data-start=\"1292\" data-end=\"1337\">\n<p data-start=\"1294\" data-end=\"1337\">Create an <span class=\"katex\"><span class=\"katex-mathml\">n\u00d7nn \\times n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> matrix <span class=\"katex\"><span class=\"katex-mathml\">MRM_R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">R<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"1338\" data-end=\"1405\">\n<p data-start=\"1340\" data-end=\"1405\"><span class=\"katex\"><span class=\"katex-mathml\">Mij=1M_{ij} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> if <span class=\"katex\"><span class=\"katex-mathml\">(ai,aj)\u2208R(a_i, a_j) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">j<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span>, else <span class=\"katex\"><span class=\"katex-mathml\">Mij=0M_{ij} = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1407\" data-end=\"1410\" \/>\n<h3 data-start=\"1412\" data-end=\"1431\">\ud83d\udccc <strong data-start=\"1419\" data-end=\"1431\">Example:<\/strong><\/h3>\n<p data-start=\"1433\" data-end=\"1458\">Let <span class=\"katex\"><span class=\"katex-mathml\">A={1,2,3}A = \\{1, 2, 3\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1460\" data-end=\"1516\">Define relation <span class=\"katex\"><span class=\"katex-mathml\">R={(1,1),(2,2),(3,3),(1,2)}R = \\{(1,1), (2,2), (3,3), (1,2)\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"1518\" data-end=\"1546\">\u2705 Matrix Representation:<\/h3>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1548\" data-end=\"1637\">\n<thead data-start=\"1548\" data-end=\"1565\">\n<tr data-start=\"1548\" data-end=\"1565\">\n<th data-start=\"1548\" data-end=\"1552\" data-col-size=\"sm\"><\/th>\n<th data-start=\"1552\" data-end=\"1556\" data-col-size=\"sm\">1<\/th>\n<th data-start=\"1556\" data-end=\"1560\" data-col-size=\"sm\">2<\/th>\n<th data-start=\"1560\" data-end=\"1565\" data-col-size=\"sm\">3<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1584\" data-end=\"1637\">\n<tr data-start=\"1584\" data-end=\"1601\">\n<td data-start=\"1584\" data-end=\"1588\" data-col-size=\"sm\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1588\" data-end=\"1592\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1592\" data-end=\"1596\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1596\" data-end=\"1601\">0<\/td>\n<\/tr>\n<tr data-start=\"1602\" data-end=\"1619\">\n<td data-start=\"1602\" data-end=\"1606\" data-col-size=\"sm\">2<\/td>\n<td data-start=\"1606\" data-end=\"1610\" data-col-size=\"sm\">0<\/td>\n<td data-start=\"1610\" data-end=\"1614\" data-col-size=\"sm\">1<\/td>\n<td data-start=\"1614\" data-end=\"1619\" data-col-size=\"sm\">0<\/td>\n<\/tr>\n<tr data-start=\"1620\" data-end=\"1637\">\n<td data-start=\"1620\" data-end=\"1624\" data-col-size=\"sm\">3<\/td>\n<td data-start=\"1624\" data-end=\"1628\" data-col-size=\"sm\">0<\/td>\n<td data-col-size=\"sm\" data-start=\"1628\" data-end=\"1632\">0<\/td>\n<td data-col-size=\"sm\" data-start=\"1632\" data-end=\"1637\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"1639\" data-end=\"1642\" \/>\n<h2 data-start=\"1644\" data-end=\"1687\">\ud83d\udd37 <strong data-start=\"1650\" data-end=\"1687\">4. Checking Properties via Matrix<\/strong><\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1689\" data-end=\"2119\">\n<thead data-start=\"1689\" data-end=\"1774\">\n<tr data-start=\"1689\" data-end=\"1774\">\n<th data-start=\"1689\" data-end=\"1704\" data-col-size=\"sm\">Property<\/th>\n<th data-start=\"1704\" data-end=\"1774\" data-col-size=\"md\">Matrix Condition<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1862\" data-end=\"2119\">\n<tr data-start=\"1862\" data-end=\"1947\">\n<td data-start=\"1862\" data-end=\"1877\" data-col-size=\"sm\">Reflexive<\/td>\n<td data-start=\"1877\" data-end=\"1947\" data-col-size=\"md\">All diagonal elements <span class=\"katex\"><span class=\"katex-mathml\">Mii=1M_{ii} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ii<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"1948\" data-end=\"2033\">\n<td data-start=\"1948\" data-end=\"1963\" data-col-size=\"sm\">Antisymmetric<\/td>\n<td data-col-size=\"md\" data-start=\"1963\" data-end=\"2033\">If <span class=\"katex\"><span class=\"katex-mathml\">Mij=1M_{ij} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">Mji=1M_{ji} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ji<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">i=ji = j<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">i<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">j<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2034\" data-end=\"2119\">\n<td data-start=\"2034\" data-end=\"2049\" data-col-size=\"sm\">Transitive<\/td>\n<td data-start=\"2049\" data-end=\"2119\" data-col-size=\"md\">If <span class=\"katex\"><span class=\"katex-mathml\">Mij=1M_{ij} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ij<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">Mjk=1M_{jk} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">jk<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">Mik=1M_{ik} = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ik<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2121\" data-end=\"2124\" \/>\n<h2 data-start=\"2126\" data-end=\"2166\">\ud83e\udde0 <strong data-start=\"2132\" data-end=\"2166\">5. GATE Perspective Questions:<\/strong><\/h2>\n<ol data-start=\"2168\" data-end=\"2389\">\n<li data-start=\"2168\" data-end=\"2219\">\n<p data-start=\"2171\" data-end=\"2219\"><strong data-start=\"2171\" data-end=\"2219\">Identify whether a relation is partial order<\/strong><\/p>\n<\/li>\n<li data-start=\"2220\" data-end=\"2262\">\n<p data-start=\"2223\" data-end=\"2262\"><strong data-start=\"2223\" data-end=\"2262\">Find the matrix of a given relation<\/strong><\/p>\n<\/li>\n<li data-start=\"2263\" data-end=\"2334\">\n<p data-start=\"2266\" data-end=\"2334\"><strong data-start=\"2266\" data-end=\"2334\">Check whether a relation is reflexive, antisymmetric, transitive<\/strong><\/p>\n<\/li>\n<li data-start=\"2335\" data-end=\"2389\">\n<p data-start=\"2338\" data-end=\"2389\"><strong data-start=\"2338\" data-end=\"2389\">Count number of relations with given properties<\/strong><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"2391\" data-end=\"2394\" \/>\n<h2 data-start=\"2396\" data-end=\"2408\">\u2705 Summary<\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"2410\" data-end=\"2697\">\n<thead data-start=\"2410\" data-end=\"2441\">\n<tr data-start=\"2410\" data-end=\"2441\">\n<th data-start=\"2410\" data-end=\"2430\" data-col-size=\"sm\">Term<\/th>\n<th data-start=\"2430\" data-end=\"2441\" data-col-size=\"md\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2474\" data-end=\"2697\">\n<tr data-start=\"2474\" data-end=\"2553\">\n<td data-start=\"2474\" data-end=\"2494\" data-col-size=\"sm\">Partial Order<\/td>\n<td data-start=\"2494\" data-end=\"2553\" data-col-size=\"md\">A relation that is reflexive, antisymmetric, transitive<\/td>\n<\/tr>\n<tr data-start=\"2554\" data-end=\"2613\">\n<td data-start=\"2554\" data-end=\"2574\" data-col-size=\"sm\">Poset<\/td>\n<td data-start=\"2574\" data-end=\"2613\" data-col-size=\"md\">A set with a partial order relation<\/td>\n<\/tr>\n<tr data-start=\"2614\" data-end=\"2697\">\n<td data-start=\"2614\" data-end=\"2638\" data-col-size=\"sm\">Matrix Representation<\/td>\n<td data-start=\"2638\" data-end=\"2697\" data-col-size=\"md\">A binary matrix showing which pairs are in the relation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2699\" data-end=\"2702\" \/>\n<p data-start=\"2704\" data-end=\"2719\">Would you like:<\/p>\n<ul data-start=\"2720\" data-end=\"2854\">\n<li data-start=\"2720\" data-end=\"2761\">\n<p data-start=\"2722\" data-end=\"2761\">\ud83d\udcc4 A <strong data-start=\"2727\" data-end=\"2746\">PDF cheat sheet<\/strong> of this topic?<\/p>\n<\/li>\n<li data-start=\"2762\" data-end=\"2798\">\n<p data-start=\"2764\" data-end=\"2798\">\ud83d\udcca <strong data-start=\"2767\" data-end=\"2784\">Practice MCQs<\/strong> with answers?<\/p>\n<\/li>\n<li data-start=\"2799\" data-end=\"2854\">\n<p data-start=\"2801\" data-end=\"2854\">\ud83c\udfa5 A <strong data-start=\"2806\" data-end=\"2833\">video-style explanation<\/strong> in Hindi or English?<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2856\" data-end=\"2894\" data-is-last-node=\"\" data-is-only-node=\"\">Let me know how you&#8217;d like to proceed!<\/p>\n<h3 data-start=\"2856\" data-end=\"2894\"><a href=\"https:\/\/dpvipracollege.ac.in\/wp-content\/uploads\/2023\/01\/Discrete-Mathematical-Structures-2nd-Ed.pdf\" target=\"_blank\" rel=\"noopener\">Part 01-Discrete mathematics for gate-Partial Order Relations and it&#8217;s matrix representation<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.dbscience.org\/wp-content\/uploads\/2020\/03\/Discrete_Mathematical_Structures-Kolman.pdf\" target=\"_blank\" rel=\"noopener\">Bernard Kolman_ Robert C. Busby_ Sharon Cutler Ross- &#8230;<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Part 01-Discrete mathematics for gate-Partial Order Relations and it&#8217;s matrix representation [fvplayer id=&#8221;250&#8243;] Partial Order Relations and Its Matrix Representation (Discrete Mathematics for GATE) 1. Introduction to Partial Order RelationsA partial order relation (poset) is a binary relation RRR on a set SSS that satisfies the following properties: Reflexivity: aRaaRaaRa for all a\u2208Sa \\in Sa\u2208S [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-3095","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3095","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=3095"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3095\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}