{"id":3041,"date":"2025-06-07T14:59:22","date_gmt":"2025-06-07T14:59:22","guid":{"rendered":"https:\/\/diznr.com\/?p=3041"},"modified":"2025-06-07T14:59:22","modified_gmt":"2025-06-07T14:59:22","slug":"day-03part-03-discrete-mathematics-for-gate-how-to-convert-partial-order-relations-to-diagram-hasse","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-03part-03-discrete-mathematics-for-gate-how-to-convert-partial-order-relations-to-diagram-hasse\/","title":{"rendered":"Day 03Part 03-Discrete mathematics for gate-How to Convert Partial order Relations to hasse diagram."},"content":{"rendered":"<p>Day 03Part 03-Discrete mathematics for gate-How to Convert Partial order Relations to hasse diagram.<\/p>\n<p>[fvplayer id=&#8221;228&#8243;]<\/p>\n<p data-start=\"0\" data-end=\"147\">\u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 (Hasse Diagram) \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 (Partial Order Relations) \u0915\u094b \u0915\u0948\u0938\u0947 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u090f, \u0907\u0938\u0915\u093e \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 \u0924\u0930\u0940\u0915\u093e \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948:<\/p>\n<h3 data-start=\"149\" data-end=\"216\"><strong data-start=\"153\" data-end=\"214\">\u091a\u0930\u0923 1: \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 (Partial Order Relation) \u0915\u094b \u0938\u092e\u091d\u0947\u0902<\/strong><\/h3>\n<p data-start=\"217\" data-end=\"295\">\u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 \u090f\u0915 \u0938\u0947\u091f \u092a\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0924\u0940\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/p>\n<ol data-start=\"296\" data-end=\"626\">\n<li data-start=\"296\" data-end=\"392\"><strong data-start=\"299\" data-end=\"329\">\u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e (Reflexivity)<\/strong>: \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 <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> \u0915\u0947 \u0932\u093f\u090f, <span class=\"katex\"><span class=\"katex-mathml\">(a,a)(a, a)<\/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><\/span><\/span> \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/li>\n<li data-start=\"393\" data-end=\"510\"><strong data-start=\"396\" data-end=\"424\">\u092a\u093e\u0930\u0917\u092e\u094d\u092f\u0924\u093e (Transitivity)<\/strong>: \u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">(a,b)(a, b)<\/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><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(b,c)(b, c)<\/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><\/span><\/span> \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0939\u0948\u0902, \u0924\u094b <span class=\"katex\"><span class=\"katex-mathml\">(a,c)(a, c)<\/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><\/span><\/span> \u092d\u0940 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0939\u094b\u0917\u093e\u0964<\/li>\n<li data-start=\"511\" data-end=\"626\"><strong data-start=\"514\" data-end=\"542\">\u092a\u094d\u0930\u0924\u093f\u0938\u092e\u0924\u093e (Antisymmetry)<\/strong>: \u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">(a,b)(a, b)<\/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><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(b,a)(b, a)<\/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><\/span><\/span> \u0926\u094b\u0928\u094b\u0902 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0939\u0948\u0902, \u0924\u094b <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> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/li>\n<\/ol>\n<h3 data-start=\"628\" data-end=\"715\"><strong data-start=\"632\" data-end=\"713\">\u091a\u0930\u0923 2: \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 \u0915\u094b \u0928\u093f\u0930\u094d\u0926\u0947\u0936\u093f\u0924 \u0917\u094d\u0930\u093e\u092b\u093c (Directed Graph) \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0947\u0902<\/strong><\/h3>\n<p data-start=\"716\" data-end=\"850\">\u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0938\u092d\u0940 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u094b \u0928\u094b\u0921 (node) \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u090f\u0901 \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(a,b)(a, b)<\/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><\/span><\/span> \u0938\u0902\u092c\u0902\u0927 \u0939\u094b\u0928\u0947 \u092a\u0930 \u090f\u0915 \u0928\u093f\u0930\u094d\u0926\u0947\u0936\u093f\u0924 \u0915\u093f\u0928\u093e\u0930\u093e (directed edge) \u091c\u094b\u0921\u093c\u0947\u0902\u0964<\/p>\n<h3 data-start=\"852\" data-end=\"915\"><strong data-start=\"856\" data-end=\"913\">\u091a\u0930\u0923 3: \u0905\u0928\u093e\u0935\u0936\u094d\u092f\u0915 \u0915\u093f\u0928\u093e\u0930\u0947 \u0939\u091f\u093e\u090f\u0901 (Remove Redundant Edges)<\/strong><\/h3>\n<ul data-start=\"916\" data-end=\"1117\">\n<li data-start=\"916\" data-end=\"1044\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">(a,b)(a, b)<\/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><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(b,c)(b, c)<\/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><\/span><\/span> \u0939\u0948\u0902 \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(a,c)(a, c)<\/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><\/span><\/span> \u092d\u0940 \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0924\u094b \u092a\u093e\u0930\u0917\u092e\u094d\u092f\u0924\u093e \u0915\u0947 \u0915\u093e\u0930\u0923 <span class=\"katex\"><span class=\"katex-mathml\">(a,c)(a, c)<\/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><\/span><\/span> \u0915\u0947 \u0915\u093f\u0928\u093e\u0930\u0947 \u0915\u094b \u0939\u091f\u093e \u0926\u0947\u0902\u0964<\/li>\n<li data-start=\"1045\" data-end=\"1117\">\u092a\u0930\u093e\u0935\u0930\u094d\u0924\u0928 (Reflexive) \u0939\u094b\u0928\u0947 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0938\u092d\u0940 <span class=\"katex\"><span class=\"katex-mathml\">(a,a)(a, a)<\/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><\/span><\/span> \u0932\u0942\u092a\u094d\u0938 \u0915\u094b \u0939\u091f\u093e \u0926\u0947\u0902\u0964<\/li>\n<\/ul>\n<h3 data-start=\"1119\" data-end=\"1151\"><strong data-start=\"1123\" data-end=\"1149\">\u091a\u0930\u0923 4: \u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 \u092c\u0928\u093e\u090f\u0902<\/strong><\/h3>\n<ul data-start=\"1152\" data-end=\"1397\">\n<li data-start=\"1152\" data-end=\"1224\">\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0915\u094b <strong data-start=\"1171\" data-end=\"1202\">\u0938\u094d\u0924\u0930\u0940\u092f \u0915\u094d\u0930\u092e (Layered Order)<\/strong> \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0947\u0902\u0964<\/li>\n<li data-start=\"1225\" data-end=\"1303\">\u0915\u0947\u0935\u0932 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0906\u0930\u0947\u0916 \u092c\u0928\u093e\u090f \u0930\u0916\u0947\u0902, \u092f\u093e\u0928\u0940, \u0938\u092c\u0938\u0947 \u0915\u092e \u0906\u0935\u0936\u094d\u092f\u0915 \u0915\u093f\u0928\u093e\u0930\u094b\u0902 \u0915\u094b \u092c\u0928\u093e\u090f \u0930\u0916\u0947\u0902\u0964<\/li>\n<li data-start=\"1304\" data-end=\"1397\">\u0906\u0930\u0947\u0916 \u0915\u094b \u092c\u093f\u0928\u093e \u0926\u093f\u0936\u093e (undirected edges) \u0915\u0947 \u092c\u0928\u093e\u090f\u0902 \u0914\u0930 \u0907\u0938\u0947 \u090a\u092a\u0930 \u0938\u0947 \u0928\u0940\u091a\u0947 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0947\u0902\u0964<\/li>\n<\/ul>\n<h3 data-start=\"1399\" data-end=\"1416\"><strong data-start=\"1403\" data-end=\"1414\">\u0909\u0926\u093e\u0939\u0930\u0923:<\/strong><\/h3>\n<p data-start=\"1417\" data-end=\"1560\">\u092e\u093e\u0928\u093e \u0915\u093f \u090f\u0915 \u0938\u0947\u091f <span class=\"katex\"><span class=\"katex-mathml\">S={1,2,3,4}S = \\{1, 2, 3, 4\\}<\/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=\"mpunct\">,<\/span><span class=\"mord\">4<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> \u092a\u0930 \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 <span class=\"katex\"><span class=\"katex-mathml\">R={(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(2,4),(3,4)}R = \\{(1,1), (2,2), (3,3), (4,4), (1,2), (1,3), (2,4), (3,4)\\}<\/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\">4<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">4<\/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\">1<\/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\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">4<\/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\">4<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span> \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964<\/p>\n<ol data-start=\"1562\" data-end=\"1797\">\n<li data-start=\"1562\" data-end=\"1648\">\n<p data-start=\"1565\" data-end=\"1590\">\u0928\u093f\u0930\u094d\u0926\u0947\u0936\u093f\u0924 \u0917\u094d\u0930\u093e\u092b\u093c \u092c\u0928\u093e\u090f\u0901:<\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary dark:bg-gray-950\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between rounded-t-[5px] h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none\"><\/div>\n<div class=\"sticky top-9 md:top-[5.75rem]\">\n<div class=\"absolute bottom-0 right-2 flex h-9 items-center\">\n<div class=\"flex items-center rounded bg-token-sidebar-surface-primary px-2 font-sans text-xs text-token-text-secondary dark:bg-token-main-surface-secondary\"><span class=\"\" data-state=\"closed\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><\/span><span class=\"\" data-state=\"closed\"><button class=\"flex select-none items-center gap-1\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">1 \u2192 2<br \/>\n1 \u2192 3<br \/>\n2 \u2192 4<br \/>\n3 \u2192 4<br \/>\n<\/code><\/div>\n<\/div>\n<\/li>\n<li data-start=\"1650\" data-end=\"1711\">\n<p data-start=\"1653\" data-end=\"1711\">\u0905\u0928\u093e\u0935\u0936\u094d\u092f\u0915 \u0915\u093f\u0928\u093e\u0930\u094b\u0902 \u0915\u094b \u0939\u091f\u093e\u090f\u0901 (\u0905\u0917\u0930 \u0915\u094b\u0908 \u092a\u093e\u0930\u0917\u093e\u092e\u0940 \u0938\u0902\u092c\u0902\u0927 \u0939\u094b \u0924\u094b)\u0964<\/p>\n<\/li>\n<li data-start=\"1713\" data-end=\"1797\">\n<p data-start=\"1716\" data-end=\"1734\">\u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 \u092c\u0928\u093e\u090f\u0901:<\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary dark:bg-gray-950\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between rounded-t-[5px] h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none\"><\/div>\n<div class=\"sticky top-9 md:top-[5.75rem]\">\n<div class=\"absolute bottom-0 right-2 flex h-9 items-center\">\n<div class=\"flex items-center rounded bg-token-sidebar-surface-primary px-2 font-sans text-xs text-token-text-secondary dark:bg-token-main-surface-secondary\"><span class=\"\" data-state=\"closed\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><\/span><span class=\"\" data-state=\"closed\"><button class=\"flex select-none items-center gap-1\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"!whitespace-pre\">  4<br \/>\n\/ \\<br \/>\n2   3<br \/>\n\\ \/<br \/>\n1<br \/>\n<\/code><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<p data-start=\"1798\" data-end=\"1896\">\u092f\u0939 \u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 <strong data-start=\"1811\" data-end=\"1826\">\u090a\u092a\u0930 \u0938\u0947 \u0928\u0940\u091a\u0947<\/strong> \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u094b \u0926\u093f\u0916\u093e\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0906\u0935\u0936\u094d\u092f\u0915 \u0915\u093f\u0928\u093e\u0930\u0947 \u0930\u0916\u0947 \u0917\u090f \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"1898\" data-end=\"2005\" data-is-last-node=\"\" data-is-only-node=\"\">\u0905\u0917\u0930 \u0906\u092a\u0915\u094b \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u0909\u0926\u093e\u0939\u0930\u0923 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0928\u0940 \u0939\u0948 \u092f\u093e \u0915\u093f\u0938\u0940 \u091c\u091f\u093f\u0932 \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0915\u094b \u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 \u092e\u0947\u0902 \u092c\u0926\u0932\u0928\u093e \u0939\u0948, \u0924\u094b \u092c\u0924\u093e\u0907\u090f!<\/p>\n<p class=\"\" data-start=\"0\" data-end=\"160\">Here&#8217;s a clear and concise explanation for <strong data-start=\"43\" data-end=\"159\">Day 03 Part 03 \u2013 Discrete Mathematics for GATE CSE\/IT: How to Convert Partial Order Relations to a Hasse Diagram<\/strong>:<\/p>\n<hr class=\"\" data-start=\"162\" data-end=\"165\" \/>\n<h2 class=\"\" data-start=\"167\" data-end=\"228\">\ud83d\udcd8 <strong data-start=\"173\" data-end=\"183\">Topic:<\/strong> Partial Order Relations and Hasse Diagrams<\/h2>\n<p class=\"\" data-start=\"229\" data-end=\"321\"><strong data-start=\"229\" data-end=\"241\">Subject:<\/strong> Discrete Mathematics<br data-start=\"262\" data-end=\"265\" \/><strong data-start=\"265\" data-end=\"276\">Target:<\/strong> GATE CSE\/IT, UGC NET, and B.Tech CS Students<\/p>\n<hr class=\"\" data-start=\"323\" data-end=\"326\" \/>\n<h3 class=\"\" data-start=\"328\" data-end=\"374\">\u2705 <strong data-start=\"334\" data-end=\"374\">1. What is a Partial Order Relation?<\/strong><\/h3>\n<p class=\"\" data-start=\"376\" data-end=\"467\">A <strong data-start=\"378\" data-end=\"404\">partial order relation<\/strong> (denoted \u2264) on a set <strong data-start=\"426\" data-end=\"431\">S<\/strong> is a binary relation <strong data-start=\"453\" data-end=\"458\">R<\/strong> that is:<\/p>\n<ul data-start=\"469\" data-end=\"628\">\n<li class=\"\" data-start=\"469\" data-end=\"506\">\n<p class=\"\" data-start=\"471\" data-end=\"506\"><strong data-start=\"471\" data-end=\"485\">Reflexive:<\/strong> \u2200a \u2208 S, (a, a) \u2208 R<\/p>\n<\/li>\n<li class=\"\" data-start=\"507\" data-end=\"566\">\n<p class=\"\" data-start=\"509\" data-end=\"566\"><strong data-start=\"509\" data-end=\"527\">Antisymmetric:<\/strong> If (a, b) \u2208 R and (b, a) \u2208 R \u21d2 a = b<\/p>\n<\/li>\n<li class=\"\" data-start=\"567\" data-end=\"628\">\n<p class=\"\" data-start=\"569\" data-end=\"628\"><strong data-start=\"569\" data-end=\"584\">Transitive:<\/strong> If (a, b) \u2208 R and (b, c) \u2208 R \u21d2 (a, c) \u2208 R<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"630\" data-end=\"755\">\ud83d\udccc Example:<br data-start=\"641\" data-end=\"644\" \/>Let <strong data-start=\"648\" data-end=\"671\">A = {1, 2, 3, 4, 6}<\/strong>, and define <strong data-start=\"684\" data-end=\"714\">R = {(a, b) | a divides b}<\/strong><br data-start=\"714\" data-end=\"717\" \/>This relation is a partial order on A.<\/p>\n<hr class=\"\" data-start=\"757\" data-end=\"760\" \/>\n<h3 class=\"\" data-start=\"762\" data-end=\"833\">\ud83e\uddee <strong data-start=\"769\" data-end=\"833\">2. Steps to Draw a Hasse Diagram from Partial Order Relation<\/strong><\/h3>\n<p class=\"\" data-start=\"835\" data-end=\"951\">The <strong data-start=\"839\" data-end=\"856\">Hasse Diagram<\/strong> is a graphical representation of a partial order that simplifies the relation by <strong data-start=\"938\" data-end=\"950\">removing<\/strong>:<\/p>\n<ul data-start=\"953\" data-end=\"1012\">\n<li class=\"\" data-start=\"953\" data-end=\"989\">\n<p class=\"\" data-start=\"955\" data-end=\"989\"><strong data-start=\"955\" data-end=\"974\">Reflexive pairs<\/strong> (i.e., (a, a))<\/p>\n<\/li>\n<li class=\"\" data-start=\"990\" data-end=\"1012\">\n<p class=\"\" data-start=\"992\" data-end=\"1012\"><strong data-start=\"992\" data-end=\"1012\">Transitive edges<\/strong><\/p>\n<\/li>\n<\/ul>\n<hr class=\"\" data-start=\"1014\" data-end=\"1017\" \/>\n<h3 class=\"\" data-start=\"1019\" data-end=\"1043\">\ud83e\ude9c <strong data-start=\"1026\" data-end=\"1043\">Step-by-Step:<\/strong><\/h3>\n<h4 class=\"\" data-start=\"1045\" data-end=\"1103\">\ud83d\udfe9 Step 1: List the elements and define the relation.<\/h4>\n<p class=\"\" data-start=\"1104\" data-end=\"1143\">Use the relation to form ordered pairs.<\/p>\n<h4 class=\"\" data-start=\"1145\" data-end=\"1195\">\ud83d\udfe8 Step 2: Draw the directed graph (digraph).<\/h4>\n<p class=\"\" data-start=\"1196\" data-end=\"1247\">Connect elements with arrows based on the relation.<\/p>\n<h4 class=\"\" data-start=\"1249\" data-end=\"1288\">\ud83d\udfe5 Step 3: Remove reflexive pairs.<\/h4>\n<p class=\"\" data-start=\"1289\" data-end=\"1318\">Remove all loops like (a, a).<\/p>\n<h4 class=\"\" data-start=\"1320\" data-end=\"1360\">\ud83d\udfe6 Step 4: Remove transitive edges.<\/h4>\n<p class=\"\" data-start=\"1361\" data-end=\"1435\">If a \u2192 b and b \u2192 c exist, then a \u2192 c is <strong data-start=\"1401\" data-end=\"1415\">transitive<\/strong> and can be removed.<\/p>\n<h4 class=\"\" data-start=\"1437\" data-end=\"1479\">\ud83d\udfeb Step 5: Replace arrows with lines.<\/h4>\n<p class=\"\" data-start=\"1480\" data-end=\"1573\">Make lines go <strong data-start=\"1494\" data-end=\"1504\">upward<\/strong> (from lower to higher elements). This becomes the <strong data-start=\"1555\" data-end=\"1572\">Hasse Diagram<\/strong>.<\/p>\n<hr class=\"\" data-start=\"1575\" data-end=\"1578\" \/>\n<h3 class=\"\" data-start=\"1580\" data-end=\"1599\">\ud83d\udcc9 <strong data-start=\"1587\" data-end=\"1599\">Example:<\/strong><\/h3>\n<p class=\"\" data-start=\"1601\" data-end=\"1652\">Set <strong data-start=\"1605\" data-end=\"1625\">A = {1, 2, 4, 8}<\/strong>, Relation R: \u201ca divides b\u201d<\/p>\n<p class=\"\" data-start=\"1654\" data-end=\"1743\">| Ordered Pairs: | (1,1), (1,2), (1,4), (1,8), (2,2), (2,4), (2,8), (4,4), (4,8), (8,8) |<\/p>\n<ul data-start=\"1745\" data-end=\"1909\">\n<li class=\"\" data-start=\"1745\" data-end=\"1793\">\n<p class=\"\" data-start=\"1747\" data-end=\"1793\">Remove reflexive: (1,1), (2,2), (4,4), (8,8)<\/p>\n<\/li>\n<li class=\"\" data-start=\"1794\" data-end=\"1862\">\n<p class=\"\" data-start=\"1796\" data-end=\"1862\">Remove transitive: (1,4) and (1,8), since (1,2) and (2,4) exist.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1863\" data-end=\"1909\">\n<p class=\"\" data-start=\"1865\" data-end=\"1909\">Final pairs for diagram: (1,2), (2,4), (4,8)<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"1911\" data-end=\"1953\">\ud83d\udccc Final Hasse Diagram (drawn vertically):<\/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=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\"><\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\">8<br \/>\n|<br \/>\n4<br \/>\n|<br \/>\n2<br \/>\n|<br \/>\n1<br \/>\n<\/code><\/div>\n<\/div>\n<hr class=\"\" data-start=\"1978\" data-end=\"1981\" \/>\n<h3 class=\"\" data-start=\"1983\" data-end=\"2013\">\ud83e\udde0 <strong data-start=\"1990\" data-end=\"2013\">Tips for GATE Exam:<\/strong><\/h3>\n<ul data-start=\"2014\" data-end=\"2211\">\n<li class=\"\" data-start=\"2014\" data-end=\"2088\">\n<p class=\"\" data-start=\"2016\" data-end=\"2088\">Understand <strong data-start=\"2027\" data-end=\"2038\">lattice<\/strong> and <strong data-start=\"2043\" data-end=\"2052\">poset<\/strong> (partially ordered set) connections<\/p>\n<\/li>\n<li class=\"\" data-start=\"2089\" data-end=\"2134\">\n<p class=\"\" data-start=\"2091\" data-end=\"2134\">Be quick in identifying transitive closures<\/p>\n<\/li>\n<li class=\"\" data-start=\"2135\" data-end=\"2211\">\n<p class=\"\" data-start=\"2137\" data-end=\"2211\">Draw Hasse Diagrams carefully in less time \u2014 good for MSQ or NAT questions<\/p>\n<\/li>\n<\/ul>\n<hr class=\"\" data-start=\"2213\" data-end=\"2216\" \/>\n<p class=\"\" data-start=\"2218\" data-end=\"2233\">Would you like:<\/p>\n<ul data-start=\"2234\" data-end=\"2375\">\n<li class=\"\" data-start=\"2234\" data-end=\"2276\">\n<p class=\"\" data-start=\"2236\" data-end=\"2276\">\ud83d\udcdd PDF of solved Hasse diagram problems?<\/p>\n<\/li>\n<li class=\"\" data-start=\"2277\" data-end=\"2318\">\n<p class=\"\" data-start=\"2279\" data-end=\"2318\">\ud83c\udfa5 Video lecture link in Hindi\/English?<\/p>\n<\/li>\n<li class=\"\" data-start=\"2319\" data-end=\"2375\">\n<p class=\"\" data-start=\"2321\" data-end=\"2375\">\ud83d\udcca Practice MCQs based on partial orders and lattices?<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"2377\" data-end=\"2415\">Let me know and I\u2019ll help you further!<\/p>\n<h3 data-start=\"2377\" data-end=\"2415\"><a href=\"https:\/\/littleflowercollege.edu.in\/upload\/e_contents\/files\/023f77699503fd8c0c5ece28e5203670.pdf\" target=\"_blank\" rel=\"noopener\">Day 03Part 03-Discrete mathematics for gate-How to Convert Partial order Relations to hasse diagram.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/faculty.ksu.edu.sa\/sites\/default\/files\/4.3_partial_ordering_0.pdf\" target=\"_blank\" rel=\"noopener\">(4.3) Partial Orderings<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/users.pja.edu.pl\/~msyd\/mad-lectures\/order.pdf\" target=\"_blank\" rel=\"noopener\">Discrete Mathematics &#8211; Order Relation<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 03Part 03-Discrete mathematics for gate-How to Convert Partial order Relations to hasse diagram. [fvplayer id=&#8221;228&#8243;] \u0939\u093e\u0938\u0947 \u0906\u0930\u0947\u0916 (Hasse Diagram) \u092c\u0928\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 (Partial Order Relations) \u0915\u094b \u0915\u0948\u0938\u0947 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u090f, \u0907\u0938\u0915\u093e \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 \u0924\u0930\u0940\u0915\u093e \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948: \u091a\u0930\u0923 1: \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 (Partial Order Relation) \u0915\u094b \u0938\u092e\u091d\u0947\u0902 \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 \u090f\u0915 \u0938\u0947\u091f [&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-3041","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3041","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=3041"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3041\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3041"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3041"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3041"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}