{"id":2957,"date":"2025-06-06T07:25:49","date_gmt":"2025-06-06T07:25:49","guid":{"rendered":"https:\/\/diznr.com\/?p=2957"},"modified":"2025-06-06T07:25:49","modified_gmt":"2025-06-06T07:25:49","slug":"discrete-mathematics-tutorial-in-hindi-previous-year-question-gate-2020-equivalence-abcd-let","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/discrete-mathematics-tutorial-in-hindi-previous-year-question-gate-2020-equivalence-abcd-let\/","title":{"rendered":"Discrete mathematics tutorial in Hindi &#8211; Previous year question &#8211; GATE 2025 -equivalence- Let a,b,c,d."},"content":{"rendered":"<p>Discrete mathematics tutorial in Hindi &#8211; Previous year question &#8211; GATE 2025 -equivalence- Let a,b,c,d.<\/p>\n<p>[fvplayer id=&#8221;189&#8243;]<\/p>\n<p data-start=\"0\" data-end=\"222\">It looks like you&#8217;re looking for <strong data-start=\"33\" data-end=\"76\">Discrete Mathematics tutorials in Hindi<\/strong> for <strong data-start=\"81\" data-end=\"94\">GATE 2025<\/strong>, specifically focusing on <strong data-start=\"121\" data-end=\"170\">equivalence relations and logical equivalence<\/strong> with previous year questions. Here\u2019s a breakdown:<\/p>\n<h3 data-start=\"229\" data-end=\"287\"><strong data-start=\"233\" data-end=\"285\">\u00a0Equivalence Relations in Discrete Mathematics<\/strong><\/h3>\n<p data-start=\"289\" data-end=\"390\">An <strong data-start=\"292\" data-end=\"316\">equivalence relation<\/strong> 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> is 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> that satisfies three properties:<\/p>\n<p data-start=\"392\" data-end=\"573\"><strong data-start=\"396\" data-end=\"410\">Reflexive:<\/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> (Each element is related to itself)<br data-start=\"456\" data-end=\"459\" \/><strong data-start=\"463\" data-end=\"477\">Symmetric:<\/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>, then <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><br data-start=\"506\" data-end=\"509\" \/><strong data-start=\"513\" data-end=\"528\">Transitive:<\/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<p data-start=\"575\" data-end=\"592\"><strong data-start=\"578\" data-end=\"590\">Example:<\/strong><\/p>\n<ul data-start=\"593\" data-end=\"849\">\n<li data-start=\"593\" data-end=\"678\">Let <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> be a relation on set <strong data-start=\"628\" data-end=\"645\">A = {1,2,3,4}<\/strong> defined as <strong data-start=\"657\" data-end=\"676\">&#8220;a \u2261 b (mod 2)&#8221;<\/strong><\/li>\n<li data-start=\"679\" data-end=\"748\">Here, <strong data-start=\"687\" data-end=\"698\">1 and 3<\/strong> are equivalent, and <strong data-start=\"719\" data-end=\"730\">2 and 4<\/strong> are equivalent.<\/li>\n<li data-start=\"749\" data-end=\"849\">This relation is <strong data-start=\"768\" data-end=\"808\">reflexive, symmetric, and transitive<\/strong>, so it is an <strong data-start=\"822\" data-end=\"846\">equivalence relation<\/strong>.<\/li>\n<\/ul>\n<h3 data-start=\"856\" data-end=\"888\"><strong data-start=\"860\" data-end=\"886\">\u00a0Logical Equivalence<\/strong><\/h3>\n<p data-start=\"890\" data-end=\"1005\">Two logical statements <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span> are <strong data-start=\"937\" data-end=\"961\">logically equivalent<\/strong> if they always have the same truth value.<\/p>\n<p data-start=\"1007\" data-end=\"1042\"><strong data-start=\"1009\" data-end=\"1040\">Common Logical Equivalences<\/strong><\/p>\n<ul data-start=\"1043\" data-end=\"1393\">\n<li data-start=\"1043\" data-end=\"1096\"><strong data-start=\"1045\" data-end=\"1065\">Double Negation:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">\u00ac(\u00acP)\u2261P\\neg (\\neg P) \\equiv P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2261<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1097\" data-end=\"1232\"><strong data-start=\"1099\" data-end=\"1120\">De Morgan\u2019s Laws:<\/strong>\n<ul data-start=\"1125\" data-end=\"1232\">\n<li data-start=\"1125\" data-end=\"1177\"><span class=\"katex\"><span class=\"katex-mathml\">\u00ac(P\u2228Q)\u2261\u00acP\u2227\u00acQ\\neg (P \\lor Q) \\equiv \\neg P \\land \\neg Q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2261<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1180\" data-end=\"1232\"><span class=\"katex\"><span class=\"katex-mathml\">\u00ac(P\u2227Q)\u2261\u00acP\u2228\u00acQ\\neg (P \\land Q) \\equiv \\neg P \\lor \\neg Q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2261<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1233\" data-end=\"1308\"><strong data-start=\"1235\" data-end=\"1257\">Implication as OR:<\/strong>\n<ul data-start=\"1262\" data-end=\"1308\">\n<li data-start=\"1262\" data-end=\"1308\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2192Q\u2261\u00acP\u2228QP \\rightarrow Q \\equiv \\neg P \\lor Q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2261<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1309\" data-end=\"1393\"><strong data-start=\"1311\" data-end=\"1330\">Contrapositive:<\/strong>\n<ul data-start=\"1335\" data-end=\"1393\">\n<li data-start=\"1335\" data-end=\"1393\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2192Q\u2261\u00acQ\u2192\u00acPP \\rightarrow Q \\equiv \\neg Q \\rightarrow \\neg P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2261<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3 data-start=\"1400\" data-end=\"1459\"><strong data-start=\"1404\" data-end=\"1457\">\u00a0GATE 2025 Previous Year Question &#8211; Equivalence<\/strong><\/h3>\n<h4 data-start=\"1461\" data-end=\"1481\"><strong data-start=\"1466\" data-end=\"1479\">Question:<\/strong><\/h4>\n<p data-start=\"1482\" data-end=\"1549\">Let <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> be a relation on set <strong data-start=\"1515\" data-end=\"1535\">A = {a, b, c, d}<\/strong> defined as:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">R={(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(c,d),(d,c)}R = \\{(a,a), (b,b), (c,c), (d,d), (a,b), (b,a), (c,d), (d,c)\\}<\/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 mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><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=\"mpunct\">,<\/span><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=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1621\" data-end=\"1678\">Which of the following properties does <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> satisfy?<\/p>\n<p data-start=\"1680\" data-end=\"1814\">(A) Reflexive only<br data-start=\"1698\" data-end=\"1701\" \/>(B) Reflexive and Symmetric only<br data-start=\"1733\" data-end=\"1736\" \/>(C) Reflexive, Symmetric, and Transitive<br data-start=\"1776\" data-end=\"1779\" \/>(D) Symmetric and Transitive only<\/p>\n<h4 data-start=\"1816\" data-end=\"1836\"><strong data-start=\"1821\" data-end=\"1834\">Solution:<\/strong><\/h4>\n<p data-start=\"1838\" data-end=\"1864\"><strong data-start=\"1842\" data-end=\"1862\">Reflexive Check:<\/strong><\/p>\n<ul data-start=\"1865\" data-end=\"1941\">\n<li data-start=\"1865\" data-end=\"1941\">Since <span class=\"katex\"><span class=\"katex-mathml\">(a,a),(b,b),(c,c),(d,d)(a,a), (b,b), (c,c), (d,d)<\/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=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> are present, <strong data-start=\"1919\" data-end=\"1937\">R is reflexive<\/strong><\/li>\n<li data-start=\"1865\" data-end=\"1941\"><strong data-start=\"1947\" data-end=\"1967\">Symmetric Check:<\/strong><\/li>\n<\/ul>\n<ul data-start=\"1970\" data-end=\"2096\">\n<li data-start=\"1970\" data-end=\"2020\">If <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>, then <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><\/li>\n<li data-start=\"2021\" data-end=\"2071\">If <span class=\"katex\"><span class=\"katex-mathml\">(c,d)\u2208R(c,d) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">(d,c)\u2208R(d,c) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/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><\/li>\n<li data-start=\"2072\" data-end=\"2096\"><strong data-start=\"2074\" data-end=\"2092\">R is symmetric<\/strong><\/li>\n<\/ul>\n<p data-start=\"2098\" data-end=\"2125\"><strong data-start=\"2102\" data-end=\"2123\">Transitive Check:<\/strong><\/p>\n<ul data-start=\"2126\" data-end=\"2321\">\n<li data-start=\"2126\" data-end=\"2210\"><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>, but <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>, so no violation.<\/li>\n<li data-start=\"2211\" data-end=\"2295\"><span class=\"katex\"><span class=\"katex-mathml\">(c,d)\u2208R(c,d) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/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\">(d,c)\u2208R(d,c) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">d<\/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>, but <span class=\"katex\"><span class=\"katex-mathml\">(c,c)\u2208R(c,c) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/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>, so no violation.<\/li>\n<li data-start=\"2296\" data-end=\"2321\"><strong data-start=\"2298\" data-end=\"2317\">R is transitive<\/strong><\/li>\n<\/ul>\n<p data-start=\"2323\" data-end=\"2400\"><strong data-start=\"2325\" data-end=\"2344\">Correct Answer:<\/strong> <strong data-start=\"2345\" data-end=\"2398\">Option (C) &#8211; Reflexive, Symmetric, and Transitive<\/strong><\/p>\n<h3 data-start=\"2407\" data-end=\"2446\"><strong data-start=\"2411\" data-end=\"2444\">\u00a0Conclusion &amp; Key Takeaways<\/strong><\/h3>\n<p data-start=\"2447\" data-end=\"2728\"><strong data-start=\"2449\" data-end=\"2474\">Equivalence relations<\/strong> are reflexive, symmetric, and transitive.<br data-start=\"2516\" data-end=\"2519\" \/><strong data-start=\"2521\" data-end=\"2544\">Logical equivalence<\/strong> follows algebraic rules like <strong data-start=\"2574\" data-end=\"2634\">De Morgan\u2019s Laws, Contrapositive, and Implication as OR.<\/strong><br data-start=\"2634\" data-end=\"2637\" \/><strong data-start=\"2639\" data-end=\"2726\">GATE questions often test understanding through set relations &amp; logical identities.<\/strong><\/p>\n<p data-start=\"2730\" data-end=\"2792\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"2733\" data-end=\"2792\" data-is-last-node=\"\">Need more solved questions or explanations in Hindi?<\/strong><\/p>\n<h3 data-start=\"2730\" data-end=\"2792\"><a href=\"https:\/\/home.iitd.ac.in\/uploads\/course-of-study\/Courses%20of%20Study%202023-24.pdf\" target=\"_blank\" rel=\"noopener\">Discrete mathematics tutorial in Hindi &#8211; Previous year question &#8211; GATE 2025 -equivalence- Let a,b,c,d.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/web.stanford.edu\/~jurafsky\/slp3\/ed3book_Jan25.pdf\" target=\"_blank\" rel=\"noopener\">Speech and Language Processing<\/a><\/h3>\n<p data-start=\"0\" data-end=\"105\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"24\" data-is-only-node=\"\">\u0921\u093f\u0938\u094d\u0915\u094d\u0930\u0940\u091f \u092e\u0948\u0925\u092e\u0947\u091f\u093f\u0915\u094d\u0938<\/strong> \u092e\u0947\u0902 <strong data-start=\"29\" data-end=\"68\">\u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation)<\/strong> \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0937\u092f \u0939\u0948, \u091c\u094b GATE \u091c\u0948\u0938\u0940 \u092a\u0930\u0940\u0915\u094d\u0937\u093e\u0913\u0902 \u092e\u0947\u0902 \u0905\u0915\u094d\u0938\u0930 \u092a\u0942\u091b\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span> \u0906\u0907\u090f \u0907\u0938\u0947 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u0938\u092e\u091d\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<hr data-start=\"107\" data-end=\"110\" \/>\n<h3 data-start=\"112\" data-end=\"143\">\ud83d\udd39 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e:<\/h3>\n<p data-start=\"145\" data-end=\"219\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0915\u093f\u0938\u0940 \u0938\u0947\u091f <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> \u092a\u0930 \u090f\u0915 \u092c\u093e\u0907\u0928\u0930\u0940 \u0930\u093f\u0932\u0947\u0936\u0928 <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> \u0915\u094b <strong data-start=\"48\" data-end=\"64\">\u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u092f\u0926\u093f \u0935\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0940\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/span><\/p>\n<ol data-start=\"221\" data-end=\"474\">\n<li data-start=\"221\" data-end=\"293\">\n<p data-start=\"224\" data-end=\"293\"><strong data-start=\"224\" data-end=\"255\">\u092a\u0930\u093e\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e (Reflexivity):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 <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> \u0915\u0947 \u0932\u093f\u090f, <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> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"294\" data-end=\"360\">\n<p data-start=\"297\" data-end=\"360\"><strong data-start=\"297\" data-end=\"320\">\u0938\u092e\u092e\u093f\u0924\u0924\u093e (Symmetry):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <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>, \u0924\u094b <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> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"361\" data-end=\"474\">\n<p data-start=\"364\" data-end=\"474\"><strong data-start=\"364\" data-end=\"395\">\u0938\u093e\u0902\u0915\u094d\u0930\u093e\u092e\u0915\u0924\u093e (Transitivity):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <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> \u0914\u0930 <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>, \u0924\u094b <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> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"476\" data-end=\"479\" \/>\n<h3 data-start=\"481\" data-end=\"495\">\ud83d\udd39 \u0909\u0926\u093e\u0939\u0930\u0923:<\/h3>\n<p data-start=\"497\" data-end=\"575\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f, \u0938\u0947\u091f <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> \u0939\u0948, \u0914\u0930 \u0930\u093f\u0932\u0947\u0936\u0928 <span class=\"katex\"><span class=\"katex-mathml\">R={(1,1),(2,2),(3,3),(1,2),(2,1)}R = \\{(1,1), (2,2), (3,3), (1,2), (2,1)\\}<\/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\">1<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span> \u0939\u0948\u0964<\/span><\/p>\n<ul data-start=\"577\" data-end=\"789\">\n<li data-start=\"577\" data-end=\"636\">\n<p data-start=\"579\" data-end=\"636\"><strong data-start=\"579\" data-end=\"596\">\u092a\u0930\u093e\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0938\u092d\u0940 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f <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> \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"637\" data-end=\"691\">\n<p data-start=\"639\" data-end=\"691\"><strong data-start=\"639\" data-end=\"651\">\u0938\u092e\u092e\u093f\u0924\u0924\u093e:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">(1,2)\u2208R(1,2) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span>, \u0924\u094b <span class=\"katex\"><span class=\"katex-mathml\">(2,1)\u2208R(2,1) \\in R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> \u092d\u0940 \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"692\" data-end=\"789\">\n<p data-start=\"694\" data-end=\"789\"><strong data-start=\"694\" data-end=\"710\">\u0938\u093e\u0902\u0915\u094d\u0930\u093e\u092e\u0915\u0924\u093e:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0939\u093e\u0901 <span class=\"katex\"><span class=\"katex-mathml\">(1,2)(1,2)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><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> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">(2,1)(2,1)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> \u0939\u0948\u0902, \u0932\u0947\u0915\u093f\u0928 <span class=\"katex\"><span class=\"katex-mathml\">(1,1)(1,1)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><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><\/span><\/span> \u092a\u0939\u0932\u0947 \u0938\u0947 \u0939\u0940 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u0917\u0941\u0923 \u092d\u0940 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"791\" data-end=\"869\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0938\u0932\u093f\u090f, <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> \u090f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948\u0964<\/span><\/p>\n<hr data-start=\"871\" data-end=\"874\" \/>\n<h3 data-start=\"876\" data-end=\"923\">\ud83d\udd39 GATE \u092e\u0947\u0902 \u092a\u0942\u091b\u0947 \u0917\u090f \u092a\u093f\u091b\u0932\u0947 \u0935\u0930\u094d\u0937\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0936\u094d\u0928:<\/h3>\n<p data-start=\"925\" data-end=\"1003\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">GATE \u092a\u0930\u0940\u0915\u094d\u0937\u093e\u0913\u0902 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0915\u0908 \u092a\u094d\u0930\u0936\u094d\u0928 \u092a\u0942\u091b\u0947 \u0917\u090f \u0939\u0948\u0902\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f:<\/span><\/p>\n<ul data-start=\"1005\" data-end=\"1285\">\n<li data-start=\"1005\" data-end=\"1144\">\n<p data-start=\"1007\" data-end=\"1144\"><strong data-start=\"1007\" data-end=\"1025\">GATE CSE 2009:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">&#8220;The number of ordered pairs in the largest and the smallest equivalence relations on S is ______.&#8221;<\/span><\/p>\n<\/li>\n<li data-start=\"1146\" data-end=\"1285\">\n<p data-start=\"1148\" data-end=\"1285\"><strong data-start=\"1148\" data-end=\"1166\">GATE CSE 1997:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">&#8220;The number of equivalence relations on the set {1,2,3,4} is ______.&#8221;<\/span> <span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">BYJU&#8217;S<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1287\" data-end=\"1365\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0928 \u092a\u094d\u0930\u0936\u094d\u0928\u094b\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0914\u0930 \u0909\u0928\u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u0940 \u0917\u0939\u0930\u0940 \u0938\u092e\u091d \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964<\/span><\/p>\n<hr data-start=\"1367\" data-end=\"1370\" \/>\n<h3 data-start=\"1372\" data-end=\"1400\">\ud83d\udd39 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0947 \u0932\u093f\u090f \u0938\u0902\u0938\u093e\u0927\u0928:<\/h3>\n<p data-start=\"1402\" data-end=\"1480\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f \u0906\u092a \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u092a\u0930 \u0914\u0930 \u0905\u0927\u093f\u0915 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0935\u0940\u0921\u093f\u092f\u094b \u0938\u0939\u093e\u092f\u0915 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/span><\/p>\n<div class=\"not-prose mb-3 flex flex-col gap-4 text-base\">\n<div><\/div>\n<\/div>\n<div class=\"not-prose mb-3 flex flex-col gap-4 text-base\">\n<div><\/div>\n<\/div>\n<p data-start=\"1570\" data-end=\"1648\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0947 \u0935\u0940\u0921\u093f\u092f\u094b \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924\u094b\u0902 \u0914\u0930 GATE \u092e\u0947\u0902 \u092a\u0942\u091b\u0947 \u0917\u090f \u092a\u094d\u0930\u0936\u094d\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n<hr data-start=\"1650\" data-end=\"1653\" \/>\n<p data-start=\"1655\" data-end=\"1769\">\u092f\u0926\u093f \u0906\u092a \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 \u092f\u093e \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0915\u0943\u092a\u092f\u093e \u092c\u0924\u093e\u090f\u0902\u0964 \u092e\u0948\u0902 \u0906\u092a\u0915\u0940 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092f\u0939\u093e\u0901 \u0939\u0942\u0901\u0964<\/p>\n<h3 data-start=\"1655\" data-end=\"1769\"><a href=\"https:\/\/www.vidyalankar.org\/gate\/assets\/docs\/notes\/maths.pdf\" target=\"_blank\" rel=\"noopener\">Discrete mathematics tutorial in Hindi &#8211; Previous year question &#8211; GATE 2025 -equivalence- Let a,b,c,d.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.islamiahcollege.edu.in\/download\/downloads\/2210241329307890.pdf\" target=\"_blank\" rel=\"noopener\">Calendar &amp; Student Handbook 2025-2025<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Discrete mathematics tutorial in Hindi &#8211; Previous year question &#8211; GATE 2025 -equivalence- Let a,b,c,d. [fvplayer id=&#8221;189&#8243;] It looks like you&#8217;re looking for Discrete Mathematics tutorials in Hindi for GATE 2025, specifically focusing on equivalence relations and logical equivalence with previous year questions. Here\u2019s a breakdown: \u00a0Equivalence Relations in Discrete Mathematics An equivalence relation on [&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-2957","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2957","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=2957"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2957\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}