Discrete Mathematics Gate Question-GATE 2025 equivalence class A relation R is defined on the set<\/p>\n
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It looks like you’re looking for a GATE 2025 Discrete Mathematics<\/strong> question related to Equivalence Classes<\/strong> and Relations<\/strong>. Here\u2019s a general breakdown to help you understand how to solve such problems.<\/p>\n A relation R<\/strong> on a set S<\/strong> is an equivalence relation<\/strong> if it satisfies three properties: If a relation R<\/strong> is an equivalence relation<\/strong>, then the equivalence class<\/strong> of an element a<\/strong> is the set:<\/p>\n [a]={x\u2208S\u2223xRa}[a] = \\{ x \\in S \\mid xRa \\}<\/span>[<\/span>a<\/span>]<\/span>=<\/span><\/span>{<\/span>x<\/span>\u2208<\/span><\/span>S<\/span>\u2223<\/span><\/span>x<\/span>R<\/span>a<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n This means all elements in an equivalence class are related to each other under R<\/strong>.<\/p>\n Let S={1,2,3,4,5,6}S = \\{1, 2, 3, 4, 5, 6\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span>,<\/span>6<\/span>}<\/span><\/span><\/span><\/span> and define a relation R<\/strong> as:<\/p>\n aRb\u2005\u200a\u27fa\u2005\u200aa\u2261b(mod3)aRb \\iff a \\equiv b \\pmod{3}<\/span>a<\/span>R<\/span>b<\/span>\u27fa<\/span><\/span>a<\/span>\u2261<\/span><\/span>b<\/span><\/span>(<\/span>mod<\/span><\/span>3<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n (i.e., a is related to b if they have the same remainder when divided by 3<\/strong>).<\/p>\n We check numbers modulo 3:<\/p>\n Final Answer:<\/strong> The equivalence classes are {1, 4}, {2, 5}, {3, 6}<\/strong>.<\/p>\n Check Reflexivity, Symmetry, and Transitivity<\/strong> to confirm R is an equivalence relation<\/strong>. \u00a0Want me to solve a specific GATE 2025 question for you? Share the full question!<\/strong><\/p>\n Let\u2019s break down and solve a GATE-style Discrete Mathematics question<\/strong> related to equivalence relations and equivalence classes<\/strong>, as might appear in GATE 2025<\/strong>:<\/p>\n Question (Sample Format):<\/strong>\u00a0Equivalence Class & Relation Definition<\/strong><\/h3>\n
Reflexivity<\/strong>: aRaaRa<\/span>a<\/span>R<\/span>a<\/span><\/span><\/span><\/span> for all a\u2208Sa \\in S<\/span>a<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span>
Symmetry<\/strong>: If aRbaRb<\/span>a<\/span>R<\/span>b<\/span><\/span><\/span><\/span>, then bRabRa<\/span>b<\/span>R<\/span>a<\/span><\/span><\/span><\/span>
Transitivity<\/strong>: If aRbaRb<\/span>a<\/span>R<\/span>b<\/span><\/span><\/span><\/span> and bRcbRc<\/span>b<\/span>R<\/span>c<\/span><\/span><\/span><\/span>, then aRcaRc<\/span>a<\/span>R<\/span>c<\/span><\/span><\/span><\/span><\/p>\n\u00a0Example GATE Question:<\/strong><\/h3>\n
\u00a0Find the Equivalence Classes:<\/strong><\/h3>\n
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\u00a0How to Solve Similar GATE Questions?<\/strong><\/h3>\n
Find elements that are related<\/strong> to each other based on the given rule.
Group elements into disjoint equivalence classes<\/strong>.<\/p>\nDiscrete Mathematics Gate Question-GATE 2025 equivalence class A relation R is defined on the set<\/a><\/h3>\n
Discrete Mathematics for Computer Science<\/a><\/h3>\n
\n\ud83e\uddee \ud83d\udccc Topic: Equivalence Class and Equivalence Relation<\/strong><\/h3>\n
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Let RR<\/span>R<\/span><\/span><\/span><\/span> be a relation on the set A={1,2,3,4,5,6}A = \\{1, 2, 3, 4, 5, 6\\}<\/span>