{"id":3054,"date":"2025-06-07T09:03:53","date_gmt":"2025-06-07T09:03:53","guid":{"rendered":"https:\/\/diznr.com\/?p=3054"},"modified":"2025-06-07T09:03:53","modified_gmt":"2025-06-07T09:03:53","slug":"discrete-mathematics-previous-year-gate-2021-equivalence-relation-let-r-and-s-be-any-equivalence-two","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/discrete-mathematics-previous-year-gate-2021-equivalence-relation-let-r-and-s-be-any-equivalence-two\/","title":{"rendered":"Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence"},"content":{"rendered":"

Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence<\/p>\n

In the context of GATE (Graduate Aptitude Test in Engineering) examinations, understanding the properties of equivalence relations is crucial. An equivalence relation<\/strong> on a set AA<\/span>A<\/span><\/span><\/span><\/span> is a relation that is reflexive, symmetric, and transitive.<\/p>\n

Key Properties of Equivalence Relations:<\/strong><\/p>\n

    \n
  1. \n

    Reflexivity:<\/strong> Every element is related to itself. For all a\u2208Aa \\in A<\/span>a<\/span>\u2208<\/span><\/span>A<\/span><\/span><\/span><\/span>, (a,a)\u2208R(a, a) \\in R<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n

  2. \n

    Symmetry:<\/strong> If an element aa<\/span>a<\/span><\/span><\/span><\/span> is related to an element bb<\/span>b<\/span><\/span><\/span><\/span>, then bb<\/span>b<\/span><\/span><\/span><\/span> is also related to aa<\/span>a<\/span><\/span><\/span><\/span>. For all a,b\u2208Aa, b \\in A<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>A<\/span><\/span><\/span><\/span>, if (a,b)\u2208R(a, b) \\in R<\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span>, then (b,a)\u2208R(b, a) \\in R<\/span>(<\/span>b<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n

  3. \n

    Transitivity:<\/strong> If an element aa<\/span>a<\/span><\/span><\/span><\/span> is related to bb<\/span>b<\/span><\/span><\/span><\/span>, and bb<\/span>b<\/span><\/span><\/span><\/span> is related to cc<\/span>c<\/span><\/span><\/span><\/span>, then aa<\/span>a<\/span><\/span><\/span><\/span> is related to cc<\/span>c<\/span><\/span><\/span><\/span>. For all a,b,c\u2208Aa, b, c \\in A<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>A<\/span><\/span><\/span><\/span>, if (a,b)\u2208R(a, b) \\in R<\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span> and (b,c)\u2208R(b, c) \\in R<\/span>(<\/span>b<\/span>,<\/span>c<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span>, then (a,c)\u2208R(a, c) \\in R<\/span>(<\/span>a<\/span>,<\/span>c<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<\/ol>\n

    Intersection of Equivalence Relations:<\/strong><\/p>\n

    When considering two equivalence relations RR<\/span>R<\/span><\/span><\/span><\/span> and SS<\/span>S<\/span><\/span><\/span><\/span> on a set AA<\/span>A<\/span><\/span><\/span><\/span>, their intersection<\/strong> R\u2229SR \\cap S<\/span>R<\/span>\u2229<\/span><\/span>S<\/span><\/span><\/span><\/span> is also an equivalence relation. This is because the intersection of two reflexive, symmetric, and transitive relations retains these properties.<\/p>\n

    Union of Equivalence Relations:<\/strong><\/p>\n

    However, the union<\/strong> R\u222aSR \\cup S<\/span>R<\/span>\u222a<\/span><\/span>S<\/span><\/span><\/span><\/span> of two equivalence relations is not necessarily<\/strong> an equivalence relation. While the union of two reflexive and symmetric relations remains reflexive and symmetric, it may fail to be transitive.<\/p>\n

    Example Illustrating Non-Transitivity:<\/strong><\/p>\n

    Consider a set A={1,2,3}A = \\{1, 2, 3\\}<\/span>A<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>}<\/span><\/span><\/span><\/span> with two equivalence relations RR<\/span>R<\/span><\/span><\/span><\/span> and SS<\/span>S<\/span><\/span><\/span><\/span>:<\/p>\n