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Smallest equivalence (or symmetric and transitive) relation containing given pairs

This type of question, asked 5 times: 24 Jan 2023, 30 Jan 2023, 29 Jan 2024, 31 Jan 2024, 23 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    24 Jan 2023

    178
    The minimum number of elements that must be added to the relation R={(a,b),(b,c),(b,d)}\mathbf{R = \{(a,b),(b,c),(b,d)\}} on the set {a,b,c,d}\mathbf{\{ a,b,c,d\}} so that it is an equivalence relation, is

  2. Question 2

    30 Jan 2023

    62
    The minimum number of elements that must be added to the relation R={(a,b),(b,c)}\mathbf{R = \{(a,b),(b,c)\}} on the set {a,b,c}\mathbf{\{ a,b,c\}} so that is becomes symmetric and transitive is:

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  3. Question 3

    29 Jan 2024

    168
    If RR is the smallest equivalence relation on the set {1,2,3,4}\mathbf{\{ 1,2,3,4\}} such that {(1,2),(1,3)}⊂R\mathbf{\{(1,2),(1,3)\} \subset R}, then the number of elements in RR is

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  4. Question 4

    31 Jan 2024

    89
    Let A={1,2,3,4}\mathbf{A = \{ 1,2,3,4\}} and R={(1,2),(2,3),(1,4)}\mathbf{R = \{(1,2),(2,3),(1,4)\}} be a relation on AA. Let SS be the equivalence relation on A such that R⊂SR \subset S and the number of elements in SS is nn. Then, the minimum value of nn is

  5. Question 5

    23 Jan 2025

    61
    Let R={(1,2),(2,3),(3,3)}\mathbf{R = \{(1,2),(2,3),(3,3)\}} be a relation defined on the set {1,2,3,4}\mathbf{\{ 1,2,3,4\}}. Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :

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