Let S be the set of a b c d e a Find a smallest possible equ

Let S be the set of {a, b, c, d, e}

(a) Find a smallest possible equivalence relation on S.

(b) Find a smallest possible relation on S which is transitive and symmetric, but not reflexive.

(d) Find a smallest possible relation on S which is reflexive and symmetric, but not transitive

Given S ={a, b, c, d, e}

Ans(a):

R = {(a, a)} is an equivalence relation in S.

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Ans(b):

R={(a,a),(b,b),(a,b),(b,a)} is transitive and symmetric, but not reflexive.

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Ans(c):

R={(a,a),(b,b),(c,c),(a,b),(b,c),(c,a)} is transitive and reflexive, but not symmetric.

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Ans(d):

R={(a,a),(b,b),(c,c),(a,b),(b,c),(b,a),(c,b)} is reflexive and symmetric, but not transitive