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Solution


Q.

Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 4)} be a relation on A. Let S be the equivalence relation on A such that R⊂S and the number of elements in S is n. Then, the minimum value of n is ________.         [2024]

Ans.

(16)

Given A = {1, 2, 3, 4}

R = {(1, 2), (2, 3), (1, 4)}

S to be equivalence, it should be reflexive, symmetric and transitive.

For reflexive add (1, 1), (2, 2), (3, 3), (4, 4).

For symmetric add (2, 1), (3, 2), (4, 1).

For transitive (1, 2), (2, 3) ⇒ (1, 3), so add (1, 3) also add (3, 1) for symmetric

and (4, 1), (1, 2) ⇒ (4, 2), so add (4, 2) also add (2, 4) for symmetric.

∴ S = {(1, 2), (2, 3), (1, 4), (1, 1), (2, 2), (3, 3), (4, 4), (2, 1), (3, 2), (4, 1), (3, 1), (1, 3), (2, 4), (4, 2), (3, 4), (4, 3)}

∴ n(S) = 16