The number of non-empty equivalence relations on the set {1, 2, 3} is

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Question 1:- The number of non-empty equivalence relations on the set {1, 2, 3} is :

(i)  6                   (ii) 7

(iii) 5                  (iv) 4                        [JEE 22 JAN 2025]

Answer :- (iii) 5

Explanation:- The equavalence relation are reflexive, Symmetric and Transitive

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

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

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

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

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

These above five relation are reflexive, symmetric and transitive so the above relation are equivalence relation.

Question 2:- Let f : R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y ∈ R. If f'(0) = 4a and f satisfies f”(x) – 3a f'(x) – f(x) = 0, a > 0, then the area of the region

R = {(x, y) | 0 ≤ y ≤ f(ax), 0 ≤ x ≤ 2} is :

(i) e^2 - 1              (ii) e^4 + 1

(iii) e^4 - 1              (iv) e^2 + 1               [JEE 22 JAN 2025]

Solution:- See full solution

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