Let A= {1, 2, 3}, then the possible equivalence relations on A are:
(A) {(1, 1), (2, 2), (3, 3)}
(B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
(C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)}
(D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
An equivalence relation must be reflexive, symmetric and transitive. All four relations contain (1, 1), (2, 2), (3, 3), so all are reflexive; only symmetry and transitivity need checking.
(A) {(1, 1), (2, 2), (3, 3)} — Correct. This is the identity relation: it is trivially symmetric and transitive.
(B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} — Correct. (1, 2) and (2, 1) are both present, so it is symmetric. The only new chains are (1, 2), (2, 1) giving (1, 1), and (2, 1), (1, 2) giving (2, 2), both present, so it is transitive.
(C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} — Incorrect. (1, 3) is present but (3, 1) is not, so it is not symmetric.
(D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} — Correct. This is (B) with 3 in place of 2, so it is symmetric and transitive in the same way.