Which of the relations on {0, 1, 2, 3} is an equivalence relation?
(A) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}
(B) { (0, 0) (1, 1) (2, 2) (3, 3)}
(C) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}
(D) {(0, 0), (0, 2), (2, 3), (1, 1), (2, 2)}


Answer: (B)

Explanation: A relation is equivalence only iff relation has reflexive, symmetric, and transitive property. Therefore,

(A) Because of (0, 2) and (2, 3), then it should have (0, 3) which is not the relation. Also, (3, 2) and (2, 0), then it should have (3, 0) which is not the relation. Therefore, this given relation is transitive, so can be equivalence relation.

(B) It is reflexive, symmetric, and transitive, so equivalence relation. It is diagonal relation and a diagonal relation is always equivalence relation.

(C) This relation is neither reflexive, nor symmetric, so it can not be reflexive relation.

(D) It is not reflexive, not symmetric and not transitive, so it can not be reflexive relation.

UGC has taken this question from Kenneth Rosen-6th edition (Ques-1, Chapter- 8.5).

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