Q. 17 Let A = \{1, 2, 3\}. Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4

Answers (1)

A = \{1, 2, 3\}

The number of equivalence relations containing (1, 2) is given by 

R = \left \{ (1,1),(2,2),(3,3),(1,2),(2,1) \right \}

We are left with four  pairs (2,3) ,(3,2),(1,3),(3,1) .

(1,1),(2,2),(3,3) \in R  , so relation R is reflexive.

(1,2),(2,1) \in R    and   (2,3),(3,2) \notin R  , so relation R is not symmetric.

 (1,3),(3,1) \notin R  , so realation R is not transitive.

Hence , equivalence relation is bigger than R  is universal relation.

Thus the total number of equivalence relations cotaining  (1,2) is two.

Thus, option B is correct.

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