Q. 173.8( 12 Votes )

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is
A. 1

B. 2

C. 3

D. 4

Answer :

It is given that A = {1, 2, 3}.

An equivalence relation is reflexive, symmetric and transitive.


The smallest equivalence relations containing (1, 2) is equal to


R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}


Now, only four pairs are left (2, 3), (3, 2), (1, 3) and (3, 1).


So, if we add one pair to R, then for symmetry we must add (3, 2).


Also, for transitivity we required to add (1, 3) and (3, 1).


Thus, the only equivalence relation is the universal relation.


Therefore, the total number of equivalence relations containing (1, 2) is 2.

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