Question

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

Answer 1

Given set is A={ 1,2,3 }.

The total number of possible pairs will be,

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

For the symmetric relation in set A, if ( 1,2 )is in relation then ( 2,1 ) should also be in relation. And, ( 1,3 )is in relation then ( 3,1 ) should also be in relation.

For reflexive relation, ( 1,1 ),( 2,2 ),( 3,3 ) should be in relation.

For transitivity, if ( 1,2 )is in relation and ( 2,1 )is in relation then ( 1,1 )should be in relation.

The smallest possible equivalence containing ( 1,2 )is given by,

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

Thus, there are only two possible equivalence relations containing ( 1,2 ).

Therefore, option (B) is correct.