If \(\mathrm{A}=\\{1,2,3\\}\), then the number of equivalence relation containing \((1,2)\) is (a) 1 (b) 2 (c) 3 (d) 8

List all possible relations in the set A, including the trivial case where every element is related to itself: 1. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} 2. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (1, 3), (3, 2), (2, 3), (3, 3)} We will now analyze each of these relations to see if they satisfy the properties of equivalence relations.

For a relation to be an equivalence relation, it must be reflexive. This means that (a, a) must be in the relation for every element a in the set A. Both relations have this property, so both can continue to the next step.

For a relation to be an equivalence relation, it must be symmetric, which means (a, b) is in the relation if and only if (b, a) is in the relation. We are given that (1, 2) must be included in the relation, so both relations must also include (2, 1). Looking at the relations we listed in Step 1, both relations have this property, so both can continue to the next step.

For a relation to be an equivalence relation, it must be transitive. This means that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. Let's check both relations: 1. Relation 1: Since (1, 2) and (2, 1) are in the relation, (1, 1) must also be in the relation (which it is). (1, 2) and (2, 3) are not in the relation, so the transitivity property is satisfied. 2. Relation 2: Since (1, 2) and (2, 3) are in the relation, (1, 3) must also be in the relation (which it is). Since (1, 3) and (3, 2) are in the relation, (1, 2) must also be in the relation (which it is). Thus, the transitivity property is satisfied. Both relations satisfy the transitivity property, so both are equivalence relations.

We have found 2 equivalence relations including the pair (1, 2). Thus, the answer to the problem is: (b) 2