#### Question

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

(A) 1

(B) 2

(C) 3

(D) 4

#### Solution

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

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

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

Now, we are left with only four pairs i.e., (2, 3), (3, 2), (1, 3), and (3, 1).

If we odd any one pair [say (2, 3)] to R_{1}, then for symmetry we must add (3, 2). Also, for transitivity we are required to add (1, 3) and (3, 1).

Hence, the only equivalence relation (bigger than R_{1}) is the universal relation.

This shows that the total number of equivalence relations containing (1, 2) is two.

The correct answer is B.

Is there an error in this question or solution?

Solution Let a = {1, 2, 3}. Then Number of Equivalence Relations Containing (1, 2) is Concept: Types of Relations.