Question

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

Options

• 1

• 2

• 3

• 4

Answer 1

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

Explanation:

It is given that A = {1, 2, 3}.

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

R₁ = {(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 R₁, 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 R₁) is the universal relation.

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