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

Solution:

The given set is

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 add anyone 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.

The correct answer is B

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 17

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

Summary:

Given that A = {1, 2, 3}, Then number of equivalence relations containing (1, 2) is 2. The correct answer is B