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

Answer: (B)

2

Answer 2

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 (2).