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.