Question

The number of equivalence relations on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is

• A. 3
• B. 1
• C. 2
• D. None of these

Answer 1

Answer: (C)

The smallest equivalence relation $R_1$ containing $(1,2)$ and $(2,1)$ is
$R_1=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$
Now, we are left with four ordered pairs namely $(2,3),(3,2),(1,3)$ and $(3,1)$. If we add any one say $(2,3)$ to $R_1$, then for symmetry, we must add $(3,2)$ and then for transitivity, we are forced to add $(1,3)$ and $(3,1)$. Thus, the only equivalence relation other than $R_1$ is the universal relation. Hence, the total number of equivalence relations containing $(1,2)$ and $(2,1)$ is 2 .