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Q.
The number of equivalence relations on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is
AMUAMU 2015Relations and Functions - Part 2
Solution:
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 .