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Q.
Let $A=\{1,2,3\}$. Then, the number of equivalence relations containing $(1,2)$ is
Relations and Functions - Part 2
Solution:
It is given that $A=\{1,2,3\}$.
An equivalence relation is reflexive, symmetric and transitive.
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 add 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.