Question

Let $A=\{1,2,3\}$. Then, the number of equivalence relations containing $(1,2)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

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.