Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$1$

$A=\{1,2,3\}$

For equivalence relation containing $(1,2)$

For symmetric, it must consists $(1,2)$ and $(2,1)$.

For transitivity, it must consists $(1,3)$ and $(3,2)$ and $(1,2),(2,1),(2,3),(3,1)$

For reflexivity, it must consists $(1,1)$ and $(2,2),(3,3)$

$\Rightarrow R=\left\{\right(1,1),(2,2),(2,3),(3,3),(1,2\left)\right(1,3),(2,1),(3,1),(3,1\left)\right\}$

$\Rightarrow$ Only $1$ such relation is possible.