Question

Let $A=\{1,2,3\}$. Then show that the smallest equivalence relation on $A$ is $\left\{\right(1,1),(2,2),(3,3\left)\right\}$

Answer 1

Given set $A=\{1,2,3\}$
Given $R=\left\{\right(1,1),(2,2),(3,3\left)\right\}$
Clearly, the relation $R$ is reflexive on $A$, as $(x,x)\in R$ for all $x\in A$
Since, there are no ordered pairs of the form $(x,y)$, so no need to check for symmetric and transitive.
Hence, the given relation is smallest equivalence relation on $A.$