Prove that equality is the smallest equilivence relation on a set A.
Given a set $\displaystyle A$ then $\displaystyle \Delta _A = \left\{ {(x,x):x \in A} \right\}$ is known as the diagonal.
Also $\displaystyle \Delta _A $ is also the equality relation.
Also if $\displaystyle \mathcal{E}$ is an equivalence on $\displaystyle A$ then it must be the case that $\displaystyle \Delta_A\subseteq\mathcal{E}$.