Equivalence closure ($\mathrm{Eq}(R)=(R\cup R^{-1}\cup {\Delta }_A{)}^{+}$)

The equivalence closure of a binary relation $R$ is the least equivalence relation containing it. Add diagonal and reversed pairs, then connect points by finite chains. Constant chains prove reflexivity, reversed chains prove symmetry, and concatenated chains prove transitivity. Any equivalence relation extending $R$ contains every such chain endpoint by induction.