Equivalence closure
= Equivalence closure
{title2=$\operatorname{Eq}(R)=(R\cup R^{-1}\cup\Delta_A)^+$}
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.