Intersection of equivalence relations
= Intersection of equivalence relations
{title2=$R=\bigcap_j R_j$}
The intersection of any family of <equivalence relations> on one <set> is an <equivalence relation>. Reflexivity and symmetry hold in every component. A pair of consecutive related points belongs to every component, so each component's <transitivity> supplies the required final pair. Its <equivalence classes> refine the component partitions. The empty intersection is the universal relation.