Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 5 / 11 / iii / Solution

The preceding proofs show that this inclusion forces both symmetry and transitivity of $T$; its reflexivity already follows from that of $R,S$. It actually forces equality of the composites, not just a one-way inclusion:This is the composition of commuting equivalence relations criterion. It remains to establish the asserted smallest-extension characterization, rather than merely to identify an equivalence relation.