= Solution
An <equivalence relation> on a <set> $A$ is a <binary relation> that is <reflexive>, <symmetric> and <transitive>: every element is related to itself, related pairs may be reversed, and two consecutive related pairs imply the pair of endpoints is related.
Write $T=R\circ S$, using the printed left-to-right <composition of relations> convention. <Reflexivity> implies $R\subseteq T$ and $S\subseteq T$, and hence $T$ is <reflexive>. Reversing a two-step chain gives
$$
T^{-1}=S^{-1}\circ R^{-1}=S\circ R,
$$
since $R,S$ are <symmetric>. Also $R\circ R=R$ and $S\circ S=S$: one inclusion is <transitivity>, and the reverse follows by inserting a reflexive step. These elementary identities supply the equivalences below, without presuming that a composite of <equivalence relations> is automatically an <equivalence relation>.
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