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

If $T$ is transitive, then $T\circ T\subseteq T$. Since $S,R\subseteq T$, monotonicity of composition of relations giveswhich is condition (iii).

Conversely, condition (iii), associativity of composition of relations, and the identities $R\circ R=R$, $S\circ S=S$ giveFor a direct chain description, start with $xRaSbRcSz$. Replace its middle $aSbRc$ by $aRdSc$ using condition (iii), then use transitivity inside $R$ and $S$ to obtain $xRdSz$. Hence $T$ is transitive. Conditions (ii) and (iii) are equivalent.