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

A binary relation $T$ is symmetric exactly when $T=T^{-1}$. By the reversed-chain identity, condition (i) therefore gives $R\circ S=S\circ R$, in particular condition (iii). Conversely, if $S\circ R\subseteq R\circ S$, reversing both sides gives $R\circ S\subseteq S\circ R$. Thus condition (iii) gives equality and therefore symmetry. Conditions (i) and (iii) are equivalent.