= Solution
By the inverse identity established in the root solution, the composite $T$ is a <symmetric relation> precisely when
$$
T=T^{-1}=S\circ R.
$$
Thus condition (i) implies condition (iii). Conversely, if $S\circ R\subseteq R\circ S$, take inverse relations. Since $R,S$ are symmetric, this gives the reverse inclusion $R\circ S\subseteq S\circ R$. The two composites are equal, making $T$ symmetric. Therefore \b[(i) and (iii) are equivalent], and either one <forces> commutation.
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