Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 1 / c / iii / Solution

False. Use the same two equivalence relations as in part (ii). They are both symmetric relations. The composite contains $1S3$, since $1R_{1}2$ and $2R_{2}3$. It does not contain $3S1$, as shown there. Hence composition of relations need not preserve symmetry of a relation, even when both factors are also reflexive and transitive.