How can I show that $R \circ S = S \circ R$ if $S, R$ are symmetric?
I'm working on a longer proof and this is one of the lemmas I think I need to show, since in general a composition of relations isn't necessarily commutative. I'm struggling with how to use composition in a proof in general so I'm not really sure where to start with this.
This is not true. If $R$ and $S$ are functions, then $R$ is symmetric iff it is its own inverse, as a function, and we know that composition of functions (even of order $2$) is not commutative. Concretely, take the relations $R=\left\{(1,2),(2,1),(3,3)\right\}$ and $S=\left\{(1,3),(2,2),(3,1)\right\}$ on $\left\{1,2,3\right\}$. Then $R\circ S=\left\{(1,2),(2,3),(3,1)\right\}$ but $S\circ R=\left\{(1,3),(2,1),(3,2)\right\}$.