Let R and S be reflexive relations on A. Assume R is also transitive. Prove S ⊆ R if and only if (S ◦ R) = R.

Question

I am really stuck on how to show (x,y) is in (S o R). I am assuming S is a subset of R and trying to show (S o R)=R and I am using what my professor calls the double subset strategy, so I proved the first direction, letting (x,y) be in the composition and showing it is in R, but am very stuck on how to go the other way.

Answer 1

Suppose that $(x,y)\in R$. Since $S$ is reflexive, $(x,x)\in S$. But$$(x,x)\in S\text{ and }(x,y)\in R\implies(x,y)\in S\circ R.$$So, $S\circ R\supset R$.