Reflexive closure Proof

Question

I have this problem I can't figure out.

Suppose R is a relation on A, and let S be the reflexive closure of R. Prove that if R is symmetric, also is S.

Could you suggest me how to do it?

Thanks

Answer 1

Consider $(x,y)\in S$. You need to show that $(y,x) \in S$.

• Case $x=y$. Trivial.

• Case $x \neq y$. Then we observe $(x,y) \in R$ and finish the proof based on the symmetry of $R$.

Answer 2

Taking the reflexive closure, we are only adding points (from the diagonal) that cannot be a counterexample to symmetry..