Relation, union, intersection

Question

Let $R$ be a relation on a set $X$. Then prove that $R\cup R^{-1}$ is the smallest symmetric relation containing $R$ and $R\cap R^{-1}$ is the largest symmetric relation contained in $R$.

Answer 1

The largest symmetric relation containing R is X×X.
If R = {(0,1)}: R$^{-1}$ = {(1,0)}; R $\cap$ R$^{-1}$ is empty; does not contain R.

On the other hand if R is a relation, then Rs = R $\cup$ R$^{-1}$ is symmetric and contains R.
There are two cases to prove Rs is symmetric. What are they?
To prove Rs is the smallest symmetric relation containing R,
assume Q is a symmetric relation containg R.
Prove Rs is a subset of Q.