Prove: R∩R−1 is symmetric. The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric."
If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, whatever elements are in the intersection must have both (x,y) and (y,x); making it symmetric, but how do I go about formally proving it?

Choose any $(x, y) \in R \cap R^{-1}$. To show that $R \cap R^{-1}$ is symmetric, it suffices to show that $(y, x) \in R \cap R^{-1}$. Indeed, since $(x, y) \in R \cap R^{-1}$, we know that $(x, y) \in R$ and $(x, y) \in R^{-1}$. The former implies that $(y, x) \in R^{-1}$ and the latter implies that $(y, x) \in R$. Hence, we conclude that $(y, x) \in R \cap R^{-1}$, as desired. $~~\blacksquare$
