Show that any relation R on any set A is symmetric if and only if $R = R^{-1}$

Question

For phone users,

Prove that any relation $R$ on any set $A$ is symmetric if and only if $R = R^{-1}$.

In fact, it is, but how?

Answer 1

I assume that you are looking for a way to write out a formal proof of this rather self-evident result.

First suppose that $R=R^{-1}$. Then

$(a,b)\in R \implies (b,a)\in R^{-1}\implies (b,a)\in R$. Therefore $R$ is symmetric.

Next suppose that $R$ is symmetric. Then we have both:

$(a,b)\in R \implies (b,a)\in R\implies (a,b)\in R^{-1}$ $(a,b)\in R^{-1} \implies (b,a)\in R\implies (a,b)\in R$

Therefore $R=R^{-1}$.