Chapter 1.7, Problem 24E

For any relation $R$ on the nonempty set $A$, the inverse of $R$ is the relation $R^{-1}$ defined by $x{R}^{-1}y$ if and only if $yRx$. Prove the following statements.

$R$ is symmetric if and only if $R=R^{-1}$.

$R$ is antisymmetric if and only if $R\cap R^{-1}$ is a subset of $\{(a,a)|a\in A\}$.

$R$ is asymmetric if and only if $R\cap R^{-1}=\varnothing$.