Exercise 8, § 3

Let A, B be two sets, let $\Gamma$ be a correspondence between A and B, and let $\Gamma'$ be a correspondence between B and A. Show that if $\Gamma'(\Gamma(x)) = \{x\}$ for all $x \in \mathrm{A}$ and if $\Gamma(\Gamma'(y)) = \{y\}$ for all $y \in \mathrm{B}$, then $\Gamma$ is a bijection of A onto B, and $\Gamma'$ is the inverse mapping.