Exercise 1, § 3

Let E and F be two sets, let $f$ be an injection of E into F, and let $g$ be a mapping of F into E. Show that there exist two subsets A, B of E such that $\mathrm{B} = \mathrm{E} - \mathrm{A}$, and two subsets $\mathrm{A}'$, $\mathrm{B}'$ of F such that $\mathrm{B}' = \mathrm{F} - \mathrm{A}'$, for which $\mathrm{A}' = f(\mathrm{A})$ and $\mathrm{B} = g(\mathrm{B}')$. (Let $\mathrm{R} = \mathrm{E} - g(\mathrm{F})$ and put $h = g \circ f$; take A to be the intersection of the subsets M of E such that $\mathrm{M} \supset \mathrm{R} \cup h(\mathrm{M})$.)