Exercise 14, § 1

Let A and B be two sets, and let R be any subset of $\mathrm{A} \times \mathrm{B}$. For each subset X of A (resp. each subset Y of B) let $\rho(\mathrm{X})$ (resp. $\sigma(\mathrm{Y})$) denote the set of all $y \in \mathrm{B}$ (resp. $x \in \mathrm{A}$) such that $(x, y) \in \mathrm{R}$ for all $x \in \mathrm{A}$ (resp. $(x, y) \in \mathrm{R}$ for all $y \in \mathrm{B}$). Show that $\rho$ and $\sigma$ are decreasing mappings, and that the mappings $\mathrm{X} \to \sigma(\rho(\mathrm{X}))$ and $\mathrm{Y} \to \rho(\sigma(\mathrm{Y}))$ are closures (Exercise 13) in $\mathfrak{P}(\mathrm{A})$ and $\mathfrak{P}(\mathrm{B})$ respectively (ordered by inclusion).