Exercise 1, § 1

Let E be a set, $A \subset E \times E$ and $(x, y) \mapsto x \top y,\ (x, y) \in A$, an internal law of composition not everywhere defined on E. Given two subsets X and Y of E, let $X \top Y$ denote the set of elements of E of the form $x \top y$ with $x \in X$, $y \in Y$ and $(x, y) \in A$. A law of composition is thus defined which is everywhere defined on $\mathscr{P}(E)$.

If $(X_\alpha)_{\alpha \in A}$ and $(Y_\beta)_{\beta \in B}$ are any two families of subsets of E, then