Exercise 4, § 1

(a) Given a set E, let $\Phi$ be the set of mappings into E of any subset of E; if $f, g, h$ are three elements of $\Phi$, show that if the composition $(f \circ g) \circ h$ is defined so is $f \circ (g \circ h)$, but that the converse is not true; if these two compositions are defined they are equal.

(b) Let $\mathscr{F}$ be a family of non-empty subsets of E such that no two have an element in common and let $\Psi$ be the subset of $\Phi$ consisting of the bijective mappings of a set of $\mathscr{F}$ onto a set of $\mathscr{F}$. Show that, under the law induced by the law $f \circ g$ on $\Psi$, the condition of Exercise 3(a) holds.