Exercise 1, § 4

(a) Let E be a set and let $\mathfrak{F}(\mathrm{E})$ be the set of finite subsets of E. Show that $\mathfrak{F}(\mathrm{E})$ is the smallest subset $\mathfrak{G}$ of $\mathfrak{P}(\mathrm{E})$ satisfying the following conditions : (i) $\emptyset \in \mathfrak{G}$; (ii) the relations $\mathrm{X} \in \mathfrak{G}$ and $x \in \mathrm{E}$ imply

(b) Deduce from (a) that the union of two finite subsets A, B of E is finite (consider the set of subsets X of E such that $\mathrm{X} \cup \mathrm{A}$ is finite; cf. § 5, no. 1, Proposition 1, Corollary 1).

(c) Deduce from (a) and (b) that for every finite set E the set $\mathfrak{P}(\mathrm{E})$ is finite (consider the set of subsets X of E such that $\mathfrak{P}(\mathrm{X})$ is finite; cf. § 5, no. 1, Proposition 1, Corollary 4).