Let \(\mathscr{A}\) be a \(\sigma\)-algebra, \(\mathscr{D}\) be a Dynkin system and \(\mathscr{G} \subset \mathscr{H} \subset \mathscr{P}(X)\) two collections of subsets of \(X\). Show that

(i) \(\delta(\mathscr{A})=\mathscr{A}\) and \(\delta(\mathscr{D})=\mathscr{D}\);

(ii) \(\delta(\mathscr{G}) \subset \delta(\mathscr{H})\);

(iii) \(\delta(\mathscr{G}) \subset \sigma(\mathscr{G})\).