Let \(\mathscr{A}\) be a \(\sigma\)-algebra. Show that

(i). if \(A_{1}, A_{2}, \ldots, A_{N} \in \mathscr{A}\), then \(A_{1} \cap A_{2} \cap \cdots \cap A_{N} \in \mathscr{A}\);

(ii). \(A \in \mathscr{A}\) if, and only if, \(A^{c} \in \mathscr{A}\);

(iii). if \(A, B \in \mathscr{A}\), then \(A \backslash B \in \mathscr{A}\) and \(A \triangle B \in \mathscr{A}\).