Let \(\mathcal{E}, \mathscr{F} \subset \mathscr{P}(X)\) be two families of subsets of \(X\). One usually uses the notation (as we do in this book)

\[\mathcal{E} \cup \mathscr{F}=\{A: A \in \mathcal{E} \text { or } A \in \mathscr{F}\} \quad \text { and } \quad \mathcal{E} \cap \mathscr{F}=\{A: A \in \mathcal{E} \text { and } A \in \mathscr{F}\} .\]

Let us, for this problem, also introduce the families

\[\mathcal{E} \uplus \mathscr{F}=\{E \cup F: E \in \mathcal{E}, F \in \mathscr{F}\} \quad \text { and } \quad \mathcal{E} \cap \mathscr{F}=\{E \cap F: E \in \mathcal{E}, F \in \mathscr{F}\} \text {. }\]

Assume now that \(\mathcal{E}\) and \(\mathscr{F}\) are \(\sigma\)-Algebras.

(i) Show that \(\mathcal{E} \uplus \mathscr{F} \supset \mathcal{E} \cup \mathscr{F}\) and \(\mathcal{E} \cap \mathscr{F} \supset \mathcal{E} \cup \mathscr{F}\).

(ii) Show that, in general, we have no equality in part (i).

(iii) Show that \(\sigma(\mathcal{E} \uplus \mathscr{F})=\sigma(\mathcal{E} \cap \mathscr{F})=\sigma(\mathcal{E} \cup \mathscr{F})\).