Verify that \(\mathcal{F}=\left\{\emptyset, \omega_{1}, \omega_{2}, \omega_{3}, \omega_{1} \cup \omega_{2}, \omega_{1} \cup \omega_{3}, \omega_{2} \cup \omega_{3}, \omega_{1} \cup \omega_{2} \cup \omega_{3}ight\}\) is a \(\sigma\)-field containing the sets \(\omega_{1}, \omega_{2}\), and \(\omega_{3}\). Prove that this is the smallest possible \(\sigma\)-field containing these sets by showing that \(\mathcal{F}\) is no longer a \(\sigma\)-field if any of the sets are eliminated from \(\mathcal{F}\).