Question: Given a non-empty collection \(\mathcal{A}\) of sets, we defined \(\mathcal{F}(\mathcal{A})\) as the intersection of all fields containing \(\mathcal{A}\). Show that \(\mathcal{F}(\mathcal{A})\) is the class of sets of the form \(\cup_{i=1}^{m} \cap_{j=1}^{n_{i}} A_{i j}\), where for each \(i\) and \(j\) either \(A_{i, j} \in \mathcal{A}\) or \(A_{i j}^{c} \in \mathcal{A}\), and where the \(m\) sets \(\cap_{j=1}^{n_{i}} A_{i j}, 1 \leq i \leq m\) are disjoint.

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