2014-21 Duality

Let \(\mathcal F\) be a non-empty collection of subsets of a finite set \(U\). Let \(D(\mathcal F)\)  be the collection of subsets of \(U\) that are subsets of an odd number of members of \(\mathcal F\). Prove that \(D(D(\mathcal F))=\mathcal F\).

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