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(Axiom of the) Power Set**

Given a set \(X\), its 'power set' \(\mathcal{P}(X)\) is the set of all subsets of \(X\); \(\mathcal{P}(X)=\{U|U\subseteq X\}\). The 'axiom of the power set', one of the Zermelo-Fraenkel axioms, asserts that for any set \(X\), there exists a set whose elements are precisely the subsets of \(X\) (i.e. the power set), in symbols: \(\forall X:\exists\mathcal{P}(X):\forall U:(U\in\mathcal{P}(X)\iff \forall a:a\in U\Rightarrow a\in X)\).