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Given a set \(X\), the union of \(X\), denoted \(\cup X\), is the set which contains the elements of all the sets in \(X\) - in symbols: \(a\in \cup X\Leftrightarrow \exists x\in X: a\in x\), or equivalently \(\cup X=\{a|\exists x\in X:a\in x\}\). If \(X=\{x_1,...,x_n\}\) consists of a finite number of sets then we often write the union of \(X\) as \(\cup X=x_1\cup\cdots\cup x_n\). If \(X=\{x_i|i\in I\}\) is an indexed collection of sets then we often denote the union as \(\cup X=\cup_{i\in I}X\). The axiom of union, one of the axioms of Zermelo-Fraenkel set theory, states that for any set \(X\), its union is also a set: \(\forall X:\exists\cup X:\forall a:(a\in\cup X\Leftrightarrow \exists x: a\in x\land x\in X)\).