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Partition**

A partition of a set \(X\) is a collection \(P\subseteq\mathcal{P}(X)\) of non-empty subsets of \(X\) such that their union is all of \(X\), \(\cup P=\cup_{A\in P}A=X\) and they are pairwise disjoint: \(\forall A,B\in P: A\cap B=\emptyset\Leftrightarrow A\neq B\). Equivalently, each element of \(x\) is contained in a unique element of \(P\), \(\forall x\in X:\exists! A\in P:x\in A\).