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Subset**

A 'subset' of a set \(X\) is a set \(U\) whose elements are all elements of \(X\). We denote this relation by \(U\subseteq X\); \(U\subseteq X\Leftrightarrow\forall a:a\in U\Rightarrow a\in X\). If \(U\neq X\), then we call \(U\) a 'proper' subset of \(X\).