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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 $U\subseteq X$; $U\subseteq X\iff \mathrm{\forall }a:a\in U\Rightarrow a\in X$. If $U\ne X$, then we call $U$ a 'proper' subset of $X$.