> [!NOTE] **Definition** (Subset) > Given [[Set|sets]] $a$ and $b,$ we say that $a$ is a *subset* of $b,$ written $a \subset b,$ provided every member $a$ is also a member of $b.$ > [!NOTE] Definition (Proper subset) > If $X$ is a subset of $Y$ and not equal to $Y$, then we say $X$ is a *proper subset* of $Y$ and write $X\subseteq Y$ or $X \subsetneq Y$. # Properties Subset relation is a [[Partial Order|partial order relation]] on the [[Power Set Axiom|power set]] of a set.