# Definition:Membership Relation

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.
Let $\mathcal R \subseteq S \times \mathcal P \left({S}\right)$ be the relation defined as:
$\left({x, A}\right) \in \mathcal R \iff x \in A$
Thus $\mathcal R$ is the relation between elements of $S$ and subsets of $S$ expressing membership.