Subset Systems
Why
We are often interested in a set of subsets of a given set.
Definition
Let $A$ be a non-empty set. A subset system (or set system) is a pair $(A, \mathcal{A} )$ in which $\mathcal{A} \subset \powerset{A}$. In this common case we call the first set the base set and the second set the distinguished subsets. A subset of $B \subset A$ which is not distinguished (i.e., $B \not\in \mathcal{A} $) is called undistinguished.
Let $A$ be a nonempty set.
Let $\mathcal{A} $ be $\powerset{A}$.
Then $(A, \mathcal{A} )$ is a subset system.
Other terminology
Other terminology refers to $(U, \mathcal{F} )$ as a set system when $U$ is a nonempty finite set and $\mathcal{F} $ is a family of subsets of $U$. Set systems are also known as hypergraphs.