Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

Examples

* The power set P(S) is a family of sets over S.
* The k-subsets S^(k) of an n-set S form a family of sets.
* An abstract simplicial complex is a family of sets.
* The class Ord of all ordinal numbers is a "large" family of sets; that is, it is not itself a set but instead a proper class.

Properties

* Any family of subsets of S is itself a subset of the power set P(S).
* Any family of sets whatsoever is a subclass of the proper class V of all sets (the universe).
* A hypergraph is a set V (the set of "vertices") together with a nonempty family of subsets of V (the "edges").

See also

* Indexed family
* Set system
* Hypergraph
* Incidence structure