Family of sets

From Wikipedia, the free encyclopedia

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.

[edit] 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.

[edit] 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).

[edit] See also

• Indexed family
• Set system
• Hypergraph
• Incidence structure