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.
• 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).

Related concepts

Certain types of objects from other areas of mathematics are equivalent to families of sets, in that they can be described purely as a collection of sets of objects of some type:

• A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges, each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any family of sets can be interpreted as a hypergraph that has the union of the sets as its vertices.
• An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line segments, triangles, tetrahedra, and higher dimensional simplices, joined face to face. In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family of finite sets in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex.
• An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation specifying which points belong to which lines. If no two lines contain the same set of points, an incidence structure can be specified by a family of sets, the sets of points belonging to each line, and any family of sets can be interpreted as an incidence structure in this way.

See also

• Indexed family
• Class (set theory)

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