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 or a set-family or a set-system.

The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

A finite family of subsets of a finite set S is also called a hypergraph.

Examples

Properties

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:

Special types of set families

A Sperner family is a set-family in which none of the sets contains any of the others. Sperner's theorem bounds the maximum size of a Sperner family.

A Helly family is a set-family such that any minimal subfamily with empty intersection has bounded size. Helly's theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

An abstract simplicial complex is a set-family F that is downward-closed, i.e., every subset of a set in F is also in F. A matroid is an abstract simplicial complex with an additional property called the augmentation property.

A convexity space is a set-family closed under arbitrary intersections and unions of chains (with respect to the inclusion relation).

Families of sets over
Is necessarily true of
or, is closed under:
Directed
by
F.I.P.
π-system Yes Yes No No No No No No No No
Semiring Yes Yes No No No No No No Yes Never
Semialgebra (Semifield) Yes Yes No No No No No Yes Yes Never
Monotone class No No No No No only if only if No No No
𝜆-system (Dynkin System) Yes Yes No only if
Yes No only if or
they are disjoint
Yes Yes Never
Ring (Order theory) Yes Yes Yes No No No No No No No
Ring (Measure theory) Yes Yes Yes Yes No No No No Yes Never
δ-Ring Yes Yes Yes Yes No Yes No No Yes Never
𝜎-Ring Yes Yes Yes Yes No Yes Yes No Yes Never
Algebra (Field) Yes Yes Yes Yes Yes No No Yes Yes Never
𝜎-Algebra (𝜎-Field) Yes Yes Yes Yes Yes Yes Yes Yes Yes Never
Dual ideal Yes Yes Yes No No No Yes Yes No No
Filter Yes Yes Yes Never Never No Yes Yes Yes
Prefilter (Filter base) Yes No No Never Never No No No Yes
Filter subbase No No No Never Never No No No Yes
Topology Yes Yes Yes No No No

(even arbitrary unions)
Yes Yes Never
Is necessarily true of
or, is closed under:
directed
downward
finite
intersections
finite
unions
relative
complements
complements
in
countable
intersections
countable
unions
contains contains Finite
Intersection
Property

Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in
A semialgebra is a semiring that contains
are arbitrary elements of and it is assumed that


See also

Notes

  1. ^ Brualdi 2010, pg. 322
  2. ^ Roberts & Tesman 2009, pg. 692
  3. ^ Biggs 1985, pg. 89

References