In set theory and related branches of mathematics, a collection of subsets of a given set is called a family of subsets of , or a family of sets over More generally, a collection of any sets whatsoever is called a family of sets, 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 is also called a hypergraph. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions.

Examples

The set of all subsets of a given set is called the power set of and is denoted by The power set of a given set is a family of sets over

A subset of having elements is called a -subset of The -subsets of a set form a family of sets.

Let An example of a family of sets over (in the multiset sense) is given by where and

The class 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 a set is itself a subset of the power set if it has no repeated members.

Any family of sets without repetitions is a subclass of the proper class of all sets (the universe).

Hall's marriage theorem, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a system of distinct representatives.

If is any family of sets then denotes the union of all sets in where in particular, Any family of sets is a family over and also a family over any superset of

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 (consisting of finite sets) that is downward closed; that is, every subset of a set in is also in A matroid is an abstract simplicial complex with an additional property called the augmentation property.

Every filter is a family of sets.

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

Other examples of set families are independence systems, greedoids, antimatroids, and bornological spaces.

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 No Yes Never
Monotone class No No No No No only if only if No No No
𝜆-system (Dynkin System) Yes No 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
Open Topology Yes Yes Yes No No No
Green check.svg

(even arbitrary )
Yes Yes Never
Closed Topology Yes Yes Yes No No
Green check.svg

(even arbitrary )
No 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