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.
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.
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
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 family of sets is said to cover a set if every point of belongs to some member of the family. A subfamily of a cover that continues to cover is called a subcover. A family is called a point-finite collection if every point of lies in only finitely many family members. If every point lies in exactly one member then the cover is called a partition.
When is a topological space, then a cover whose members are all open sets is called and open cover. A family is called locally finite if each point in the space has a neighborhood that intersects only finitely many family members. A σ-locally finite or countably locally finite collection is any family that is equal to a union of countably many locally finite families.
One cover is said to refine another (coarser) cover if every member of is contained in some member of A star refinement is a particular type of refinement.
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 | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
𝜆-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
δ-Ring | Never | |||||||||
𝜎-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
𝜎-Algebra (𝜎-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
Open Topology |
(even arbitrary ) |
Never | ||||||||
Closed Topology |
(even arbitrary ) |
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 |