In set theory and related branches of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used in phrases like "family of sets" because if one instead uses "set of sets" then the subsequent use of "set" can be confusing as to whether it is the containing set or one of the member sets. A common use is "family of subsets of some set S". A family of sets is also called a set family or a set system. 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 collection 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 .
The trace of a family of subsets of on a subset is .
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 of that is also a cover of is called a subcover . A family is called a point-finite collection if every point of lies in only finitely many members of the family. If every point of a cover lies in exactly one member of , the cover is a partition of .
When is a topological space, a cover whose members are all open sets is called an open cover . A family is called locally finite if each point in the space has a neighborhood that intersects only finitely many members of the family. A σ-locally finite or countably locally finite collection is a family that is the union of countably many locally finite families.
A 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 | ||||||||||
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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 |
Filter | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Proper filter | ![]() | ![]() | ![]() | Never | Never | ![]() | ![]() | ![]() | Never | ![]() |
Prefilter(Filter base) | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Filter subbase | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
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 |