Family of sets

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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.

Contents

Examples

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.

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 .

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:

Covers and topologies

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.

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 Check-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svg
Semiring Check-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svgNever
Semialgebra(Semifield) Check-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svgNever
Monotone class Dark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgonly if only if Dark Red x.svgDark Red x.svgDark Red x.svg
𝜆-system(Dynkin System) Check-green.svgDark Red x.svgDark Red x.svgonly if
Check-green.svgDark Red x.svgonly if or
they are disjoint
Check-green.svgCheck-green.svgNever
Ring (Order theory) Check-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svg
Ring (Measure theory) Check-green.svgCheck-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svgNever
δ-Ring Check-green.svgCheck-green.svgCheck-green.svgCheck-green.svgDark Red x.svgCheck-green.svgDark Red x.svgDark Red x.svgCheck-green.svgNever
𝜎-Ring Check-green.svgCheck-green.svgCheck-green.svgCheck-green.svgDark Red x.svgCheck-green.svgCheck-green.svgDark Red x.svgCheck-green.svgNever
Algebra (Field) Check-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svgCheck-green.svgCheck-green.svgNever
𝜎-Algebra(𝜎-Field) Check-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgCheck-green.svgNever
Filter Check-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svg
Proper filter Check-green.svgCheck-green.svgCheck-green.svgNeverNeverDark Red x.svgCheck-green.svgCheck-green.svgNeverCheck-green.svg
Prefilter(Filter base) Check-green.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svg
Filter subbase Dark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgDark Red x.svgCheck-green.svg
Open Topology Check-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svgDark Red x.svg Green check.svg
(even arbitrary )
Check-green.svgCheck-green.svgNever
Closed Topology Check-green.svgCheck-green.svgCheck-green.svgDark Red x.svgDark Red x.svg Green check.svg
(even arbitrary )
Dark Red x.svgCheck-green.svgCheck-green.svgNever
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 where every complement is equal to a finite disjoint union of sets in
are arbitrary elements of and it is assumed that


See also

Notes

    References