Family of sets
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 ". 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.
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 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.
Hall's marriage theorem, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets 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 of a family of subsets of on a subset is.
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:- A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges, each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any family of sets can be interpreted as a hypergraph that has the union of the sets as its vertices.
- An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face. In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family of finite sets without repetitions in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex.
- An incidence structure consists of a set of points, a set of lines, and an binary relation, called the incidence relation, specifying which points belong to which lines. An incidence structure can be specified by a family of sets, the sets of points belonging to each line, and any family of sets can be interpreted as an incidence structure in this way.
- A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length. When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A block code can also be described as a family of sets, by describing each codeword as the set of positions at which it contains a 1.
- A topological space consists of a pair where is a set and is a on, which is a family of sets over that contains both the empty set and itself, and is closed under arbitrary set unions and finite set intersections.
Covers and topologies
A subfamily of a cover of that is also a cover of is called a.
A family is called a 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.
A family is called if each point in the space has a neighborhood that intersects only finitely many members of the family.
A or is a family that is the union of countably many locally finite families.
A cover is said to another cover if every member of is contained in some member of. A 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 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.
Other examples of set families are independence systems, greedoids, antimatroids, and bornological spaces.