In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. 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. Additionally, a family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of .[1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member,[2][3][4] and in other contexts it may form a proper class.
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.
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:
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 (arbitrary) binary relation, called the incidence relation, specifying which points belong to which lines. An incidence structure can be specified by a family of sets (even if two distinct lines contain the same set of points), 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 (whose elements are called points) and is a topology on which is a family of sets (whose elements are called open sets) over that contains both the empty set and itself, and is closed under arbitrary set unions and finite set intersections.
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, the cover is a partition of
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.
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
Algebra of sets – Identities and relationships involving sets
Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members