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, 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
-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
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:
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.
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.
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.
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.
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 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.