In
set theory and related branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a collection
of
subsets of a given
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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.
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.
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
In mathematics, particularly in combinatorics, given a family of sets, here called a collection ''C'', a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the ...
.
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 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.
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 space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s 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
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
(with respect to the
inclusion relation).
Other examples of set families are
independence systems,
greedoid
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of optimization pro ...
s,
antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids ...
s, and
bornological spaces.
See also
*
*
*
*
*
*
*
*
*
*
* (or ''Set of sets that do not contain themselves'')
*
*
Notes
References
*
*
*
External links
*
{{Set theory
Basic concepts in set theory