In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
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
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
is also called a ''
hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, an undirected hypergraph H is a pair H = (X,E) wh ...
''. 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
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of
and is denoted by
The
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
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
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
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
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
if it has no repeated members.
Any family of sets without repetitions is a
subclass of the
proper class of all sets (the
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
).
Hall's marriage theorem
In mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations:
* The combinatorial formulation deals with a collection of finite sets. It gives a necessary and sufficient condition for being able to select a di ...
, due to
Philip Hall
Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups.
Biography
He was educated first at Christ's Hospital, where he won the Thomps ...
, 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
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
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, an undirected hypergraph H is a pair H = (X,E) wh ...
, also called a set system, is formed by a set of
vertices together with another set of ''
hyperedges
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
Symbols
A
B
...
'', 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
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
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
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, 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
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
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
In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks.
There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract defini ...
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
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
, it can be used as an
error-correcting code
In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...
. 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
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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 In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family ''F'' of subsets of a finite set ''E'' in which none of the sets contains another. Equivalently, a Sperner family is an antichain i ...
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
In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non ...
is a set family such that any minimal subfamily with empty intersection has bounded size.
Helly's theorem
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's t ...
states that
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s of bounded dimension form Helly families.
An
abstract simplicial complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
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
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...
is an abstract simplicial complex with an additional property called the ''
augmentation property''.
Every
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
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. A c ...
(with respect to the
inclusion relation
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
).
Other examples of set families are
independence system In combinatorial mathematics, an independence system is a pair (V, \mathcal), where is a finite set and is a collection of subsets of (called the independent sets or feasible sets) with the following properties:
# The empty set is independent, i ...
s,
greedoid
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Hassler Whitney, Whitney in 1935 to study planar graphs and was later used by Jack Edmonds, Edmonds to characterize a ...
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. Antimatroi ...
s, and
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
s.
See also
*
*
*
*
*
*
*
*
*
*
* (or ''Set of sets that do not contain themselves'')
*
*
Notes
References
*
*
*
External links
*
{{Set theory
Basic concepts in set theory