Family of sets explained
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, and in other contexts it may form a
proper class.
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
F=\left\{A1,A2,A3,A4\right\},
where
A1=\{a,b,c\},A2=\{1,2\},A3=\{1,2\},
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.
If
is any family of sets then
\cupl{F}:={stylecup\limitsF
}} F denotes the union of all sets in
where in particular,
\cup\varnothing=\varnothing.
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.
Covers and topologies
See also: Filters in topology.
A family of sets is said to 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 . 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, 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 (coarser) 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
(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 (with respect to the inclusion relation).
Other examples of set families are independence systems, greedoids, antimatroids, and bornological spaces.
See also
- (or Set of sets that do not contain themselves)
Notes and References
- P. Halmos, Naive Set Theory, p.34. The University Series in Undergraduate Mathematics, 1960. Litton Educational Publishing, Inc.
