View source for Family of sets

{{Short description|Any collection of sets, or subsets of a set}}

In [[set theory]] and related branches of [[mathematics]], a '''family''' (or '''collection''') can mean, depending upon the context, any of the following: [[Set (mathematics)|set]], [[indexed set]], [[multiset]], or [[Class (set theory)|class]].  A collection <math>F</math> of [[subset]]s of a given [[Set (mathematics)|set]] <math>S</math> is called a '''family of subsets''' of <math>S</math>, or a '''family of sets''' over <math>S.</math> 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 <math>I</math>, known as the index set, to <math>F</math>, in which case the sets of the family are indexed by members of <math>I</math>.<ref>P. Halmos, ''Naive Set Theory'', p.34. The University Series in Undergraduate Mathematics, 1960. Litton Educational Publishing, Inc.</ref> In some contexts, a family of sets may be allowed to contain repeated copies of any given member,<ref>{{harvnb|Brualdi|2010|loc=pg. 322}}</ref><ref>{{harvnb|Roberts|Tesman|2009|loc=pg. 692}}</ref><ref>{{harvnb|Biggs|1985|loc=pg. 89}}</ref> and in other contexts it may form a [[Class (set theory)|proper class]].

A finite family of subsets of a [[finite set]] <math>S</math> 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 <math>S</math> is called the [[power set]] of <math>S</math> and is denoted by <math>\wp(S).</math> The [[power set]] <math>\wp(S)</math> of a given set <math>S</math> is a family of sets over <math>S.</math> 

A subset of <math>S</math> having <math>k</math> elements is called a [[Finite set#Definition and terminology|<math>k</math>-subset]] of <math>S.</math> 
The [[Finite set#Definition and terminology|<math>k</math>-subsets]] <math>S^{(k)}</math> of a set <math>S</math> form a family of sets. 

Let <math>S = \{a, b, c, 1, 2\}.</math> An example of a family of sets over <math>S</math> (in the [[multiset]] sense) is given by <math>F = \left\{A_1, A_2, A_3, A_4\right\},</math> where <math>A_1 = \{a, b, c\}, A_2 = \{1, 2\}, A_3 = \{1, 2\},</math> and <math>A_4 = \{a, b, 1\}.</math>

The class <math>\operatorname{Ord}</math> of all [[ordinal number]]s is a ''large'' family of sets. That is, it is not itself a set but instead a [[Class (set theory)|proper class]].

==Properties==

Any family of subsets of a set <math>S</math> is itself a subset of the [[power set]] <math>\wp(S)</math> if it has no repeated members. 

Any family of sets without repetitions is a [[Subclass (sets)|subclass]] of the [[Class (set theory)|proper class]] of all sets (the [[universe (math)|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 [[Transversal (combinatorics)|system of distinct representatives]].

If <math>\mathcal{F}</math> is any family of sets then <math>\cup \mathcal{F} := {\textstyle \bigcup\limits_{F \in \mathcal{F}}} F</math> denotes the union of all sets in <math>\mathcal{F},</math> where in particular, <math>\cup \varnothing = \varnothing.</math> 
Any family <math>\mathcal{F}</math> of sets is a family over <math>\cup \mathcal{F}</math> and also a family over any superset of <math>\cup \mathcal{F}.</math>

==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 [[Vertex (graph theory)|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 [[Simplex|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 (computer science)|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&nbsp;1.
* A [[topological space]] consists of a pair <math>(X, \tau)</math> where <math>X</math> is a set (whose elements are called ''points'') and <math>\tau</math> is a {{em|[[Topology (structure)|topology]]}} on <math>X,</math> which is a family of sets (whose elements are called ''open sets'') over <math>X</math> that contains both the [[empty set]] <math>\varnothing</math> and <math>X</math> itself, and is closed under arbitrary set unions and finite set intersections.

===Covers and topologies===

{{See also|List of partition topics#Set partitions|Filters in topology}}

A family of sets is said to {{em|[[Cover (topology)|cover]]}} a set <math>X</math> if every point of <math>X</math> belongs to some member of the family. 
A subfamily of a cover of <math>X</math> that is also a cover of <math>X</math> is called a {{em|[[subcover]]}}. 
A family is called a {{em|[[point-finite collection]]}} if every point of <math>X</math> lies in only finitely many members of the family. If every point of a cover lies in exactly one member of <math>X</math>, the cover is a [[Partition of a set|partition]] of <math>X.</math>

When <math>X</math> is a [[topological space]], a cover whose members are all [[open set]]s is called an {{em|[[open cover]]}}. 
A family is called {{em|[[Locally finite collection|locally finite]]}} if each point in the space has a [[Neighbourhood (mathematics)|neighborhood]] that intersects only finitely many members of the family.
A {{em|[[σ-locally finite]]}} or {{em|[[countably locally finite collection]]}} is a family that is the union of countably many locally finite families.

A cover <math>\mathcal{F}</math> is said to {{em|[[Refinement of an open cover|refine]]}} another (coarser) cover <math>\mathcal{C}</math> if every member of <math>\mathcal{F}</math> is contained in some member of <math>\mathcal{C}.</math> A {{em|[[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.

A '''[[Helly family]]''' is a set family such that any minimal subfamily with empty intersection has bounded size. [[Helly's theorem]] states that [[convex set]]s in [[Euclidean space]]s of bounded dimension form Helly families.

An '''[[abstract simplicial complex]]''' is a set family <math>F</math> (consisting of finite sets) that is [[downward closed]]; that is, every subset of a set in <math>F</math> is also in <math>F.</math> 
A '''[[matroid]]''' is an abstract simplicial complex with an additional property called the ''[[augmentation property]]''. 

Every [[Filter (set theory)|filter]] is a family of sets.

A '''[[convexity space]]''' is a set family closed under arbitrary intersections and unions of [[Total order|chains]] (with respect to the [[inclusion relation]]).

Other examples of set families are [[independence system]]s, [[greedoid]]s, [[antimatroid]]s, and [[bornological space]]s.

{{Families of sets|expanded}}
<br/>
{{clear}}

==See also==

* {{annotated link|Algebra of sets}}
* {{annotated link|Class (set theory)}}
* {{annotated link|Combinatorial design}}
* {{annotated link|Delta-ring|δ-ring}}
* {{annotated link|Field of sets}}
* {{annotated link|Generalized quantifier}}
* {{annotated link|Indexed family}}
* {{annotated link|Dynkin system|λ-system (Dynkin system)}}
* {{annotated link|Pi-system|π-system}}
* {{annotated link|Ring of sets}}
* {{annotated link|Russell's paradox}} (or ''Set of sets that do not contain themselves'')
* {{annotated link|σ-algebra}}
* {{annotated link|Sigma-ring|σ-ring}}

==Notes==

{{reflist}}

==References==

* {{citation|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Clarendon Press|place=Oxford|year=1985|isbn=0-19-853252-0}}
* {{citation|last=Brualdi|first=Richard A.|title=Introductory Combinatorics|edition=5th|year=2010|publisher=Prentice Hall|place=Upper Saddle River, NJ |isbn=978-0-13-602040-0}}
* {{citation|last1=Roberts|first1=Fred S.|last2=Tesman|first2=Barry|title=Applied Combinatorics|edition=2nd|publisher=CRC Press|place=Boca Raton|year=2009|isbn=978-1-4200-9982-9}}

==External links==

* {{Commonscatinline|Set families}}

{{Set theory}}

[[Category:Basic concepts in set theory]]
[[Category:Families of sets| ]]