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,[2][3][4] and in other contexts it may form a proper class.

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 edit

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 edit

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   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 edit

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 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.

Covers and topologies edit

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 of   that is also a cover of   is called a subcover. A family is called a point-finite collection 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 open cover. A family is called locally finite if each point in the space has a neighborhood that intersects only finitely many members of the family. A σ-locally finite or countably locally finite collection is a family that is the union of countably many locally finite families.

A cover   is said to refine another (coarser) cover   if every member of   is contained in some member of   A star refinement is a particular type of refinement.

Special types of set families edit

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.

Families   of sets over  
Is necessarily true of  
or, is   closed under:
Semiring                   Never
Semialgebra (Semifield)                   Never
Monotone class           only if   only if        
𝜆-system (Dynkin System)       only if
    only if   or
they are disjoint
Ring (Order theory)                    
Ring (Measure theory)                   Never
δ-Ring                   Never
𝜎-Ring                   Never
Algebra (Field)                   Never
𝜎-Algebra (𝜎-Field)                   Never
Dual ideal                    
Filter       Never Never          
Prefilter (Filter base)       Never Never          
Filter subbase       Never Never          
Open Topology              
(even arbitrary  )
Closed Topology            
(even arbitrary  )
Is necessarily true of  
or, is   closed under:
contains   contains   Finite

Additionally, a semiring is a π-system where every complement   is equal to a finite disjoint union of sets in  
A semialgebra is a semiring where every complement   is equal to a finite disjoint union of sets in  
  are arbitrary elements of   and it is assumed that  

See also edit

  • Algebra of sets – Identities and relationships involving sets
  • Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members
  • Combinatorial design – Symmetric arrangement of finite sets
  • δ-ring – Ring closed under countable intersections
  • Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
  • Generalized quantifier – type of expression in linguistic semantics
  • Indexed family – Collection of objects, each associated with an element from some index set
  • λ-system (Dynkin system) – Family closed under complements and countable disjoint unions
  • π-system – Family of sets closed under intersection
  • Ring of sets – Family closed under unions and relative complements
  • Russell's paradox – Paradox in set theory (or Set of sets that do not contain themselves)
  • σ-algebra – Algebraic structure of set algebra
  • σ-ring – Ring closed under countable unions

Notes edit

  1. ^ P. Halmos, Naive Set Theory, p.34. The University Series in Undergraduate Mathematics, 1960. Litton Educational Publishing, Inc.
  2. ^ Brualdi 2010, pg. 322
  3. ^ Roberts & Tesman 2009, pg. 692
  4. ^ Biggs 1985, pg. 89

References edit

  • Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0
  • Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-602040-2
  • Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN 978-1-4200-9982-9

External links edit