Family of sets

Inset theoryand related branches ofmathematics,afamily(orcollection) can mean, depending upon the context, any of the following:set,indexed set,multiset,orclass.A collection$F$ofsubsetsof a givenset$S$is called afamily of subsetsof$S$,or afamily of setsover$S.$More generally, a collection of any sets whatsoever is called afamily of sets,set family,or aset system.Additionally, a family of sets may be defined as a function from a set$I$,known as the index set, to$F$,in which case the sets of the family are indexed by members of$I$.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 aproper class.

A finite family of subsets of afinite set$S$is also called ahypergraph.The subject ofextremal set theoryconcerns the largest and smallest examples of families of sets satisfying certain restrictions.

Examples

The set of all subsets of a given set$S$is called thepower setof$S$and is denoted by$\wp (S).$Thepower set$\wp (S)$of a given set$S$is a family of sets over$S.$

A subset of$S$having$k$elements is called a$k$-subsetof$S.$ The$k$-subsets$S^{(k)}$of a set$S$form a family of sets.

Let$S=\{a,b,c,1,2\}.$An example of a family of sets over$S$(in themultisetsense) is given by$F=\left\{A_{1},A_{2},A_{3},A_{4}\right\},$where$A_{1}=\{a,b,c\},A_{2}=\{1,2\},A_{3}=\{1,2\},$and$A_{4}=\{a,b,1\}.$

The class$\operatorname {Ord}$of allordinal numbersis alargefamily of sets. That is, it is not itself a set but instead aproper class.

Properties

Any family of subsets of a set$S$is itself a subset of thepower set$\wp (S)$if it has no repeated members.

Any family of sets without repetitions is asubclassof theproper classof all sets (theuniverse).

Hall's marriage theorem,due toPhilip Hall,gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have asystem of distinct representatives.

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

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:

• Ahypergraph,also called a set system, is formed by a set ofverticestogether with another set ofhyperedges,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.
• Anabstract simplicial complexis a combinatorial abstraction of the notion of asimplicial complex,a shape formed by unions of line segments, triangles, tetrahedra, and higher-dimensionalsimplices,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.
• Anincidence structureconsists of a set ofpoints,a set oflines,and an (arbitrary)binary relation,called theincidence 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 binaryblock codeconsists of a set of codewords, each of which is astringof 0s and 1s, all the same length. When each pair of codewords has largeHamming distance,it can be used as anerror-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.
• Atopological spaceconsists of a pair$(X,\tau )$where$X$is a set (whose elements are calledpoints) and$\tau$is atopologyon$X,$which is a family of sets (whose elements are calledopen sets) over$X$that contains both theempty set$\varnothing$and$X$itself, and is closed under arbitrary set unions and finite set intersections.

Covers and topologies

See also:List of partition topics § Set partitions,andFilters in topology

A family of sets is said tocovera set$X$if every point of$X$belongs to some member of the family. A subfamily of a cover of$X$that is also a cover of$X$is called asubcover. A family is called apoint-finite collectionif every point of$X$lies in only finitely many members of the family. If every point of a cover lies in exactly one member, the cover is apartitionof$X.$

When$X$is atopological space,a cover whose members are allopen setsis called anopen cover. A family is calledlocally finiteif each point in the space has aneighborhoodthat intersects only finitely many members of the family. Aσ-locally finiteorcountably locally finite collectionis a family that is the union of countably many locally finite families.

A cover${\mathcal {F}}$is said torefineanother (coarser) cover${\mathcal {C}}$if every member of${\mathcal {F}}$is contained in some member of${\mathcal {C}}.$Astar refinementis a particular type of refinement.

Special types of set families

ASperner familyis a set family in which none of the sets contains any of the others.Sperner's theorembounds the maximum size of a Sperner family.

AHelly familyis a set family such that any minimal subfamily with empty intersection has bounded size.Helly's theoremstates thatconvex setsinEuclidean spacesof bounded dimension form Helly families.

Anabstract simplicial complexis a set family$F$(consisting of finite sets) that isdownward closed;that is, every subset of a set in$F$is also in$F.$ Amatroidis an abstract simplicial complex with an additional property called theaugmentation property.

Everyfilteris a family of sets.

Aconvexity spaceis a set family closed under arbitrary intersections and unions ofchains(with respect to theinclusion relation).

Other examples of set families areindependence systems,greedoids,antimatroids,andbornological spaces.

Families${\mathcal {F}}$of setsover$\Omega$
Is necessarily true of${\mathcal {F}}\colon$ or, is${\mathcal {F}}$closed under:	Directed by$\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra(Semifield)										Never
Monotone class						only if$A_{i}\searrow$	only if$A_{i}\nearrow$
𝜆-system(Dynkin System)				only if $A\subseteq B$			only if$A_{i}\nearrow$or they aredisjoint			Never
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				$\varnothing \not \in {\mathcal {F}}$
Prefilter(Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary$\cup$)			Never
Closed Topology						(even arbitrary$\cap$)				Never
Is necessarily true of${\mathcal {F}}\colon$ or, is${\mathcal {F}}$closed under:	directed downward	finite intersections	finite unions	relative complements	complements in$\Omega$	countable intersections	countable unions	contains$\Omega$	contains$\varnothing$	Finite Intersection Property
Additionally, asemiringis aπ-systemwhere every complement$B\setminus A$is equal to a finitedisjoint unionof sets in${\mathcal {F}}.$ Asemialgebrais a semiring where every complement$\Omega \setminus A$is equal to a finitedisjoint unionof sets in${\mathcal {F}}.$ $A,B,A_{1},A_{2},\ldots$are arbitrary elements of${\mathcal {F}}$and it is assumed that${\mathcal {F}}\neq \varnothing.$

See also

• 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– Expression denoting a set of sets in formal 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 (orSet of sets that do not contain themselves)
• σ-algebra– Algebraic structure of set algebra
• σ-ring– Family of sets closed under countable unions

References

• Biggs, Norman L. (1985),Discrete Mathematics,Oxford: Clarendon Press,ISBN0-19-853252-0
• Brualdi, Richard A. (2010),Introductory Combinatorics(5th ed.), Upper Saddle River, NJ: Prentice Hall,ISBN0-13-602040-2
• Roberts, Fred S.; Tesman, Barry (2009),Applied Combinatorics(2nd ed.), Boca Raton: CRC Press,ISBN978-1-4200-9982-9

External links

• Media related toSet familiesat Wikimedia Commons