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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, indexed set, multiset, or class. A collection F {\displaystyle F} of subsets of a given set S {\displaystyle S} is called a family of subsets of S {\displaystyle S} , or a family of sets over S . {\displaystyle S.} 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 I {\displaystyle I} , known as the index set, to F {\displaystyle F} , in which case the sets of the family are indexed by members of I {\displaystyle I} .[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 S {\displaystyle S} is also called a hypergraph. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions.
The set of all subsets of a given set S {\displaystyle S} is called the power set of S {\displaystyle S} and is denoted by ℘ ( S ) . {\displaystyle \wp (S).} The power set ℘ ( S ) {\displaystyle \wp (S)} of a given set S {\displaystyle S} is a family of sets over S . {\displaystyle S.}
A subset of S {\displaystyle S} having k {\displaystyle k} elements is called a k {\displaystyle k} -subset of S . {\displaystyle S.} The k {\displaystyle k} -subsets S ( k ) {\displaystyle S^{(k)}} of a set S {\displaystyle S} form a family of sets.
Let S = { a , b , c , 1 , 2 } . {\displaystyle S=\{a,b,c,1,2\}.} An example of a family of sets over S {\displaystyle S} (in the multiset sense) is given by F = { A 1 , A 2 , A 3 , A 4 } , {\displaystyle 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 } , {\displaystyle A_{1}=\{a,b,c\},A_{2}=\{1,2\},A_{3}=\{1,2\},} and A 4 = { a , b , 1 } . {\displaystyle A_{4}=\{a,b,1\}.}
The class Ord {\displaystyle \operatorname {Ord} } of all ordinal numbers is a large family of sets. That is, it is not itself a set but instead a proper class.
Any family of subsets of a set S {\displaystyle S} is itself a subset of the power set ℘ ( S ) {\displaystyle \wp (S)} 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 F {\displaystyle {\mathcal {F}}} is any family of sets then ∪ F := ⋃ F ∈ F F {\displaystyle \cup {\mathcal {F}}:={\textstyle \bigcup \limits _{F\in {\mathcal {F}}}}F} denotes the union of all sets in F , {\displaystyle {\mathcal {F}},} where in particular, ∪ ∅ = ∅ . {\displaystyle \cup \varnothing =\varnothing .} Any family F {\displaystyle {\mathcal {F}}} of sets is a family over ∪ F {\displaystyle \cup {\mathcal {F}}} and also a family over any superset of ∪ F . {\displaystyle \cup {\mathcal {F}}.}
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 family of sets is said to cover a set X {\displaystyle X} if every point of X {\displaystyle X} belongs to some member of the family. A subfamily of a cover of X {\displaystyle X} that is also a cover of X {\displaystyle X} is called a subcover. A family is called a point-finite collection if every point of X {\displaystyle X} lies in only finitely many members of the family. If every point of a cover lies in exactly one member of X {\displaystyle X} , the cover is a partition of X . {\displaystyle X.}
When X {\displaystyle X} 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 F {\displaystyle {\mathcal {F}}} is said to refine another (coarser) cover C {\displaystyle {\mathcal {C}}} if every member of F {\displaystyle {\mathcal {F}}} is contained in some member of C . {\displaystyle {\mathcal {C}}.} 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 F {\displaystyle F} (consisting of finite sets) that is downward closed; that is, every subset of a set in F {\displaystyle F} is also in F . {\displaystyle F.} 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 F {\displaystyle {\mathcal {F}}} of sets over Ω {\displaystyle \Omega } Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon }Additionally, a semiring is a π-system where every complement B ∖ A {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement Ω ∖ A {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A , B , A 1 , A 2 , … {\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of F {\displaystyle {\mathcal {F}}} and it is assumed that F ≠ ∅ . {\displaystyle {\mathcal {F}}\neq \varnothing .}
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