In set theory and related branches of mathematics, 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.
The term "collection" is used here because, 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 rather than a set.
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.
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.
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
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 topologiesEdit
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 that continues to cover is called a subcover. A family is called a point-finite collection if every point of lies in only finitely many family members. If every point lies in exactly one member then the cover is called a partition.
When is a topological space, then a cover whose members are all open sets is called and open cover. A family is called locally finite if each point in the space has a neighborhood that intersects only finitely many family members. A σ-locally finite or countably locally finite collection is any family that is equal to a union of countably many locally finite families.
One 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 familiesEdit
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:
|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|
|Prefilter (Filter base)||Never||Never|
(even arbitrary )
(even arbitrary )
|Is necessarily true of
or, is closed under:
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in
- 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 – Algebric structure of set algebra
- σ-ring – Ring closed under countable unions
- ^ Brualdi 2010, pg. 322
- ^ Roberts & Tesman 2009, pg. 692
- ^ Biggs 1985, pg. 89
- 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
- Media related to Set families at Wikimedia Commons