In set theory and related branches of mathematics, a collection *F* of subsets of a given set *S* is called a **family of subsets** of *S*, or a **family of sets** over *S*. More generally, a collection of any sets whatsoever is called a **family of sets** or a **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,^{[1]}^{[2]}^{[3]} and in other contexts it may form a proper class rather than a set.

A finite family of subsets of a finite set S is also called a **hypergraph**.

## Examples

- The power set
**P**(*S*) is a family of sets over*S*. - The
*k*-subsets*S*^{(k)}of a set*S*(i.e., a subset of*S*with the number of the subset elements as*k*) form a family of sets. - Let
*S*= {a,b,c,1,2}. An example of a family of sets over*S*(in the multiset sense) is given by*F*= {A_{1}, A_{2}, A_{3}, A_{4}}, where A_{1}= {a,b,c}, A_{2}= {1,2}, A_{3}= {1,2} and A_{4}= {a,b,1}. - The class Ord of all ordinal numbers is a
*large*family of sets. That is, it is not itself a set but instead a proper class.

## Properties

- Any family of subsets of
*S*is itself a subset of the power set**P**(*S*) if it has no repeated members. - Any family of sets without repetitions is a subclass of the proper class
**V**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.

## 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 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 (X, τ) where X is a set (called
*points*) and τ is a family of sets (called*open sets*) over X. τ must contain both the empty set and X itself, and is closed under set union and finite set intersection.

## 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 sets in Euclidean spaces of bounded dimension form Helly families.

An **abstract simplicial complex** is a set-family *F* that is downward-closed, i.e., every subset of a set in *F* is also in *F*. A **matroid** is an abstract simplicial complex with an additional property called the *augmentation property*.

Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||

π-system | ||||||||||

Semiring | Never | |||||||||

Semialgebra (Semifield) | Never | |||||||||

Monotone class | only if | only if | ||||||||

𝜆-system (Dynkin System) | only if |
only if or they are disjoint |
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 | ||||||||

Prefilter (Filter base) | Never | Never | ||||||||

Filter subbase | Never | Never | ||||||||

Topology | (even arbitrary unions) |
Never | ||||||||

Is necessarily true of or, is closed under: |
directed downward |
finite intersections |
finite unions |
relative complements |
complements in |
countable intersections |
countable unions |
contains | contains | Finite Intersection Property |

Additionally, a A is a semiring that contains semialgebraAll families are assumed to be non-empty. are arbitrary elements of |

## See also

- Algebra of sets
- Class (set theory)
- Combinatorial design
- δ-ring
- Field of sets
- Indexed family
- λ-system (Dynkin system)
- π-system
- Ring of sets
- Russell's paradox (or
*Set of sets that do not contain themselves*) - σ-algebra
- σ-ring

## Notes

**^**Brualdi 2010, pg. 322**^**Roberts & Tesman 2009, pg. 692**^**Biggs 1985, pg. 89

## References

- 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