Family of sets
Encyclopedia
In set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a collection F of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s 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. The term "collection" is used here (rather than the term "set") because a family of sets may contain repeated copies of any given member. That is to say, a family of sets is a multiset
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...

.

Examples

  • The power set P(S) is a family of sets over S.
  • The k-subsets
    N-set
    In mathematics, an n-set is a set containing exactly n elements, where n is a natural number. Thus, every finite set is an n-set for some specific natural number n. If S is any set, then a subset of S containing k elements is called a k-subset, or a k-combination...

     S(k) of an n-set
    N-set
    In mathematics, an n-set is a set containing exactly n elements, where n is a natural number. Thus, every finite set is an n-set for some specific natural number n. If S is any set, then a subset of S containing k elements is called a k-subset, or a k-combination...

     S form a family of sets.
  • The class Ord of all ordinal number
    Ordinal number
    In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

    s 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 whatsoever is a subclass of the proper class V of all sets (the universe).

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
    Hypergraph
    In mathematics, a hypergraph is a generalization of a graph, where an edge can connect any number of vertices. Formally, a hypergraph H is a pair H = where X is a set of elements, called nodes or vertices, and E is a set of non-empty subsets of X called hyperedges or links...

    , also called a set system, is formed by a set of vertices
    Vertex (graph theory)
    In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

     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
    Abstract simplicial complex
    In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...

     is a combinatorial abstraction of the notion of a simplicial complex
    Simplicial complex
    In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...

    , a shape formed by unions of line segments, triangles, tetrahedra, and higher dimensional simplices
    Simplex
    In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

    , 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 in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex.
  • An incidence structure
    Incidence structure
    In mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...

     consists of a set of points, a set of lines, and an (arbitrary) binary relation
    Binary relation
    In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

    , 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.
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