Family of sets

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. A family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of . 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.