Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Family of sets
Summary
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.