Family of sets

In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection ${\displaystyle F}$ of subsets of a given set ${\displaystyle S}$ is called a family of subsets of ${\displaystyle S}$, or a family of sets over ${\displaystyle S.}$ More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set ${\displaystyle I}$, known as the index set, to ${\displaystyle F}$, in which case the sets of the family are indexed by members of ${\displaystyle I}$.^[1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member,^[2][3][4] and in other contexts it may form a proper class.

A finite family of subsets of a finite set ${\displaystyle S}$ is also called a hypergraph. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions.