Family of sets

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a> and related branches of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a family (or collection) can mean, depending upon the context, any of the following: <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, <a href="/facts/Indexed_set/aFqfzJN0">indexed set</a>, <a href="/facts/Multiset/bPFJGMDo">multiset</a>, or <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a>. A collection 
 
 
 
 F
 
 
 {\displaystyle F}
 
 of <a href="/facts/Subset/SJGERmbA">subsets</a> of a given <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is called a family of subsets of 
 
 
 
 S
 
 
 {\displaystyle S}
 
, or a family of sets over 
 
 
 
 S
 .
 
 
 {\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 
 
 
 
 I
 
 
 {\displaystyle I}
 
, known as the index set, to 
 
 
 
 F
 
 
 {\displaystyle F}
 
, in which case the sets of the family are indexed by members of 
 
 
 
 I
 
 
 {\displaystyle I}
 
. 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 <a href="/facts/Class_(set_theory)/F5EmDd9b">proper class</a>.
A finite family of subsets of a <a href="/facts/Finite_set/za1newlP">finite set</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is also called a <a href="/facts/Hypergraph/qIR2o2I3">hypergraph</a>. The subject of <a href="/facts/Extremal_set_theory/j9sVICGG">extremal set theory</a> concerns the largest and smallest examples of families of sets satisfying certain restrictions.

Family of sets open-in-new

Family of sets