In set theory and related branches of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used in phrases like "family of sets" because if one instead uses "set of sets" then the subsequent use of "set" can be confusing as to whether it is the containing set or one of the member sets. A common use is "family of subsets of some set S". A family of sets is also called a set family or a set system. 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.

The collection of all subsets of a given set ${\displaystyle S}$ is called the power set of ${\displaystyle S}$ and is denoted by ⁠${\displaystyle \wp (S)}$⁠. The power set ${\displaystyle \wp (S)}$ of a given set ${\displaystyle S}$ is a family of sets over ⁠${\displaystyle S}$⁠.

A subset of ${\displaystyle S}$ having ${\displaystyle k}$ elements is called a ${\displaystyle k}$-subset of ⁠${\displaystyle S}$⁠. The ${\displaystyle k}$-subsets ${\displaystyle S^{(k)}}$ of a set ${\displaystyle S}$ form a family of sets.

Let ⁠${\displaystyle S=\{a,b,c,1,2\}}$⁠. An example of a family of sets over ${\displaystyle S}$ (in the multiset sense) is given by ⁠${\displaystyle F=\left\{A_{1},A_{2},A_{3},A_{4}\right\}}$⁠, where ⁠${\displaystyle A_{1}=\{a,b,c\},A_{2}=\{1,2\},A_{3}=\{1,2\}}$⁠, and ⁠${\displaystyle A_{4}=\{a,b,1\}}$⁠.

The class ${\displaystyle \operatorname {Ord} }$ 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 ${\displaystyle S}$ is itself a subset of the power set ${\displaystyle \wp (S)}$ 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.