Disjoint sets

In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.

Generalizations

This definition of disjoint sets can be extended to families of sets and to indexed families of sets.

By definition, a collection of sets is called a family of sets (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated.

An indexed family of sets

(

A

i

)

i

∈

I

,

{\displaystyle \left(A_{i}\right)_{i\in I},}

is by definition a set-valued function (that is, it is a function that assigns a set

A

i

{\displaystyle A_{i}}

to every element

i

∈

I

{\displaystyle i\in I}

in its domain) whose domain

I

{\displaystyle I}

is called its index set (and elements of its domain are called indices).

There are two subtly different definitions for when a family of sets

F

{\displaystyle {\mathcal {F}}}

is called pairwise disjoint. According to one such definition, the family is disjoint if each two sets in the family are either identical or disjoint.