Definitions

# Disjoint sets

In mathematics, two sets are said to be disjoint if they have no element in common. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets.

## Explanation

Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if

$Acap B = varnothing.,$

This definition extends to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint if any two distinct sets in the collection are disjoint.

Formally, let I be an index set, and for each i in I, let Ai be a set. Then the family of sets {Ai : iI} is pairwise disjoint if for any i and j in I with ij,

$A_i cap A_j = varnothing.,$

For example, the collection of sets { {1}, {2}, {3}, ... } is pairwise disjoint. If {Ai} is a pairwise disjoint collection (containing at least two sets), then clearly its intersection is empty:

$bigcap_\left\{iin I\right\} A_i = varnothing.,$

However, the converse is not true: the intersection of the collection {{1, 2}, {2, 3}, {3, 1}} is empty, but the collection is not pairwise disjoint - in fact, there are no two disjoint sets in the collection.

A partition of a set X is any collection of non-empty subsets {Ai : iI} of X such that {Ai} are pairwise disjoint and

$bigcup_\left\{iin I\right\} A_i = X.,$