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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## See also

- $Acap\; B\; =\; varnothing.,$

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 A_{i} be a set. Then the family of sets {A_{i} : i ∈ I} is pairwise disjoint if for any i and j in I with i ≠ j,

- $A\_i\; cap\; A\_j\; =\; varnothing.,$

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

- $bigcap\_\{iin\; I\}\; A\_i\; =\; varnothing.,$

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 {A_{i} : i ∈ I} of X such that {A_{i}} are pairwise disjoint and

- $bigcup\_\{iin\; I\}\; A\_i\; =\; X.,$

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Last updated on Sunday September 28, 2008 at 12:36:01 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 28, 2008 at 12:36:01 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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