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Disjoint)

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.

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

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 collection of sets {*A*_{i} : *i* in *I*} is pairwise disjoint if for any *i* and *j* in *I*,

For example, the collection of sets { {1}, {2}, {3}, ... } is pairwise disjoint. If {*A*_{i}} is a pairwise disjoint collection, then clearly its intersection is empty:

However, the converse is not true -- the intersection of the collection {{1, 2, 3}, {4, 5, 6}, {3, 4}} is empty, but the collection is *not* pairwise disjoint.

A collection of sets {*A*_{i} : *i* in *I*} is a partition of the set *X* if {*A*_{i}} is a pairwise disjoint collection not containing the empty set, and if

See also:

Last updated: 10-20-2005 22:05:26

Last updated: 10-29-2005 02:13:46