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Pairwise disjoint

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Pairwise 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.[1]

Explanation

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

A\cap B = \varnothing.\,

This definition extends to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint if, given any two sets in the collection, those two sets 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_{i\in I} A_i = \varnothing.\,

However, the converse is not true: the intersection of the collection is empty, but the collection is not pairwise disjoint. In fact, there are no two disjoint sets in this 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_{i\in I} A_i = X.\,

See also

References

  • MathWorld.
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