Disjoint sets

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In mathematics, two sets are said to be disjoint if they have no element in common. For example, and are disjoint sets.

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 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 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 , , , ... } is pairwise disjoint. If is a pairwise disjoint collection, then clearly its intersection is empty:

[\bigcap_ 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 on the collection.

A partition of a set X is any collection of non-empty subsets of X such that are pairwise disjoint and

[\bigcup_ A_i = X.\,]

Disjoint sets

Explanation

See also