Disjoint sets
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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.
Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if
- <math>A\cap B = \varnothing.\,<math>
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 : i ∈ I} is pairwise disjoint if for any i and j in I with i ≠ j,
- <math>A_i \cap A_j = \varnothing.\,<math>
For example, the collection of sets { {1}, {2}, {3}, ... } is pairwise disjoint. If {Ai} is a pairwise disjoint collection, then clearly its intersection is empty:
- <math>\bigcap_{i\in I} A_i = \varnothing.\,<math>
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 partition of a set X is any collection of non-empty subsets {Ai : i ∈ I} of X such that {Ai} are pairwise disjoint and
- <math>\bigcup_{i\in I} A_i = X.\,<math>