العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolDISJOINT SETS
(Redirected from 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.
Formally, two sets ''A'' and ''B'' are disjoint if their intersection is the empty set, 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 family of sets {''A''''i'' : ''i'' ∈ ''I''} is pairwise disjoint if for any ''i'' and ''j'' in ''I'' with ''i'' ≠ ''j'',
:
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}, {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
:
★ almost disjoint sets
★ connectedness
★ disjoint union
★ disjoint-set data structure
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.
| Contents |
| Explanation |
| See also |
Explanation
Formally, two sets ''A'' and ''B'' are disjoint if their intersection is the empty set, 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 family of sets {''A''''i'' : ''i'' ∈ ''I''} is pairwise disjoint if for any ''i'' and ''j'' in ''I'' with ''i'' ≠ ''j'',
:
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}, {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
:
See also
★ almost disjoint sets
★ connectedness
★ disjoint union
★ disjoint-set data structure
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
