Information AboutDisjoint |
| CATEGORIES ABOUT DISJOINT SETS | |
| basic concepts in set theory | |
| set families | |
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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 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 {''A''''i'' : ''i'' ∈ ''I''} of ''X'' such that {''A''''i''} are pairwise disjoint and : SEE ALSO |
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