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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.

The following statements are logically equivalent:

  • A and B are disjoint.
  • The intersection of A and B is the empty set.
  • AB = {} (the same as the above, but in symbols).

Two or more sets are mutually disjoint if any two of the sets in question are disjoint. For example, {1,2,3}, {4,5,6}, and {7,8,9} are mutually disjoint. However, {1,2,3}, {4,5,6}, and {3,4} are not mutually disjoint, even though there is no number that belongs to all of them.

We can also say that a set U whose elements are themselves sets is mutually disjoint if its members are mutually disjoint. In symbols:

For any A,B in U, A = B or AB = {}.

U is a partition of a set X if:

  • the union of U is X;
  • U is mutually disjoint (as above); and
  • {} does not belong to U.

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