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set: Operations on Sets

There are three basic set operations: intersection, union, and complementation. The intersection of two sets is the set containing the elements common to the two sets and is denoted by the symbol ∩. The union of two sets is the set containing all elements belonging to either one of the sets or to both, denoted by the symbol ∪. Thus, if C ={1, 2, 3, 4} and D ={3, 4, 5}, then C D ={3, 4} and C D ={1, 2, 3, 4, 5}. These two operations each obey the associative law and the commutative law , and together they obey the distributive law .

In any discussion the set of all elements under consideration must be specified, and it is called the universal set. If the universal set is U ={1, 2, 3, 4, 5} and A ={1, 2, 3}, then the complement of A (written A′ ) is the set of all elements in the universal set that are not in A, or A′ ={4, 5}. The intersection of a set and its complement is the empty set (denoted by ∅), or A A′ =∅; the union of a set and its complement is the universal set, or A A′ = U. See also symbolic logic .

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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