# 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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