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.

- Introduction
- Definition of Sets
- Operations on Sets

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