In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In set-builder notation, . For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is: A = {x is an even integer larger than 1} B = {x is an odd integer larger than 1} As another example, the number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of even numbers {2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. Algebraic properties Binary union is an associative operation; that is, for any sets Thus the parentheses may be omitted without ambiguity: either of the above can be written as Also, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, for any set Also, the union operation is idempotent: All these properties follow from analogous facts about logical disjunction. Intersection distributes over union and union distributes over intersection The power set of a set together with the operations given by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula where the superscript denotes the complement in the universal set Finite unions One can take the union of several sets simultaneously. For example, the union of three sets A, B,