In set theory, the union (denoted as ∪) of a collection of sets is the set of all distinct elements in the collection.^{[1]} The union of a collection of sets gives a set .
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Definition
The union of two sets A and B is the collection of points which are in A or in B (or in both):
A simple example:
Other more complex operations can be done including the union, if the set is for example defined by a property rather than a finite or assumed infinite enumeration of elements. As an example, a set could be defined by a property or algebraic equation, which is referred to as a solution set when resolved. An example of a property used in a union would be the following:
If we are then to refer to a single element by the variable "x", then we can say that x is a member of the union if it is an element present in set A or in set B, or both.
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. 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.
Algebraic properties
Binary union is an associative operation; that is,
The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e. either of the above can be expressed equivalently as A ∪ B ∪ C). Similarly, 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, A ∪ {} = A, for any set A. In terms of the definitions, these facts follow from analogous facts about logical disjunction.
Together with intersection and complement, union makes any power set into a Boolean algebra. For example, union and intersection distribute over each other, and all three operations are combined in De Morgan's laws. Replacing union with symmetric difference gives a Boolean ring instead of a Boolean algebra
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