Union (set theory)

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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 S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.

Contents

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  \cup B = \{ x: x \in A \,\,\,\textrm{ or }\,\,\, x \in B\}


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 ABC). 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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