Union (set theory)

Summary

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.[1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero () sets and it is by definition equal to the empty set.

Union of two sets:
Union of three sets:
The union of A, B, C, D, and E is everything except the white area.

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

Union of two sets

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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.[2] In set-builder notation,

 .[3]

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then AB = {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,[3][4] 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

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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.[5] 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[2]   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

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One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of ABC if and only if x is in at least one of A, B, and C.

A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set.[6][7]

Arbitrary unions

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The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A.[8] In symbols:

 

This idea subsumes the preceding sections—for example, ABC is the union of the collection {A, B, C}. Also, if M is the empty collection, then the union of M is the empty set.

Notations

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The notation for the general concept can vary considerably. For a finite union of sets   one often writes   or  . Various common notations for arbitrary unions include  ,  , and  . The last of these notations refers to the union of the collection  , where I is an index set and   is a set for every  . In the case that the index set I is the set of natural numbers, one uses the notation  , which is analogous to that of the infinite sums in series.[8]

When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.

Notation encoding

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In Unicode, union is represented by the character U+222A UNION.[9] In TeX,   is rendered from \cup and   is rendered from \bigcup.

See also

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Notes

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  1. ^ Weisstein, Eric W. "Union". Wolfram Mathworld. Archived from the original on 2009-02-07. Retrieved 2009-07-14.
  2. ^ a b "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". Probability Course. Retrieved 2020-09-05.
  3. ^ a b Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314.
  4. ^ deHaan, Lex; Koppelaars, Toon (2007-10-25). Applied Mathematics for Database Professionals. Apress. ISBN 9781430203483.
  5. ^ Halmos, P. R. (2013-11-27). Naive Set Theory. Springer Science & Business Media. ISBN 9781475716450.
  6. ^ Dasgupta, Abhijit (2013-12-11). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media. ISBN 9781461488545.
  7. ^ "Finite Union of Finite Sets is Finite". ProofWiki. Archived from the original on 11 September 2014. Retrieved 29 April 2018.
  8. ^ a b Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01). A Transition to Advanced Mathematics. Cengage Learning. ISBN 9781285463261.
  9. ^ "The Unicode Standard, Version 15.0 – Mathematical Operators – Range: 2200–22FF" (PDF). Unicode. p. 3.
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