Union (set theory)

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.

Union of two sets:
${\displaystyle ~A\cup B}$

Union of three sets:
${\displaystyle ~A\cup B\cup C}$

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

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 symbols,

${\displaystyle A\cup B=\{x:x\in A{\text{ or }}x\in B\}}$.[2]

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}

${\displaystyle A\cup B=\{2,3,4,5,6,\dots \}}$

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,[2][3] 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 A, B, and C,

${\displaystyle A\cup (B\cup C)=(A\cup B)\cup C.}$

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.[4]

The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction.

Since sets with unions and intersections form a Boolean algebra, intersection distributes over union

${\displaystyle A\cap (B\cup C)=(A\cap B)\cup (A\cap C)}$

and union distributes over intersection

${\displaystyle A\cup (B\cap C)=(A\cup B)\cap (A\cup C)}$ .

Within a given universal set, union can be written in terms of the operations of intersection and complement as

${\displaystyle A\cup B=\left(A^{C}\cap B^{C}\right)^{C}}$

where the superscript ^C denotes the complement with respect to the universal set.

Finite unions

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 A ∪ B ∪ C 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.[5][6]

Arbitrary unions

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.[7] In symbols:

${\displaystyle x\in \bigcup \mathbf {M} \iff \exists A\in \mathbf {M} ,\ x\in A.}$

This idea subsumes the preceding sections—for example, A ∪ B ∪ C 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

The notation for the general concept can vary considerably. For a finite union of sets ${\displaystyle S_{1},S_{2},S_{3},\dots ,S_{n}}$ one often writes ${\displaystyle S_{1}\cup S_{2}\cup S_{3}\cup \dots \cup S_{n}}$ or ${\displaystyle \bigcup _{i=1}^{n}S_{i}}$. Various common notations for arbitrary unions include ${\displaystyle \bigcup \mathbf {M} }$, ${\displaystyle \bigcup _{A\in \mathbf {M} }A}$, and ${\displaystyle \bigcup _{i\in I}A_{i}}$, the last of which refers to the union of the collection ${\displaystyle \left\{A_{i}:i\in I\right\}}$ where I is an index set and ${\displaystyle A_{i}}$ is a set for every ${\displaystyle i\in I}$. In the case that the index set I is the set of natural numbers, one uses a notation ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}}$ analogous to that of the infinite sums in series.[7]

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

See also

• Alternation (formal language theory), the union of sets of strings
• Disjoint union
• Intersection (set theory)
• Iterated binary operation
• Naive set theory
• Symmetric difference

Notes

1. Weisstein, Eric W. "Union". Wolfram's Mathworld. Archived from the original on 2009-02-07. Retrieved 2009-07-14.
2. Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314.
3. deHaan, Lex; Koppelaars, Toon (2007-10-25). Applied Mathematics for Database Professionals. Apress. ISBN 9781430203483.
4. Halmos, P. R. (2013-11-27). Naive Set Theory. Springer Science & Business Media. ISBN 9781475716450.
5. Dasgupta, Abhijit (2013-12-11). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media. ISBN 9781461488545.
6. "Finite Union of Finite Sets is Finite - ProofWiki". proofwiki.org. Archived from the original on 11 September 2014. Retrieved 29 April 2018.
7. Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01). A Transition to Advanced Mathematics. Cengage Learning. ISBN 9781285463261.

External links

• Weisstein, Eric W. "Union". MathWorld.
• "Union of sets", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.