Union (set theory)

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 nullary union refers to a union of zero (⁠${\displaystyle 0}$⁠) sets and it is by definition equal to the empty set.

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 set-builder notation,

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

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, 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 ⁠${\displaystyle A,B,{\text{ and }}C}$⁠, $${\displaystyle A\cup (B\cup C)=(A\cup B)\cup C.}$$ Thus, the parentheses may be omitted without ambiguity: either of the above can be written as ⁠${\displaystyle A\cup B\cup C}$⁠. 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, ⁠${\displaystyle A\cup \varnothing =A}$⁠, for any set ⁠${\displaystyle A}$⁠. Also, the union operation is idempotent: ⁠${\displaystyle A\cup A=A}$⁠. All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union $${\displaystyle A\cap (B\cup C)=(A\cap B)\cup (A\cap C)}$$ and union distributes over intersection$${\displaystyle A\cup (B\cap C)=(A\cup B)\cap (A\cup C).}$$ The power set of a set ⁠${\displaystyle U}$⁠, 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 $${\displaystyle A\cup B=(A^{\complement }\cap B^{\complement })^{\complement },}$$ where the superscript ${\displaystyle {}^{\complement }}$ denotes the complement in the universal set ⁠${\displaystyle U}$⁠. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: ${\displaystyle A\cap B=(A^{\complement }\cup B^{\complement })^{\complement }}$. These two expressions together are called De Morgan's laws.

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.

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. 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 ${\textstyle \bigcup _{i=1}^{n}S_{i}}$. Various common notations for arbitrary unions include ${\textstyle \bigcup \mathbf {M} }$, ${\textstyle \bigcup _{A\in \mathbf {M} }A}$, and ${\textstyle \bigcup _{i\in I}A_{i}}$. The last of these notations 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 the notation ${\textstyle \bigcup _{i=1}^{\infty }A_{i}}$, which is analogous to that of the infinite sums in series.

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

Notation encoding

In Unicode, union is represented by the character U+222A ∪ UNION. In TeX, ${\displaystyle \cup }$ is rendered from \cup and ${\textstyle \bigcup }$ is rendered from \bigcup.

See also

• Mathematics portal

• Algebra of sets – Identities and relationships involving sets
• Alternation (formal language theory) – in formal language theory and pattern matching, the union of two sets of strings or patternsPages displaying wikidata descriptions as a fallback − the union of sets of strings
• Axiom of union – Concept in axiomatic set theory
• Disjoint union – In mathematics, operation on sets
• Inclusion–exclusion principle – Counting technique in combinatorics
• Intersection (set theory) – Set of elements common to all of some sets
• Iterated binary operation – Repeated application of an operation to a sequence
• List of set identities and relations – Equalities for combinations of sets
• Naive set theory – Informal set theories
• Symmetric difference – Elements in exactly one of two sets

Notes

1. Weisstein, Eric W. "Union". Wolfram Mathworld. Archived from the original on 2009-02-07. Retrieved 2009-07-14.
2. "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". Probability Course. Retrieved 2020-09-05.
3. 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. "MathCS.org - Real Analysis: Theorem 1.1.4: De Morgan's Laws". mathcs.org. Retrieved 2024-10-22.
7. Doerr, Al; Levasseur, Ken. ADS Laws of Set Theory.
8. "The algebra of sets - Wikipedia, the free encyclopedia". www.umsl.edu. Retrieved 2024-10-22.
9. Dasgupta, Abhijit (2013-12-11). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media. ISBN 9781461488545.
10. "Finite Union of Finite Sets is Finite". ProofWiki. Archived from the original on 11 September 2014. Retrieved 29 April 2018.
11. Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01). A Transition to Advanced Mathematics. Cengage Learning. ISBN 9781285463261.
12. "The Unicode Standard, Version 15.0 – Mathematical Operators – Range: 2200–22FF" (PDF). Unicode. p. 3.

External links

• "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.