Union (set theory)

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,

${\displaystyle A\cup B=\{x:x\in A{\text{ or }}x\in B\}}$.[3]

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,[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.

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.[5] 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[2]

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

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

$${\displaystyle A\cup B=\left(A^{\text{c}}\cap B^{\text{c}}\right)^{\text{c}},}$$

where the superscript ${\displaystyle {}^{\text{c}}}$ denotes the complement in the universal set ${\displaystyle U.}$

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.[6][7]

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:

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

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.

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

• 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

• "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.