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]Inset-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 $${\displaystyle A\cup B=(A^{\complement }\cap B^{\complement })^{\complement },}$$  where the superscript ${\displaystyle {}^{\complement }}$  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.

Afinite 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 AofM 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.

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.^[8]

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

InUnicode, union is represented by the character U+222A ∪ UNION.^[9]InTeX, ${\displaystyle \cup }$  is rendered from \cup and ${\textstyle \bigcup }$  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.