In set theory
, the union (denoted by ∪) of a collection of sets
is the set of all element
s in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
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''.
For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is:
: ''A'' =
: ''B'' =
As another example, the number 9 is ''not'' contained in the union of the set of prime number
s and the set of even number
s , because 9 is neither prime nor even.
Sets cannot have duplicate elements,
so the union of the sets and is . 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 ''A'', ''B'', and ''C'',
Thus the parentheses may be omitted without ambiguity: either of the above can be written as ''A'' ∪ ''B'' ∪ ''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, ''A'' ∪ ∅ = ''A'', for any set ''A.'' Also, the union operation is idempotent: ''A'' ∪ ''A'' = ''A''. All these properties follow from analogous facts about logical disjunction
Intersection distributes over union
and union distributes over intersection
The power set
of a set ''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
where the superscript
denotes the complement in the universal set
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
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''.
This idea subsumes the preceding sections—for example, ''A'' ∪ ''B'' ∪ ''C'' is the union of the collection . 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
one often writes
. Various common notations for arbitrary unions include
The last of these notations refers to the union of the collection
, where ''I'' is an index set
is a set for every
. In the case that the index set ''I'' is the set of natural number
s, one uses the notation
, which is analogous to that of the infinite sum
s in series.
When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.
In Unicode, union is represented by the character . In TeX
is rendered from \cup.
* Algebra of sets
* Alternation (formal language theory)
, the union of sets of strings
* Axiom of union
* Disjoint union
* Intersection (set theory)
* Iterated binary operation
* List of set identities and relations
* Naive set theory
* Symmetric difference
*Infinite Union and Intersection at ProvenMath
De Morgan's laws formally proven from the axioms of set theory.
Category:Operations on sets