In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, 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 refers to a union of
zero () sets and it is by definition equal to the
empty set.
For explanation of the symbols used in this article, refer to the
table of mathematical symbols
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formul ...
.
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,
:
.
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
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and the set of
even numbers , 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.
Algebraic properties
Binary union is an
associative operation; that is, for any sets
Thus the parentheses may be omitted without ambiguity: either of the above can be written as
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,
for any set
Also, the union operation is idempotent:
All these properties follow from analogous facts about
logical disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
.
Intersection distributes over union
and union distributes over intersection
The
power set of a set
together with the operations given by union,
intersection, and
complementation, is a
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
. 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
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
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
whose elements are sets, then ''x'' is an element of the union of M
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
there is
at least one element ''A'' of M such that ''x'' is an element of ''A''.
In symbols:
:
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.
Notations
The notation for the general concept can vary considerably. For a finite union of sets
one often writes
or
. Various common notations for arbitrary unions include
,
, and
. The last of these notations refers to the union of the collection
, where ''I'' is an
index set and
is a set for every
. In the case that the index set ''I'' is the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, one uses the notation
, 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 . In
TeX,
is rendered from \cup.
See also
*
* − the union of sets of strings
*
*
*
*
*
*
*
*
Notes
External links
*
Infinite Union and Intersection at ProvenMathDe Morgan's laws formally proven from the axioms of set theory.
{{Improve categories, date=May 2021
Boolean algebra
Basic concepts in set theory
Operations on sets
Set theory