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In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
, the union (denoted by ∪) of a collection of sets is the set of all
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
s 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 (0)
 zero (<math>0</math>)
sets and it is by definition equal to the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

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 formula. ...
.


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 symbols, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup 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 number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a set or its contents.


Algebraic properties

Binary union is an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
operation; that is, for any sets A, B, \text C, 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 A \cup B \cup C. Also, union is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, so the sets can be written in any order. The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
is an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
for the operation of union. That is, A \cup \varnothing = A, for any set A. Also, the union operation is idempotent: A \cup A = A. All these properties follow from analogous facts about
logical disjunction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
. Intersection distributes over union A \cap (B \cup C) = (A \cap B)\cup(A \cap C) and union distributes over intersection A \cup (B \cap C) = (A \cup B) \cap (A \cup C). The
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of a set U, together with the operations given by union,
intersection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, and Complement (set theory), complementation, is a Boolean algebra (structure), Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula A \cup B = \left(A^\text \cap B^\text \right)^\text, where the superscript ^\text denotes the complement in the Universe (mathematics), universal set U.


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 (set theory), class whose elements are sets, then ''x'' is an element of the union of M if and only if there is existential quantification, at least one element ''A'' of M such that ''x'' is an element of ''A''. In symbols: : x \in \bigcup \mathbf \iff \exists A \in \mathbf,\ x \in 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.


Notations

The notation for the general concept can vary considerably. For a finite union of sets S_1, S_2, S_3, \dots , S_n one often writes S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n or \bigcup_^n S_i. Various common notations for arbitrary unions include \bigcup \mathbf, \bigcup_ A, and \bigcup_ A_. The last of these notations refers to the union of the collection \left\, where ''I'' is an index set and A_i is a set for every i \in I. In the case that the index set ''I'' is the set of natural numbers, one uses the notation \bigcup_^ A_, 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, \cup is rendered from \cup.


See also

* 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


Notes


External links

*
Infinite Union and Intersection at ProvenMath
De Morgan's laws formally proven from the axioms of set theory. Operations on sets Set theory Boolean algebra {{Improve categories, date=May 2021