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In
set theory illustrating the intersection of two sets 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 ...
, 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 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 an ...

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 A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that ...
.


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 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). I ...
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 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). I ...
, so the sets can be written in any order. The
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), and calculus, change (mathematical an ...

empty set
is an
identity element 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 h ...
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, disjunction is a logical connective typically notated \lor whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics of logic, seman ...
. 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 Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...
of a set U, together with the operations given by union,
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
, and
complementation A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
, is a
Boolean algebra 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 ...
. 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
universal set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
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 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 ...
.


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 f ...
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 bicondit ...
there is 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 In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The ...
and A_i is a set for every i \in I. In the case that the index set ''I'' is the set of
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s, one uses the notation \bigcup_^ A_, which is analogous to that of the
infinite sum 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). I ...
s 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 TeX (, see below), stylized within the system as TeX, is a typesetting system which was designed and mostly written by Donald Knuth and released in 1978. TeX is a popular means of typesetting complex mathematical formulae; it has been noted ...
, \cup is rendered from \cup.


See also

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Algebra of sets 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). I ...
*
Alternation (formal language theory)In formal language theory In mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedn ...
, the union of sets of strings *
Axiom of union In axiomatic set theory illustrating the intersection of two sets. Set theory is a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
*
Disjoint union 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). ...
*
Intersection (set theory) 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 ...
*
Iterated binary operationIn 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 ha ...
*
List of set identities and relations This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for ...
*
Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
*
Symmetric difference 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 ...


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