In

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

set theory
Set theory is the branch of that studies , 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 , is mostly concerned with those that are relevant to ...

, 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 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 ...

.
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\; \backslash cup\; B\; =\; \backslash $. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : $A\; \backslash cup\; B\; =\; \backslash $ As another example, the number 9 is ''not'' contained in the union of the set ofprime 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 anassociative
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 $A,\; B,\; \backslash text\; C,$
$$A\; \backslash cup\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cup\; C.$$
Thus the parentheses may be omitted without ambiguity: either of the above can be written as $A\; \backslash cup\; B\; \backslash cup\; C.$ Also, union is commutative
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 ...

, 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 ...

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, $A\; \backslash cup\; \backslash varnothing\; =\; A,$ for any set $A.$ Also, the union operation is idempotent: $A\; \backslash cup\; A\; =\; A.$ 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
$$A\; \backslash cap\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cap\; B)\backslash cup(A\; \backslash cap\; C)$$
and union distributes over intersection
$$A\; \backslash cup\; (B\; \backslash cap\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cap\; (A\; \backslash cup\; C).$$
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
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 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 t ...

. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula
$$A\; \backslash cup\; B\; =\; \backslash left(A^\backslash text\; \backslash cap\; B^\backslash text\; \backslash right)^\backslash text,$$
where the superscript $^\backslash text$ 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 afinite 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 and t ...

.
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 orclass
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
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 ...

there is at least one element ''A'' of M such that ''x'' is an element of ''A''. In symbols:
: $x\; \backslash in\; \backslash bigcup\; \backslash mathbf\; \backslash iff\; \backslash exists\; A\; \backslash in\; \backslash mathbf,\backslash \; x\; \backslash in\; A.$
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 $S\_1,\; S\_2,\; S\_3,\; \backslash dots\; ,\; S\_n$ one often writes $S\_1\; \backslash cup\; S\_2\; \backslash cup\; S\_3\; \backslash cup\; \backslash dots\; \backslash cup\; S\_n$ or $\backslash bigcup\_^n\; S\_i$. Various common notations for arbitrary unions include $\backslash bigcup\; \backslash mathbf$, $\backslash bigcup\_\; A$, and $\backslash bigcup\_\; A\_$. The last of these notations refers to the union of the collection $\backslash left\backslash $, where ''I'' is anindex 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. Th ...

and $A\_i$ is a set for every $i\; \backslash in\; I$. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s, one uses the notation $\backslash bigcup\_^\; A\_$, which is analogous to that of the infinite sum
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 ...

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 . InTeX
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 ...

, $\backslash cup$ is rendered from \cup.
See also

*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 logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογ ...

, 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 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 ...

* 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
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 ...

* 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