View source for Operation (mathematics)

{{distinguish|Operator (mathematics)}}
{{Short description|Addition, multiplication, division, ...}}
[[File:Arithmetic_operations.svg|thumb|right|[[Elementary arithmetic]] operations:{{unbulleted list
| style=margin-left:1.6em;
| +, plus (addition)
| −, minus (subtraction)
| ÷, obelus (division)
| ×, times (multiplication)
}}]]
In [[mathematics]], an '''operation''' is a [[Function (mathematics)|function]] from a [[Set (mathematics)|set]] to itself. For example, an operation on [[Real number|real numbers]] will take in real numbers and return a real number. An operation can take zero or more input values (also called "''[[operand]]s''" or "arguments") to a well-defined output value. The number of operands is the [[arity]] of the operation.

The most commonly studied operations are [[binary operation]]s (i.e., operations of arity&nbsp;2), such as [[addition]] and [[multiplication]], and [[unary operation]]s (i.e., operations of arity&nbsp;1), such as [[additive inverse]] and [[multiplicative inverse]]. An operation of [[arity]] zero, or [[nullary operation]], is a [[constant (mathematics)|constant]].<ref name=":1">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Algebraic_operation|title=Algebraic operation - Encyclopedia of Mathematics|website=www.encyclopediaofmath.org|access-date=2019-12-10}}</ref><ref>{{Cite web|url=http://www.math.hawaii.edu/~williamdemeo/latticetheory/Glossary.pdf|title=Universal Algebra Notes|last=DeMeo|first=William|date=August 26, 2010|website=math.hawaii.edu|access-date=2019-12-09|archive-date=2021-05-19|archive-url=https://web.archive.org/web/20210519000135/http://www.math.hawaii.edu/~williamdemeo/latticetheory/Glossary.pdf|url-status=dead}}</ref> The [[mixed product]] is an example of an operation of arity&nbsp;3, also called [[ternary operation]].

Generally, the arity is taken to be finite. However, [[infinitary operation]]s are sometimes considered,<ref name=":1" /> in which case the "usual" operations of finite arity are called '''finitary operations'''.

A '''partial operation''' is defined similarly to an operation, but with a [[partial function]] in place of a function.

==Types of operation==
[[File:Binary operations as black box.svg|thumb|A binary operation takes two arguments <math>x</math> and <math>y</math>, and returns the result <math>x\circ y</math>.]]
There are two common types of operations: [[unary operation|unary]] and [[binary operation|binary]]. Unary operations involve only one value, such as [[negation]] and [[trigonometric function]]s.<ref>{{mathworld|title=Unary Operation|id=UnaryOperation}}</ref> Binary operations, on the other hand, take two values, and include [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], and [[exponentiation]].<ref>{{mathworld|title=Binary Operation|id=BinaryOperation}}</ref>

Operations can involve mathematical objects other than numbers. The [[Truth value|logical values]] ''true'' and ''false'' can be combined using [[logic operation]]s, such as ''and'', ''or,'' and ''not''. [[vector (geometric)|Vectors]] can be added and subtracted.<ref>{{mathworld|title=Vector|id=Vector}} "Vectors can be added together (vector addition), subtracted (vector subtraction) ..."</ref> [[Rotation]]s can be combined using the [[function composition]] operation, performing the first rotation and then the second. Operations on [[Set (mathematics)|sets]] include the binary operations ''[[Union (mathematics)|union]]'' and ''[[intersection (mathematics)|intersection]]'' and the unary operation of ''[[Complementation (mathematics)|complementation]]''.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Union|url=https://mathworld.wolfram.com/Union.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Intersection|url=https://mathworld.wolfram.com/Intersection.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Complementation|url=https://mathworld.wolfram.com/Complementation.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> Operations on [[Function (mathematics)|function]]s include [[Function composition|composition]] and [[convolution]].<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Composition|url=https://mathworld.wolfram.com/Composition.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Convolution|url=https://mathworld.wolfram.com/Convolution.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>

Operations may not be defined for every possible value of its ''[[Domain of a function|domain]]''. For example, in the real numbers one cannot divide by zero<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Division by Zero|url=https://mathworld.wolfram.com/DivisionbyZero.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> or take square roots of negative numbers. The values for which an operation is defined form a set called its ''domain of definition'' or ''active domain''. The set which contains the values produced is called the ''[[codomain]]'', but the set of actual values attained by the operation is its codomain of definition, active codomain, ''[[Image (mathematics)|image]]'' or ''[[Range of a function|range]]''.<ref>{{mathworld|title=Coomain|id=Codomain}}</ref> For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers.

Operations can involve dissimilar objects: a vector can be multiplied by a [[scalar (mathematics)|scalar]] to form another vector (an operation known as [[scalar multiplication]]),<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Scalar Multiplication|url=https://mathworld.wolfram.com/ScalarMultiplication.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> and the [[inner product]] operation on two vectors produces a quantity that is scalar.<ref>{{Cite book|last1=Jain|first1=P. K.|url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203|title=Functional Analysis|last2=Ahmad|first2=Khalil|last3=Ahuja|first3=Om P.|date=1995|publisher=New Age International|isbn=978-81-224-0801-0|language=en}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Inner Product|url=https://mathworld.wolfram.com/InnerProduct.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> An operation may or may not have certain properties, for example it may be [[associative]], [[commutative]], [[anticommutative]], [[idempotent]], and so on.

The values combined are called ''operands'', ''arguments'', or ''inputs'', and the value produced is called the ''value'', ''result'', or ''output''. Operations can have fewer or more than two inputs (including the case of zero input and infinitely many inputs<ref name=":1" />).

An '''operator''' is similar to an operation in that it refers to the symbol or the process used to denote the operation. Hence, their point of view is different. For instance, one often speaks of "the operation of addition" or "the addition operation," when focusing on the operands and result, but one switch to "addition operator" (rarely "operator of addition"), when focusing on the process, or from the more symbolic viewpoint, the function {{nowrap|+: ''X'' × ''X'' → ''X''}} (where X is a set such as the set of real numbers).

==Definition==
An '''''n''-ary operation''' ''ω'' on a [[Set (mathematics)|set]] ''X'' is a [[Function (mathematics)|function]] {{nowrap|''ω'': ''X''<sup>n</sup> → ''X''}}. The set {{nowrap|''X''<sup>n</sup>}} is called the ''domain'' of the operation, the output set is called the ''[[codomain]]'' of the operation, and the fixed non-negative integer ''n'' (the number of operands) is called the ''[[arity]]'' of the operation. Thus a [[unary operation]] has arity one, and a [[binary operation]] has arity two. An operation of arity zero, called a ''nullary'' operation, is simply an element of the codomain ''Y''. An ''n''-ary operation can also be viewed as an {{nowrap|(''n'' + 1)}}-ary [[Finitary relation|relation]] that is [[Finitary relation#Definitions|total]] on its ''n'' input domains and [[Finitary relation#Definitions|unique]] on its output domain.

An '''''n''-ary partial operation''' ''ω'' from {{nowrap|''X''<sup>n</sup> to ''X''}}  is a [[partial function]] {{nowrap|''ω'': ''X''<sup>n</sup> → ''X''}}. An ''n''-ary partial operation can also be viewed as an {{nowrap|(''n'' + 1)}}-ary relation that is unique on its output domain.

The above describes what is usually called a '''finitary operation''', referring to the finite number of operands (the value ''n''). There are obvious extensions where the arity is taken to be an infinite [[ordinal number|ordinal]] or [[cardinal number|cardinal]],<ref name=":1" /> or even an arbitrary set indexing the operands.

Often, the use of the term ''operation'' implies that the domain of the function includes a power of the codomain (i.e. the [[Cartesian product]] of one or more copies of the codomain),<ref>{{cite book|chapter=Chapter II, Definition 1.1|first1=S. N.|last1=Burris|first2=H. P.|last2=Sankappanavar|title=A Course in Universal Algebra|publisher=Springer|date=1981|url=http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html}}</ref> although this is by no means universal, as in the case of [[dot product]], where vectors are multiplied and result in a scalar. An ''n''-ary operation {{nowrap|''ω'': ''X''<sup>''n''</sup> → ''X''}} is called an '''{{vanchor|internal operation}}'''. An ''n''-ary operation {{nowrap|''ω'': ''X''<sup>''i''</sup> × ''S'' × ''X''<sup>''n'' − ''i'' − 1</sup> → ''X''}} where {{nowrap|0 ≤ ''i'' < ''n''}} is called an '''external operation''' by the ''scalar set'' or ''operator set'' ''S''. In particular for a binary operation, {{nowrap|''ω'': ''S'' × ''X'' → ''X''}} is called a '''left-external operation''' by ''S'', and {{nowrap|''ω'': ''X'' × ''S'' → ''X''}} is called a '''right-external operation''' by ''S''. An example of an internal operation is [[vector addition]], where two vectors are added and result in a vector. An example of an external operation is [[scalar multiplication]], where a vector is multiplied by a scalar and result in a vector.

An '''''n''-ary multifunction''' or '''{{vanchor|multioperation}}''' ''ω'' is a mapping from a Cartesian power of a set into the set of subsets of that set, formally <math>\omega : X^n \rightarrow \mathcal{P}(X)</math>.<ref>{{cite journal |last1=Brunner |first1=J. |last2=Drescher |first2=Th. |last3=Pöschel |first3=R. |last4=Seidel |first4=H. |date=Jan 1993 |title=Power algebras: clones and relations |url=https://wwwpub.zih.tu-dresden.de/~poesch-r/poePUBLICATIONSpdf/1993_Brunner_Dre_Poe_Sei.pdf |journal=EIK (Elektronische Informationsverarbeitung und Kybernetik) |volume=29 |issue= |pages=293–302 |doi= |access-date=2022-10-25}}</ref>

==See also==
* [[Finitary relation]]
* [[Hyperoperation]]
* [[Infix notation]]
* [[Operator (mathematics)]]
* [[Order of operations]]

==References==
<references />

{{DEFAULTSORT:Operation (Mathematics)}}
[[Category:Elementary mathematics]]
[[Category:Operations on numbers| ]]