View source for Binary operation

{{Short description|Mathematical operation with two operands}}
{{Distinguish|Bitwise operation}}

[[File:Binary operations as black box.svg|thumb|A binary operation <math>\circ</math> is a rule for combining the arguments <math>x</math> and <math>y</math> to produce <math>x\circ y</math>]]
In [[mathematics]], a '''binary operation''' or '''dyadic operation''' is a rule for combining two [[Element (mathematics)|elements]] (called [[operands]]) to produce another element. More formally, a binary operation is an [[Operation (mathematics)|operation]] of [[arity]] two.

More specifically, a '''binary operation''' on a [[Set (mathematics)|set]] is a [[binary function]] that maps every [[ordered pair|pair]] of elements of the set to an element of the set. Examples include the familiar [[arithmetic operations]] like [[addition]], [[subtraction]], [[multiplication]], set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as [[vector addition]], [[matrix multiplication]], and [[Conjugation (group theory)|conjugation in groups]].

A binary function that involves several sets is sometimes also called a ''binary operation''. For example, [[scalar multiplication]] of [[vector space]]s takes a scalar and a vector to produce a vector, and [[scalar product]] takes two vectors to produce a scalar.

Binary operations are the keystone of most [[algebraic structure|structure]]s that are studied in [[algebra]], in particular in [[semigroup]]s, [[monoid]]s, [[group (mathematics)|groups]], [[ring (algebra)|rings]], [[field (mathematics)|fields]], and [[vector space]]s.

==Terminology==
More precisely, a binary operation on a [[Set (mathematics)|set]] <math>S</math> is a [[Map (mathematics)|mapping]] of the elements of the [[Cartesian product]] <math>S \times S</math> to <math>S</math>:<ref>{{harvnb|Rotman|1973|loc=pg. 1}}</ref><ref>{{harvnb|Hardy|Walker|2002|loc=pg. 176, Definition 67}}</ref><ref>{{harvnb|Fraleigh|1976|loc= pg. 10}}</ref>
:<math>\,f \colon S \times S \rightarrow S.</math>

If <math>f</math> is not a [[Function (mathematics)|function]] but a [[partial function]], then <math>f</math> is called a '''partial binary operation'''. For instance, division is a partial binary operation on the set of all [[real numbers]], because one cannot [[Division by zero|divide by zero]]: <math>\frac{a}{0}</math> is undefined for every real number <math>a</math>.  In both [[model theory]] and classical [[universal algebra]], binary operations are required to be defined on all elements of <math>S \times S</math>. However, [[partial algebra]]s<ref name="Gratzer2008">{{cite book|author=George A. Grätzer|title=Universal Algebra|url=https://archive.org/details/isbn_9780387774862|url-access=registration|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-77487-9|at=Chapter 2. Partial algebras|edition=2nd}}</ref> generalize [[universal algebra]]s to allow partial operations.

Sometimes, especially in [[computer science]], the term binary operation is used for any [[binary function]].

== Properties and examples ==
Typical examples of binary operations are the [[addition]] (<math>+</math>) and [[multiplication]] (<math>\times</math>) of [[number]]s and [[matrix (mathematics)|matrices]] as well as [[composition of functions]] on a single set.
For instance,
* On the set of real numbers <math>\mathbb R</math>, <math>f(a,b)=a+b</math> is a binary operation since the sum of two real numbers is a real number.
* On the set of natural numbers <math>\mathbb N</math>, <math>f(a,b)=a+b</math> is a binary operation since the sum of two natural numbers is a natural number.  This is a different binary operation than the previous one since the sets are different.
* On the set <math>M(2,\mathbb R)</math> of <math>2 \times 2</math> matrices with real entries, <math>f(A,B)=A+B</math> is a binary operation since the sum of two such matrices is a <math>2 \times 2</math> matrix.
* On the set <math>M(2,\mathbb R)</math> of <math>2 \times 2</math> matrices with real entries, <math>f(A,B)=AB</math> is a binary operation since the product of two such matrices is a <math>2 \times 2</math> matrix.
* For a given set <math>C</math>, let <math>S</math> be the set of all functions <math>h \colon C \rightarrow C</math>.  Define <math>f \colon S \times S \rightarrow S</math> by <math>f(h_1,h_2)(c)=(h_1 \circ h_2)(c)=h_1(h_2(c))</math> for all <math>c \in C</math>, the composition of the two functions <math>h_1</math> and <math>h_2</math> in <math>S</math>. Then <math>f</math> is a binary operation since the composition of the two functions is again a function on the set <math>C</math> (that is, a member of <math>S</math>).

Many binary operations of interest in both algebra and formal logic are [[commutative]], satisfying <math>f(a,b)=f(b,a)</math> for all elements <math>a</math> and <math>b</math> in <math>S</math>, or [[associative]], satisfying <math>f(f(a,b),c)=f(a,f(b,c))</math> for all <math>a</math>, <math>b</math>, and <math>c</math> in <math>S</math>.  Many also have [[identity element]]s and [[inverse element]]s.

The first three examples above are commutative and all of the above examples are associative.

On the set of real numbers <math>\mathbb R</math>, [[subtraction]], that is, <math>f(a,b)=a-b</math>, is a binary operation which is not commutative since, in general, <math>a-b \neq b-a</math>.  It is also not associative, since, in general, <math>a-(b-c) \neq (a-b)-c</math>; for instance, <math>1-(2-3)=2</math> but <math>(1-2)-3=-4</math>.

On the set of natural numbers <math>\mathbb N</math>, the binary operation [[exponentiation]], <math>f(a,b)=a^b</math>, is not commutative since, <math>a^b \neq b^a</math> (cf. [[Equation x^y = y^x|Equation x<sup>y</sup> = y<sup>x</sup>]]), and is also not associative since <math>f(f(a,b),c) \neq f(a,f(b,c))</math>.  For instance, with <math>a=2</math>, <math>b=3</math>, and <math>c=2</math>, <math>f(2^3,2)=f(8,2)=8^2=64</math>, but <math>f(2,3^2)=f(2,9)=2^9=512</math>.  By changing the set <math>\mathbb N</math> to the set of integers <math>\mathbb Z</math>, this binary operation becomes a partial binary operation since it is now undefined when <math>a=0</math> and <math>b</math> is any negative integer.  For either set, this operation has a ''right identity'' (which is <math>1</math>) since <math>f(a,1)=a</math> for all <math>a</math> in the set, which is not an ''identity'' (two sided identity) since <math>f(1,b) \neq b</math> in general.

[[division (mathematics)|Division]] (<math>\div</math>), a partial binary operation on the set of real or rational numbers, is not commutative or associative.  [[Tetration]] (<math>\uparrow\uparrow</math>), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.

==Notation==
Binary operations are often written using [[infix notation]] such as <math>a \ast b</math>, <math>a+b</math>, <math>a \cdot b</math> or (by [[Juxtaposition#Mathematics|juxtaposition]] with no symbol) <math>ab</math> rather than by functional notation of the form <math>f(a, b)</math>.  Powers are usually also written without operator, but with the second argument as [[superscript]].

Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses.  They are also called, respectively, [[Polish notation]] <math>\ast a b</math> and [[reverse Polish notation]] <math>a b \ast</math>.

== Binary operations as ternary relations ==

A binary operation <math>f</math> on a set <math>S</math> may be viewed as a [[ternary relation]] on <math>S</math>, that is, the set of triples <math>(a, b, f(a,b))</math> in <math>S \times S \times S</math> for all <math>a</math> and <math>b</math> in <math>S</math>.

== Other binary operations ==

For example, [[scalar multiplication]] in [[linear algebra]].  Here <math>K</math> is a [[field (mathematics)|field]] and <math>S</math> is a [[vector space]] over that field.

Also the [[dot product]] of two vectors maps <math>S \times S</math> to <math>K</math>, where <math>K</math> is a field and <math>S</math> is a vector space over <math>K</math>. It depends on authors whether it is considered as a binary operation.

==See also==

* [[:Category:Properties of binary operations]]
* {{annotated link|Iterated binary operation}}
* {{annotated link|Magma (algebra)}}
* {{annotated link|Operator (programming)}}
* {{annotated link|Ternary operation}}
* {{slink|Truth table#Binary operations}}
* {{annotated link|Unary operation}}

== Notes==
{{Reflist}}

== References==
* {{citation|last=Fraleigh|first=John B.|title=A First Course in Abstract Algebra|edition=2nd|publisher=Addison-Wesley|place=Reading|year=1976|isbn=0-201-01984-1}}
* {{citation|last=Hall|first= Marshall Jr.|title=The Theory of Groups|publisher=Macmillan|place=New York|year=1959}}
* {{citation|last1=Hardy|first1=Darel W.|last2=Walker|first2=Carol L.|author2-link=Carol Walker|title=Applied Algebra: Codes, Ciphers and Discrete Algorithms|publisher=Prentice-Hall|place=Upper Saddle River, NJ|year=2002|isbn=0-13-067464-8}}
* {{citation|last=Rotman|first=Joseph J.|title=The Theory of Groups: An Introduction|publisher=Allyn and Bacon|place=Boston|year=1973|edition=2nd}}

== External links ==
* {{MathWorld|title=Binary Operation|urlname=BinaryOperation}}

{{Mathematical logic}}
{{Authority control}}

{{DEFAULTSORT:Binary Operation}}
[[Category:Binary operations| ]]