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In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
of
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...
two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
are the same set. Examples include the familiar arithmetic operations of
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
, subtraction, and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
. Other examples are readily found in different areas of mathematics, such as
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
,
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example,
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vecto ...
of vector spaces takes a scalar and a vector to produce a vector, and
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s that are studied in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, in particular in semigroups,
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
s,
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
,
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
,
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
, and vector spaces.

# Terminology

More precisely, a binary operation on a set $S$ is a mapping of the elements of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
$S \times S$ to $S$: :$\,f \colon S \times S \rightarrow S.$ Because the result of performing the operation on a pair of elements of $S$ is again an element of $S$, the operation is called a closed (or internal) binary operation on $S$ (or sometimes expressed as having the property of closure). If $f$ is not a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
, but a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is d ...
, then $f$ is called a partial binary operation. For instance, division of
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
is a partial binary operation, because one can't divide by zero: $\frac$ is undefined for every real number $a$. In both
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
and
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
, binary operations are required to be defined on all elements of $S \times S$. 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 Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
($+$) and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
($\times$) of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...
s and matrices as well as composition of functions on a single set. For instance, * On the set of real numbers $\mathbb R$, $f\left(a,b\right)=a+b$ is a binary operation since the sum of two real numbers is a real number. * On the set of natural numbers $\mathbb N$, $f\left(a,b\right)=a+b$ 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 $M\left(2,\mathbb R\right)$ of $2 \times 2$ matrices with real entries, $f\left(A,B\right)=A+B$ is a binary operation since the sum of two such matrices is a $2 \times 2$ matrix. * On the set $M\left(2,\mathbb R\right)$ of $2 \times 2$ matrices with real entries, $f\left(A,B\right)=AB$ is a binary operation since the product of two such matrices is a $2 \times 2$ matrix. * For a given set $C$, let $S$ be the set of all functions $h \colon C \rightarrow C$. Define $f \colon S \times S \rightarrow S$ by $f\left(h_1,h_2\right)\left(c\right)=\left(h_1 \circ h_2\right)\left(c\right)=h_1\left(h_2\left(c\right)\right)$ for all $c \in C$, the composition of the two functions $h_1$ and $h_2$ in $S$. Then $f$ is a binary operation since the composition of the two functions is again a function on the set $C$ (that is, a member of $S$). Many binary operations of interest in both algebra and formal logic are
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, satisfying $f\left(a,b\right)=f\left(b,a\right)$ for all elements $a$ and $b$ in $S$, or
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replaceme ...
, satisfying $f\left(f\left(a,b\right),c\right)=f\left(a,f\left(b,c\right)\right)$ for all $a$, $b$, and $c$ in $S$. Many also have
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
s and
inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
s. The first three examples above are commutative and all of the above examples are associative. On the set of real numbers $\mathbb R$, subtraction, that is, $f\left(a,b\right)=a-b$, is a binary operation which is not commutative since, in general, $a-b \neq b-a$. It is also not associative, since, in general, $a-\left(b-c\right) \neq \left(a-b\right)-c$; for instance, $1-\left(2-3\right)=2$ but $\left(1-2\right)-3=-4$. On the set of natural numbers $\mathbb N$, the binary operation
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, $f\left(a,b\right)=a^b$, is not commutative since, $a^b \neq b^a$ (cf. Equation xy = yx), and is also not associative since $f\left(f\left(a,b\right),c\right) \neq f\left(a,f\left(b,c\right)\right)$. For instance, with $a=2$, $b=3$, and $c=2$, $f\left(2^3,2\right)=f\left(8,2\right)=8^2=64$, but $f\left(2,3^2\right)=f\left(2,9\right)=2^9=512$. By changing the set $\mathbb N$ to the set of integers $\mathbb Z$, this binary operation becomes a partial binary operation since it is now undefined when $a=0$ and $b$ is any negative integer. For either set, this operation has a ''right identity'' (which is $1$) since $f\left(a,1\right)=a$ for all $a$ in the set, which is not an ''identity'' (two sided identity) since $f\left(1,b\right) \neq b$ in general.
Division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
($\div$), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration ($\uparrow\uparrow$), 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 $a \ast b$, $a+b$, $a \cdot b$ or (by
juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary terms is the showing ...
with no symbol) $ab$ rather than by functional notation of the form $f\left(a, b\right)$. 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 Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast t ...
and
reverse Polish notation Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in wh ...
.

# Binary operations as ternary relations

A binary operation $f$ on a set $S$ may be viewed as a ternary relation on $S$, that is, the set of triples $\left(a, b, f\left(a,b\right)\right)$ in $S \times S \times S$ for all $a$ and $b$ in $S$.

# External binary operations

An external binary operation is a binary function from $K \times S$ to $S$. This differs from a ''binary operation on a set'' in the sense in that $K$ need not be $S$; its elements come from ''outside''. An example of an external binary operation is
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vecto ...
in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. Here $K$ is a field and $S$ is a vector space over that field. Some external binary operations may alternatively be viewed as an action of $K$ on $S$. This requires the existence of an
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replaceme ...
multiplication in $K$, and a compatibility rule of the form $a\left(bs\right)=\left(ab\right)s$, where $a,b\in K$ and $s\in S$ (here, both the external operation and the multiplication in $K$ are denoted by juxtaposition). The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
of two vectors maps $S \times S$ to $K$, where $K$ is a field and $S$ is a vector space over $K$. It depends on authors whether it is considered as a binary operation.

* :Properties of binary operations * Iterated binary operation *
Operator (programming) In computer programming, operators are constructs defined within programming languages which behave generally like functions, but which differ syntactically or semantically. Common simple examples include arithmetic (e.g. addition with ), ...
* Ternary operation * Truth table#Binary operations * Unary operation *
Magma (algebra) In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed ...
, a set equipped with a binary operation.

* * * *