TheInfoList 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 their changes (cal ...
, a binary operation or dyadic operation is a calculation that combines two elements (called
operands 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 ...
) to produce another element. More formally, a binary operation is an operation of
arity Arity () is the number of arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
two. More specifically, a binary operation ''on a set'' is an operation whose two domains and the
codomain 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 ... are the same set. Examples include the familiar
arithmetic operations Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...
of
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ... ,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ... ,
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 arithmetic, elementary Operation (mathematics), mathematical operations ... . Other examples are readily found in different areas of mathematics, such as
vector addition 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 ... ,
matrix multiplication 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 ... and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example,
scalar multiplication 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... of
vector space 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). ...
s takes a scalar and a vector to produce a vector, and
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
takes two vectors to produce a scalar. Such binary operations may be called simply
binary function In mathematics, a binary function (also called bivariate function, or function of two variables) is a function (mathematics), function that takes two inputs. Precisely stated, a function f is binary if there exists Set (mathematics), sets X, Y, Z ...
s. Binary operations are the keystone of most
algebraic structure 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 ...
s, that are studied in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ... , in particular in
semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...
s,
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
s,
groups A group is a number of people 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 identi ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...
,
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
, and
vector space 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). ...
s.

# Terminology

More precisely, a binary operation on a set ''S'' is a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
of the elements of the
Cartesian product 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 ...
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 function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, but a
partial function 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). ... , then ''f'' is called a partial binary operation. For instance, division of
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ... is a partial binary operation, because one can't divide by zero: ''a''/0 is undefined for every real number ''a''. In both
universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...
and
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
, binary operations are required to be defined on all of . Sometimes, especially in
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
, the term binary operation is used for any
binary function In mathematics, a binary function (also called bivariate function, or function of two variables) is a function (mathematics), function that takes two inputs. Precisely stated, a function f is binary if there exists Set (mathematics), sets X, Y, Z ...
.

# Properties and examples

Typical examples of binary operations are the
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ... (+) 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 arithmetic, elementary Operation (mathematics), mathematical operations ... (×) of
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ... s and
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
as well as
composition of functions 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). ...
on a single set. For instance, * On the set of real numbers R, is a binary operation since the sum of two real numbers is a real number. * On the set of natural numbers N, 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(2,R) of matrices with real entries, is a binary operation since the sum of two such matrices is a matrix. * On the set M(2,R) of matrices with real entries, is a binary operation since the product of two such matrices is a matrix. * For a given set ''C'', let ''S'' be the set of all functions . Define by for all , the composition of the two functions ''h'' and ''h'' 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 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 ...
, satisfying for all elements ''a'' and ''b'' in ''S'', or
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). ...
, satisfying for all ''a'', ''b'' and ''c'' in ''S''. Many also have
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 ...
s and
inverse element In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. The first three examples above are commutative and all of the above examples are associative. On the set of real numbers R,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ... , that is, , is a binary operation which is not commutative since, in general, . It is also not associative, since, in general, ; for instance, but . On the set of natural numbers N, the binary operation
exponentiation Exponentiation is a 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) ...
, , is not commutative since, (cf.
Equation xʸ = yˣ In general, 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 An ...
), and is also not associative since . For instance, with , and , , but . By changing the set N to the set of integers Z, this binary operation becomes a partial binary operation since it is now undefined when and ''b'' is any negative integer. For either set, this operation has a ''right identity'' (which is 1) since for all ''a'' in the set, which is not an ''identity'' (two sided identity) since 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 o ...
(/), a partial binary operation on the set of real or rational numbers, is not commutative or associative.
Tetration 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). ...
(↑↑), 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 Infix notation is the notation commonly used in arithmetical and logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...
such as , , 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 . Powers are usually also written without operator, but with the second argument as
superscript Pro; the size of the subscript is about 62% of the original characters, dropped below the baseline by about 16%. The second typeface is Myriad A myriad (from Ancient Greek Ancient Greek includes the forms of the Greek language used in a ... . Binary operations sometimes use prefix or (probably more often) 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 Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in which operators ''precede'' their ...
.

# Pair and tuple

A binary operation, ''ab'', depends on the
ordered pair 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 ... (''a, b'') and so (''ab'')''c'' (where the parentheses here mean first operate on the ordered pair (''a'', ''b'') and then operate on the result of that using the ordered pair ((''ab''), ''c'')) depends in general on the ordered pair ((''a'', ''b''), ''c''). Thus, for the general, non-associative case, binary operations can be represented with
binary tree In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ... s. However: *If the operation is associative, (''ab'')''c'' = ''a''(''bc''), then the value of (''ab'')''c'' depends only on the
tuple 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). ...
(''a'', ''b'', ''c''). *If the operation is commutative, ''ab'' = ''ba'', then the value of (''ab'')''c'' depends only on , where braces indicate
multiset 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 ...
s. *If the operation is both associative and commutative then the value of (''ab'')''c'' depends only on the multiset . *If the operation is associative, commutative and
idempotent Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
, ''aa'' = ''a'', then the value of (''ab'')''c'' depends only on the set .

# 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 (''a'', ''b'', ''f''(''a,b'')) in ''S'' × ''S'' × ''S'' for all ''a'' and ''b'' in ''S''.

# External binary operations

An external binary operation is a binary function from ''K'' × ''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 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... 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 matrix (mat ...
. Here ''K'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
and ''S'' is a
vector space 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). ...
over that field. An external binary operation may alternatively be viewed as an
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
; ''K'' is acting on ''S''. The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of two vectors maps from ''S'' × ''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.

* Truth table#Binary operations *
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 ...
*
Operator (programming) In computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, ge ...
*
Ternary 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 ...
*
Unary operation 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 ...

* * * *