In
mathematics, a binary operation or dyadic operation is a rule for combining two
elements (called
operands
In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Example
The following arithmetic expression shows an example of operators and operands:
:3 + 6 = 9
In the above exampl ...
) 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 two.
More specifically, an internal binary operation ''on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
'' 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
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
of
addition,
subtraction, and
multiplication. 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 a ...
,
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 s ...
, 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 vector b ...
of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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'' 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 structures 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 ...
, in particular in
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s,
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.
Monoid ...
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 space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s.
Terminology
More precisely, a binary operation on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is a
mapping of the elements of the
Cartesian product to
:
:
Because the result of performing the operation on a pair of elements of
is again an element of
, the operation is called a closed (or internal) binary operation on
(or sometimes expressed as having the property of
closure).
If
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-oriente ...
, 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 de ...
, then
is called a partial binary operation. For instance, division of
real numbers is a partial binary operation, because one can't
divide by zero:
is undefined for every real number
. 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 stu ...
and
model theory, binary operations are required to be defined on all elements of
.
Sometimes, especially in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, the term binary operation is used for any
binary function.
Properties and examples
Typical examples of binary operations are the
addition (
) and
multiplication (
) 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 c ...
s and
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
as well as
composition of functions
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
on a single set.
For instance,
* On the set of real numbers
,
is a binary operation since the sum of two real numbers is a real number.
* On the set of natural numbers
,
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
of
matrices with real entries,
is a binary operation since the sum of two such matrices is a
matrix.
* On the set
of
matrices with real entries,
is a binary operation since the product of two such matrices is a
matrix.
* For a given set
, let
be the set of all functions
. Define
by
for all
, the composition of the two functions
and
in
. Then
is a binary operation since the composition of the two functions is again a function on the set
(that is, a member of
).
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
for all elements
and
in
, or
associative, satisfying
for all
,
, and
in
. 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 su ...
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 is ...
s.
The first three examples above are commutative and all of the above examples are associative.
On the set of real numbers
,
subtraction, 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
, 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 r ...
,
, is not commutative since,
(cf.
Equation xy = yx), and is also not associative since
. For instance, with
,
, and
,
, but
. By changing the set
to the set of integers
, this binary operation becomes a partial binary operation since it is now undefined when
and
is any negative integer. For either set, this operation has a ''right identity'' (which is
) since
for all
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 ...
(
), a partial binary operation on the set of real or rational numbers, is not commutative or associative.
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
(
), 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
,
,
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 showin ...
with no symbol)
rather than by functional notation of the form
. 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 ...
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 whi ...
.
Binary operations as ternary relations
A binary operation
on a set
may be viewed as a
ternary relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.
Just as a binary relat ...
on
, that is, the set of triples
in
for all
and
in
.
External binary operations
An external binary operation is a binary function from
to
. This differs from a ''binary operation on a set'' in the sense in that
need not be
; 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 vector b ...
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
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 grass ...
and
is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over that field.
Some external binary operations may alternatively be viewed as an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of
on
. This requires the existence of an
associative multiplication in
, and a compatibility rule of the form
, where
and
(here, both the external operation and the multiplication in
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 alge ...
of two vectors maps
to
, where
is a field and
is a vector space over
. It depends on authors whether it is considered as a binary operation.
See also
*
:Properties of binary operations
*
Iterated binary operation
In mathematics, an iterated binary operation is an extension of a binary operation on a set ''S'' to a function on finite sequences of elements of ''S'' through repeated application. Common examples include the extension of the addition 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 ), c ...
*
Ternary operation
In mathematics, a ternary operation is an ''n''-ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''.
In computer science, a ternary operator i ...
*
Truth table#Binary operations
*
Unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
*
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.
Notes
References
*
*
*
*
External links
*
{{DEFAULTSORT:Binary Operation