In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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-Man ...
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
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 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'' ...
,
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 ...
, and
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
. Other examples are readily found in different areas of mathematics, such as
vector addition,
matrix multiplication, and
conjugation in groups.
An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example,
scalar multiplication 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 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 that takes two inputs.
Precisely stated, a function f is binary if there exists sets X, Y, Z such that
:\,f \colon X \times Y \righta ...
s.
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 ...
, 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.
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,
fields, 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
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\t ...
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, but a
partial function, then
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 re ...
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 study ...
and
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, 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 Applied science, practical discipli ...
, 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 that takes two inputs.
Precisely stated, a function f is binary if there exists sets X, Y, Z such that
:\,f \colon X \times Y \righta ...
.
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'' ...
(
) and
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
(
) 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 number ...
s and
matrices as well as
composition of functions 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, satisfying
for all elements
and
in
, 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 replacement ...
, satisfying
for all
,
, and
in
. Many also have
identity elements and
inverse elements.
The first three examples above are commutative and all of the above examples are associative.
On the set of real numbers
,
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
, 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 ...
,
, 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 (
), a partial binary operation on the set of real or rational numbers, is not commutative or associative.
Tetration (
), 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 logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in .
Usage
Binary relations are ...
such as
,
,
or (by
juxtaposition 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
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
.
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 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
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 relati ...
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
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 matrice ...
. Here
is a
field 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 of
on
. 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 replacement ...
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 alg ...
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
*
Operator (programming)
*
Ternary operation
*
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), a set equipped with a binary operation.
Notes
References
*
*
*
*
External links
*
{{DEFAULTSORT:Binary Operation