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 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,
subtraction, and
multiplication. 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 spaces 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 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
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 spaces.
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, but a
partial function, 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 study ...
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 Applied science, practical discipli ...
, 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 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, 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, 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 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.
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.
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 over that field.
Some external binary operations may alternatively be viewed as an
action 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 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
*
Magma (algebra), a set equipped with a binary operation.
Notes
References
*
*
*
*
External links
*
{{DEFAULTSORT:Binary Operation