Binary operation explained
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 of arity two.
More specifically, a binary operation on a set is a binary function 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.
A binary function 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.
Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
Terminology
is a
mapping of the elements of the
Cartesian product
to
:
The
closure property of a binary operation expresses the existence of a result for the operation given any pair of operands.
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 not
divide by zero:
is undefined for every real number
. In both
model theory and classical
universal algebra, binary operations are required to be defined on all elements of
. However,
partial algebras
[1] generalize
universal algebras to allow partial operations.
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 (
) and
multiplication (
) of
numbers 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.
of
matrices with real entries,
is a binary operation since the sum of two such matrices is a
matrix.
of
matrices with real entries,
is a binary operation since the product of two such matrices is a
matrix.
, let
be the set of all functions
. Define
by
f(h1,h2)(c)=(h1\circh2)(c)=h1(h2(c))
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,
, is not commutative since,
(cf.
Equation xy = yx), and is also not associative since
f(f(a,b),c) ≠ f(a,f(b,c))
. 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.
and
reverse Polish notation
.
Binary operations as ternary relations
A binary operation
on a set
may be viewed as a
ternary relation on
, that is, the set of triples
in
for all
and
in
.
Other binary operations
For example, scalar multiplication in linear algebra. Here
is a
field and
is a
vector space over that field.
Also the dot product 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.
Notes and References
- Book: George A. Grätzer. Universal Algebra. registration. 2008. Springer Science & Business Media. 978-0-387-77487-9. Chapter 2. Partial algebras. 2nd.