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In mathematics, a binary operation is a calculation involving two operands (an operation whose arity is two). Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division.

Let A, B and C be three sets. Then a relation * from A X B --> C is a binary relation. This is a general binary relation. Usually, but not always, the case of binary relations defined on a single set is of interest.

Terminology[link]

More precisely, a binary operation on a set Failed to parse (Missing texvc executable; please see math/README to configure.): S

is a binary relation that maps elements of the Cartesian product Failed to parse (Missing texvc executable; please see math/README to configure.): S 

×Failed to parse (Missing texvc executable; please see math/README to configure.): S

to Failed to parse (Missing texvc executable; please see math/README to configure.): S

Such a binary operation from S×S→S is called a closed binary operation on S

Failed to parse (Missing texvc executable; please see math/README to configure.): \,f \colon S \times S \rightarrow S.

If Failed to parse (Missing texvc executable; please see math/README to configure.): f

is not a function, but is instead a partial function, it is called a partial operation. For instance, division of real numbers is a partial function, because one can't divide by zero: Failed to parse (Missing texvc executable; please see math/README to configure.): a/0
is not defined for any real Failed to parse (Missing texvc executable; please see math/README to configure.): a

. Note however that both in algebra and model theory the binary operations considered are defined on the whole of Failed to parse (Missing texvc executable; please see math/README to configure.): S \times S .

Sometimes, especially in computer science, the term is used for any binary function. That Failed to parse (Missing texvc executable; please see math/README to configure.): f

takes values in the same set Failed to parse (Missing texvc executable; please see math/README to configure.): S
that provides its arguments is the property of closure.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more. Most generally, a magma is a set together with any binary operation defined on it.

Laws on operations[link]

Many binary operations of interest in both algebra and formal logic are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (×) of numbers and matrices as well as composition of functions on a single set.

An example of an operation that is not commutative is subtraction (−). Examples of partial operations that are not commutative include division (/), exponentiation(^), and tetration(↑↑).

Notation[link]

Binary operations are often written using infix notation such as Failed to parse (Missing texvc executable; please see math/README to configure.): a*b , Failed to parse (Missing texvc executable; please see math/README to configure.): a + b , Failed to parse (Missing texvc executable; please see math/README to configure.): a \cdot b

or (by juxtaposition with no symbol) Failed to parse (Missing texvc executable; please see math/README to configure.): ab
rather than by functional notation of the form Failed to parse (Missing texvc executable; please see math/README to configure.): f(a, b)

. Powers are usually also written without operator, but with the second argument as superscript.

Binary operations sometimes use prefix or (probably more often) postfix notation, both which dispense with parentheses. They are also called, respectively, Polish notation and reverse Polish notation.

Pair and tuple[link]

A binary operation, Failed to parse (Missing texvc executable; please see math/README to configure.): ab , depends on the ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): (a, b)

and so Failed to parse (Missing texvc executable; please see math/README to configure.): (ab)c
(where the parentheses here mean first operate on the ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): (a, b)
and then operate on the result of that using the ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): ((ab), c))
depends in general on the ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): ((a,b),c)

. Thus, for the general, non-associative case, binary operations can be represented with binary trees.

However:

• If the operation is associative, Failed to parse (Missing texvc executable; please see math/README to configure.): (ab)c = a(bc)

, then the value depends only on the tuple (a, b, c).

• If the operation is commutative, Failed to parse (Missing texvc executable; please see math/README to configure.): ab = ba

, then the value depends only on { {a, b}, c}, where braces indicate multisets.

• If the operation is both associative and commutative then the value depends only on the multiset {a, b, c}.
• If the operation is both associative and commutative and idempotent, Failed to parse (Missing texvc executable; please see math/README to configure.): aa = a

, then the value depends only on the set {a, b, c}.

External binary operations[link]

An external binary operation is a binary function from Failed to parse (Missing texvc executable; please see math/README to configure.): K

× Failed to parse (Missing texvc executable; please see math/README to configure.): S
to Failed to parse (Missing texvc executable; please see math/README to configure.): S

. This differs from a binary operation in the strict sense in that Failed to parse (Missing texvc executable; please see math/README to configure.): K

need not be Failed to parse (Missing texvc executable; please see math/README to configure.): S

its elements come from outside.

An example of an external binary operation is scalar multiplication in linear algebra. Here Failed to parse (Missing texvc executable; please see math/README to configure.): K

is a field and Failed to parse (Missing texvc executable; please see math/README to configure.): S
is a vector space over that field.

An external binary operation may alternatively be viewed as a action; Failed to parse (Missing texvc executable; please see math/README to configure.): K

is acting on Failed to parse (Missing texvc executable; please see math/README to configure.): S

.

Note that the dot product of two vectors is not a binary operation, external or otherwise, as it maps from Failed to parse (Missing texvc executable; please see math/README to configure.): S × Failed to parse (Missing texvc executable; please see math/README to configure.): S

to Failed to parse (Missing texvc executable; please see math/README to configure.): K

, where Failed to parse (Missing texvc executable; please see math/README to configure.): K

is a field and Failed to parse (Missing texvc executable; please see math/README to configure.): S
is a vector space over Failed to parse (Missing texvc executable; please see math/README to configure.): K

.

References[link]

• Weisstein, Eric W., "Binary Operation" from MathWorld.

See also[link]

• Iterated binary operation
• Operator (programming)
• Ternary operation
• Unary operation