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In mathematics, a binary operation or dyadic operation is a calculation that combines 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 operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.
However, a binary operation may also involve several sets. 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 algebraic structures, that are studied in algebra, and used in all mathematics, such as fields, groups, monoids, rings, algebras, and many more.
More precisely, a binary operation on a set S is a map which sends elements of the Cartesian product S × S to S:^{[1]}^{[2]}^{[3]}
Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).^{[4]} If f is not a function, but is instead a partial function, it is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can't divide by zero: a/0 is not defined for any real a. However, both in universal algebra and model theory the binary operations considered are defined on all of S × S.
Sometimes, especially in computer science, the term is used for any binary function.
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,
Many binary operations of interest in both algebra and formal logic are commutative, satisfying f(a, b) = f(b, a) for all elements a and b in S, or associative, satisfying f(f(a, b), c) = f(a, f(b, c)) for all a, b and c in S. 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 R, subtraction, that is, f(a, b) = a − b, is a binary operation which is not commutative since, in general, a − b ≠ b − a. It is also not associative, since, in general, a − (b − c) ≠ (a − b) − c; for instance, 1 − (2 − 3) = 2 but (1 − 2) − 3 = −4.
On the set of natural numbers N, the binary operation exponentiation, f(a,b) = a^{b}, is not commutative since, a^{b} ≠ b^{a} (cf. Equation xʸ = yˣ), and is also not associative since f(f(a, b), c) ≠ f(a, f(b, c)). For instance, with a = 2, b = 3 and c = 2, f(2^{3},2) = f(8,2) = 8^{2} = 64, but f(2,3^{2}) = f(2,9) = 2^{9} = 512. By changing the set N to the set of integers Z, this binary operation becomes a partial binary operation since it is now undefined when a = 0 and b is any negative integer. For either set, this operation has a right identity (which is 1) since f(a, 1) = a for all a in the set, which is not an identity (two sided identity) since f(1, b) ≠ b 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.
Binary operations are often written using infix notation such as a ∗ b, a + b, a · b or (by juxtaposition with no symbol) ab rather than by functional notation of the form 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 of which dispense with parentheses. They are also called, respectively, Polish notation and reverse Polish notation.
A binary operation, ab, depends on the ordered pair (a, b) and so (ab)c (where the parentheses here mean first operate on the ordered pair (a, b) and then operate on the result of that using the ordered pair ((ab), c)) depends in general on the ordered pair ((a, b), c). Thus, for the general, non-associative case, binary operations can be represented with binary trees.
However:
A binary operation f on a set S may be viewed as a ternary relation on S, that is, the set of triples (a, b, f(a,b)) in S × S × S for all a and b in S.
An external binary operation is a binary function from K × S to S. This differs from a binary operation on a set in the sense in that K need not be S; its elements come from outside.
An example of an external binary operation is scalar multiplication in linear algebra. Here K is a field and S is a vector space over that field.
An external binary operation may alternatively be viewed as an action; K is acting on S.
The dot product of two vectors maps from S × S to K, where K is a field and S is a vector space over K. It depends on authors whether it is considered as a binary operation.
This article uses material from the Wikipedia article "Binary operation", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
Mathematics