binary operation

TheInfoList

OR:

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 of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the
codomain In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
, 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 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 ...
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 functions. Binary operations are the keystone of most
algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
s that are studied in
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
, in particular in semigroups, monoids, groups, rings, fields, and
vector space 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 ...
s.

# Terminology

More precisely, a binary operation on a set $S$ is a mapping of the elements of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), 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 notatio ...
$S \times S$ to $S$: :$\,f \colon S \times S \rightarrow S.$ 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 (or internal) binary operation on $S$ (or sometimes expressed as having the property of closure). If $f$ is not a function, but a
partial function In mathematics, a partial function from a Set (mathematics), set to a set is a function from a subset of (possibly itself) to . The subset , that is, the Domain of a function, domain of viewed as a function, is called the domain of defini ...
, then $f$ is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can't divide by zero: $\frac$ is undefined for every real number $a$. In both universal algebra 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 $S \times S$. 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 ($\times$) 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 numbers can ...
s and matrices as well as composition of functions on a single set. For instance, * On the set of real numbers $\mathbb R$, $f\left(a,b\right)=a+b$ is a binary operation since the sum of two real numbers is a real number. * On the set of natural numbers $\mathbb N$, $f\left(a,b\right)=a+b$ 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 $M\left(2,\mathbb R\right)$ of $2 \times 2$ matrices with real entries, $f\left(A,B\right)=A+B$ is a binary operation since the sum of two such matrices is a $2 \times 2$ matrix. * On the set $M\left(2,\mathbb R\right)$ of $2 \times 2$ matrices with real entries, $f\left(A,B\right)=AB$ is a binary operation since the product of two such matrices is a $2 \times 2$ matrix. * For a given set $C$, let $S$ be the set of all functions $h \colon C \rightarrow C$. Define $f \colon S \times S \rightarrow S$ by $f\left(h_1,h_2\right)\left(c\right)=\left(h_1 \circ h_2\right)\left(c\right)=h_1\left(h_2\left(c\right)\right)$ for all $c \in C$, the composition of the two functions $h_1$ and $h_2$ in $S$. Then $f$ is a binary operation since the composition of the two functions is again a function on the set $C$ (that is, a member of $S$). Many binary operations of interest in both algebra and formal logic are commutative, satisfying $f\left(a,b\right)=f\left(b,a\right)$ for all elements $a$ and $b$ in $S$, or associative, satisfying $f\left(f\left(a,b\right),c\right)=f\left(a,f\left(b,c\right)\right)$ 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 $\mathbb R$, subtraction, that is, $f\left(a,b\right)=a-b$, is a binary operation which is not commutative since, in general, $a-b \neq b-a$. It is also not associative, since, in general, $a-\left(b-c\right) \neq \left(a-b\right)-c$; for instance, $1-\left(2-3\right)=2$ but $\left(1-2\right)-3=-4$. On the set of natural numbers $\mathbb N$, the binary operation
exponentiation Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...
, $f\left(a,b\right)=a^b$, is not commutative since, $a^b \neq b^a$ (cf. Equation xy = yx), and is also not associative since $f\left(f\left(a,b\right),c\right) \neq f\left(a,f\left(b,c\right)\right)$. For instance, with $a=2$, $b=3$, and $c=2$, $f\left(2^3,2\right)=f\left(8,2\right)=8^2=64$, but $f\left(2,3^2\right)=f\left(2,9\right)=2^9=512$. By changing the set $\mathbb N$ to the set of integers $\mathbb Z$, 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 $f\left(a,1\right)=a$ for all $a$ in the set, which is not an ''identity'' (two sided identity) since $f\left(1,b\right) \neq b$ in general. Division ($\div$), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration ($\uparrow\uparrow$), 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 $a \ast b$, $a+b$, $a \cdot b$ or (by juxtaposition with no symbol) $ab$ rather than by functional notation of the form $f\left(a, b\right)$. 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 Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which Operation (mathematics), operators ''precede'' thei ...
and reverse Polish notation.

# Binary operations as ternary relations

A binary operation $f$ on a set $S$ may be viewed as a ternary relation on $S$, that is, the set of triples $\left(a, b, f\left(a,b\right)\right)$ in $S \times S \times S$ for all $a$ and $b$ in $S$.

# External binary operations

An external binary operation is a binary function from $K \times S$ 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 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 matrix (mat ...
. Here $K$ is a field and $S$ is a
vector space 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 ...
over that field. Some external binary operations may alternatively be viewed as an action of $K$ on $S$. This requires the existence of an associative multiplication in $K$, and a compatibility rule of the form $a\left(bs\right)=\left(ab\right)s$, where $a,b\in K$ and $s\in S$ (here, both the external operation and the multiplication in $K$ are denoted by juxtaposition). The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...
of two vectors maps $S \times S$ 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.

* :Properties of binary operations *
Iterated binary operation In mathematics, an iterated binary operation is an extension of a binary operation on a Set (mathematics), set ''S'' to a function (mathematics), function on finite sequences of elements of ''S'' through repeated application. Common examples includ ...
* Operator (programming) * Ternary operation * Truth table#Binary operations * Unary operation * Magma (algebra), a set equipped with a binary operation.

* * * *