Binary operation

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 that maps every ordered pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. 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 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 , , groups, rings, fields, and .

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Terminology More precisely, a binary operation on a set $S$ is a mapping of the elements of the Cartesian product $S\; \backslash times\; S$ to $S$:

$\backslash ,f\; \backslash colon\; S\; \backslash times\; S\; \backslash rightarrow\; S.$

If $f$ is not a function but a partial function, then $f$ is called a partial binary operation. For instance, division is a partial binary operation on the set of all real numbers, because one cannot divide by zero: $\backslash frac\{a\}\{0\}$ is undefined for every real number $a$. In both model theory and classical universal algebra, binary operations are required to be defined on all elements of $S\; \backslash times\; S$. However,

generalize universal algebras to allow partial operations.

Sometimes, especially in computer science, the term binary operation is used for any binary function.

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Properties and examples Typical examples of binary operations are the addition ($+$) and multiplication ($\backslash times$) of and matrices as well as composition of functions on a single set. For instance,

• On the set of real numbers $\backslash mathbb\; R$, $f(a,b)=a+b$ is a binary operation since the sum of two real numbers is a real number.
• On the set of natural numbers $\backslash mathbb\; N$, $f(a,b)=a+b$ 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 $M(2,\backslash mathbb\; R)$ of $2\; \backslash times\; 2$ matrices with real entries, $f(A,B)=A+B$ is a binary operation since the sum of two such matrices is a $2\; \backslash times\; 2$ matrix.
• On the set $M(2,\backslash mathbb\; R)$ of $2\; \backslash times\; 2$ matrices with real entries, $f(A,B)=AB$ is a binary operation since the product of two such matrices is a $2\; \backslash times\; 2$ matrix.
• For a given set $C$, let $S$ be the set of all functions $h\; \backslash colon\; C\; \backslash rightarrow\; C$. Define $f\; \backslash colon\; S\; \backslash times\; S\; \backslash rightarrow\; S$ by $f(h\_1,h\_2)(c)=(h\_1\; \backslash circ\; h\_2)(c)=h\_1(h\_2(c))$ for all $c\; \backslash in\; C$, 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 $f(a,b)=f(b,a)$ 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 and .

The first three examples above are commutative and all of the above examples are associative.

On the set of real numbers $\backslash mathbb\; R$, subtraction, that is, $f(a,b)=a-b$, is a binary operation which is not commutative since, in general, $a-b\; \backslash neq\; b-a$. It is also not associative, since, in general, $a-(b-c)\; \backslash neq\; (a-b)-c$; for instance, $1-(2-3)=2$ but $(1-2)-3=-4$.

On the set of natural numbers $\backslash mathbb\; N$, the binary operation exponentiation, $f(a,b)=a^b$, is not commutative since, $a^b\; \backslash neq\; b^a$ (cf. Equation x^y = y^x), and is also not associative since $f(f(a,b),c)\; \backslash neq\; f(a,f(b,c))$. 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 $\backslash mathbb\; N$ to the set of integers $\backslash mathbb\; Z$, 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 $f(a,1)=a$ for all $a$ in the set, which is not an identity (two sided identity) since $f(1,b)\; \backslash neq\; b$ in general.

Division ($\backslash div$), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration ($\backslash uparrow\backslash uparrow$), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.

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Notation Binary operations are often written using infix notation such as $a\; \backslash ast\; b$, $a+b$, $a\; \backslash cdot\; b$ 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 are sometimes written using prefix or postfix notation, both of which dispense with parentheses. They are also called, respectively, Polish notation $\backslash ast\; a\; b$ and reverse Polish notation $a\; b\; \backslash ast$.

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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 $(a,\; b,\; f(a,b))$ in $S\; \backslash times\; S\; \backslash times\; S$ for all $a$ and $b$ in $S$.

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Other binary operations For example, scalar multiplication in linear algebra. Here $K$ is a field and $S$ is a vector space over that field.

Also the dot product of two vectors maps $S\; \backslash times\; S$ 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.

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See also

• Properties of binary operations

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Notes

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External links

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