# Binary operation

In

two.More specifically, a **binary operation** *on a *

*conjugation in groups*

*.*

A binary function that involves several sets is sometimes also called a *binary operation*. For example,

Binary operations are the keystone of most

*.
*

## Terminology

More precisely, a binary operation on a set is a mapping of the elements of the Cartesian product to :^{[1]}^{[2]}^{[3]}

The closure property of a binary operation expresses the existence of a result for the operation given any pair of operands.^{[4]}

If is not a function but a partial function, then is called a **partial binary operation**. For instance, division of

^{[5]}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

- 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.
- On the set of matrices with real entries, is a binary operation since the sum of two such matrices is a matrix.
- On the set of matrices with real entries, is a binary operation since the product of two such matrices is a matrix.
- For a given set , let be the set of all functions . Define by 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

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.

^{y}= y

^{x}

*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

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 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.

## See also

- Category:Properties of binary operations
- Iterated binary operation – Repeated application of an operation to a sequence
- Magma (algebra) – Algebraic structure with a binary operation
- Operator (programming)– Construct associated with a mathematical operation in computer programs
- Ternary operation – Mathematical operation that combines three elements to produce another element
- Truth table § Binary operations
- Unary operation – Mathematical operation with only one operand

## Notes

**^**Rotman 1973, pg. 1**^**Hardy & Walker 2002, pg. 176, Definition 67**^**Fraleigh 1976, pg. 10**^**Hall 1959, pg. 1- ISBN 978-0-387-77487-9.

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## References

- Fraleigh, John B. (1976),
*A First Course in Abstract Algebra*(2nd ed.), Reading: Addison-Wesley, - Hall, Marshall Jr. (1959),
*The Theory of Groups*, New York: Macmillan - Hardy, Darel W.;
- Rotman, Joseph J. (1973),
*The Theory of Groups: An Introduction*(2nd ed.), Boston: Allyn and Bacon

## External links