Binary operation

Some definitions require the two inputs and the output to be elements of the same set.

Binary operations are usually written using infix notation such as ${\displaystyle a\star b}$ rather than by functional notation of the form ${\displaystyle f(a,b)}$. Multiplication and exponentiation are frequently written without operator, but with exponents as superscript.

In abstract algebra, a binary operation on a set ${\displaystyle S}$ is a mapping of the elements of the Cartesian product ${\displaystyle S\times S}$ to ${\displaystyle S}$: $${\displaystyle f:S\times S\to S.}$$

If the mapping is total, ${\displaystyle (S,f)}$ is a magma. However, if ${\displaystyle f}$ is a partial mapping, then ${\displaystyle f}$ is a partial binary operation, which can be one of the operations in a partial algebra on ${\displaystyle S}$, such as the partial groupoid ${\displaystyle (S,f)}$.

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

Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.

Examples

Arithmetic operations like addition (${\displaystyle a+b}$), subtraction (${\displaystyle a-b}$), and multiplication (${\displaystyle a\times b}$ or, using juxtaposition, ${\displaystyle ab}$) are binary operations on numbers. Division (${\displaystyle a/b}$ or ${\displaystyle a÷b}$) is only a partial binary operation on the set of rational numbers and the set of real numbers, because division by zero is undefined, but a (total) binary operation on the set of positive real numbers.

Union (${\displaystyle X\cup Y}$), intersection (${\displaystyle X\cap Y}$), and set difference (${\displaystyle X\mathrm{\setminus }Y}$) are binary operations on sets. There are 16 binary operations on binary variables, such as 'AND', 'OR', and 'XOR'.

If a broad definition of "binary operation" is used, such that the inputs and output do not have to be from the same set, then scalar multiplication (${\displaystyle a\overrightarrow{v}}$) is an binary operator from a scalar and a vector to a vector, and dot product (${\displaystyle \overrightarrow{u}\cdot \overrightarrow{v}}$) is a binary operator from two vectors to a scalar.

Function composition (${\displaystyle g\circ h}$) is a binary operation on compatible functions.