Binary operation/Computing operation/Introduction/Section

We consider the mathematical operations addition and multiplication, within the real numbers, as mappings

$\mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,$

so we assign to each pair

${}(x,y)\in \mathbb {R} \times \mathbb {R} \,$

the real number ${}x+y$ (and ${}x\cdot y$ respectively). Such a mapping is called an operation.

Definition

An operation (or binary operation) ${}\circ$ on a set ${}M$ is a mapping

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.$

The domain of an operation is thus the product set of ${}M$ with itself, and the range is also ${}M$. Addition, multiplication and subtraction (on ${}\mathbb {Z}$, on ${}\mathbb {Q}$, or on ${}\mathbb {R}$) are operations. On ${}\mathbb {Q}$ and ${}\mathbb {R}$, the division is not an operation, since it is not defined in case the second component equals ${}0$. However, the division is an operation on ${}\mathbb {R} \setminus \{0\}$.