Operation

Intuition

An operation is a process aiming at obtaining a result through some manipulations of mathematical objects or values: addition, substraction, multiplication and division are the most commonly known operations.

Formal definition

Let $E$ be a set. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.

Remarks and examples

• If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation - like addition, substraction, multiplication...

• The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is an operation on the set of integers.

• The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.