function

A function is a relation that is total and functional.

Formally,
a function $f$ from domain $X$ to codomain $Y$ is a subset of the Cartesian product $X \times Y$ such that

every element of the domain has at least one image element:
- for all $x \in X$ there exists a $y \in Y$ such that $(x, y) \in R$
  i.e. $f$ is left-total
every element of the domain of definition is uniquely assigned:
- for all $x \in X$ and all $y_{a}, y_{b} \in Y$ , if $(x, y_{a}) \in R$ and $(x, y_{b}) \in R$ then $y_{a} = y_{b}$
  i.e. $f$ is right-unique

Thus, to be a function is for every element of the domain to have exactly one existing image element.

If $y \in Y$ is assigned to $x \in X$ by function $f$ , we say $f$ maps $x$ to $y$ , and this is commonly written $y = f (x)$ or $f : x \mapsto y$ .

The subset of $Y$ that is paired with elements of $X$ by $f$ is the image of $X$ under $f$ .

A function which is also a bijective relation is a bijection.