Functions and Mappings

Definition

Definition. Let $X$ be a set with dimension $n\in {\mathbb{N}}_{>0}$ and set $Y$ set with dimension 1, a function is relationship between these sets defined as

f:&X&\to&Y\\ &(x^1,\ldots,x^n)&\mapsto& f(x^1,\ldots,x^n)\;. \end{array}

A mapping or a vector-valued function is a tuple of functions, i.e., it is a relationship between a set $X$ having dimension $n\in {\mathbb{N}}_{>0}$ into a set $Y$ having dimension $m\in {\mathbb{N}}_{>1}$

F:&X&\to&Y\\ &(x^1,\ldots,x^n)&\mapsto&(f^1(x^1,\ldots,x^n),\ldots,f^m(x^1,\ldots,x^n))\;, \end{array} $$ where each function $f^i$ is said to be a _coordinate function_. For a mapping, the function $$\begin{array}{rrcl} \pi^i:&X&\to&\mathbb{R}\\ &(x^1,\ldots,x^i,\ldots,x^n)&\mapsto&x^i \end{array}

is said to be the projection of $X$ into $\mathbb{R}$.

References

@Boo75 pp. 25