Use of $\mapsto$ and $\to$ I'm confused as to when one uses $\mapsto$ and when one uses $\to$. From what I understand, we use $\to$ when dealing with sets and $\mapsto$ when dealing with elements but I'm not entirely sure. 
For example which of the two is used for the following? $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdots \begin{pmatrix} x+y \\ z+y \\ x+z \\ -z\end{pmatrix}$$ 

The difference is that $\mapsto$ denotes the function itself.  Thus you need not name the function.  $a\mapsto b$ fully describes the action of the function.  $\to$, on the other hand, describes only the domain and codomain.  Thus one might say $x\mapsto x+1$ is equivalent to $f(x)=x+1$, in which case we would say $f:\mathbb{R}\to\mathbb{R}$.
