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? $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdots \begin{pmatrix} x+y \\ z+y \\ x+z \\ -z\end{pmatrix}$$ 

Your operative definition of $\longrightarrow$ and $\longmapsto$ are correct. In your example you are dealing with vectors in $\mathbb{R}^n$, which are elements of a set, so you should use $\longmapsto$.
see also this wikipedia article: http://en.wikipedia.org/wiki/List_of_mathematical_symbols
