TITLE: How to determine injectivity and surjectivity for a map $\mathbb{Z}_n \to \mathbb{Z}_{n}$?
QUESTION [7 upvotes]: I know how to determine injectivity and surjectivity for maps between regular sets, but in this case I've got some problems. How can I solve this?

Given the following map $\psi:\overline{x} \in
 \mathbb{Z}_{16}\mapsto \overline{7}\overline{x}\in\mathbb{Z}_{16}$.
  Without calculating a single element's image, and just using the properties
  of $\overline{7}$ in $\mathbb{Z}_{16}$, decide if $\psi$ is injective,
  surjective or both. If possible, find the inverse of
  $\psi$.

REPLY [2 votes]: Since $7$ is relatively prime to $16$, it is a unit in the ring $\mathbb{Z}_{16}$, so $7x=7y$ implies $x=y$. Thus, multiplication by $7$ is injective. Since $\mathbb{Z}_{16}$ is a finite set, multiplication by $7$ is also bijective. The inverse of the map is also multiplication by $7$ since $7$ is its own inverse mod $16$.