Injection/Examples/2x Function on Integers

Example of Mapping which is Not an Injection
Let $f: \Z \to Z$ be the mapping defined on the set of integers as:
 * $\forall x \in \Z: f \paren x = 2 x$

Then $f$ is an injection, but not a surjection.

Proof
Let $x_1$ and $x_2$ be integers.

Then:

Hence $f$ is an injection by definition.

Now consider $y = 2 n + 1$ for some $n \in \Z$.

There exists no $x \in \Z$ such that $f \paren x = y$.

Thus by definition $f$ is not a surjection.