Injection/Examples/Non-Injection/Half Even Zero Odd

Example of Mapping which is Not an Injection
Let $f: \Z \to Z$ be the real function defined as:
 * $\forall x \in \Z: \map f x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$

Then $f$ is not an injection.

Proof
For $f$ to be an injection, it would be necessary that:
 * $\forall x_1, x_2 \in \Z: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$

But take, for example, $x_1 = 1, x_2 = 3$.

Then:
 * $\map f {x_1} = 0 = \map f {x_2}$

and it follows that $f$ is not an injection.