# Injection/Examples/2x Function on Integers

## Example of Injection which is Not a Surjection

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = 2 x$

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

## Proof

Let $x_1$ and $x_2$ be integers.

Then:

 $\ds \map f {x_1}$ $=$ $\ds \map f {x_2}$ by supposition $\ds \leadsto \ \$ $\ds 2 x_1$ $=$ $\ds 2 x_2$ Definition of $f$ $\ds \leadsto \ \$ $\ds x_1$ $=$ $\ds x_2$

Hence $f$ is an injection by definition.

$\Box$

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

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

Thus by definition $f$ is not a surjection.

$\blacksquare$