Permutation/Examples/Addition of Constant on Integers

Examples of Permutations
Let $\Z$ denote the set of integers.

Let $a \in \Z$.

Let $f: \Z \to \Z$ denote the mapping defined as:
 * $\forall x \in \Z: \map f x = x + a$

Then $f$ is a permutation on $\Z$.

Proof
demonstrating that $f$ is injective.

Then:

As $y - a \in \Z$ it follows that $f$ is a surjection

So $f$ is both an injection and a surjection.

By definition, then, $f$ is a bijection.

As the domain and codomain of $f$ is the same, $f$ is by definition a permutation/.