# 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

 $\, \displaystyle \forall x, y \in \Z: \,$ $\displaystyle \map f x$ $=$ $\displaystyle \map f y$ $\displaystyle \leadsto \ \$ $\displaystyle x + a$ $=$ $\displaystyle y + a$ Definition of $f$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$

demonstrating that $f$ is injective.

Then:

 $\, \displaystyle \forall y \in \Z: \,$ $\displaystyle y$ $=$ $\displaystyle \paren {y - a} + a$ $\displaystyle$ $=$ $\displaystyle \map f {y - a}$ Definition of $f$

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/.

$\blacksquare$