Permutation/Examples/Addition of Constant on Integers

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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/.

$\blacksquare$


Sources