Permutation/Examples/Addition of Constant on Integers

From ProofWiki
Jump to navigation Jump to search

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

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

demonstrating that $f$ is injective.


Then:

\(\ds \forall y \in \Z: \, \) \(\ds y\) \(=\) \(\ds \paren {y - a} + a\)
\(\ds \) \(=\) \(\ds \map f {y - a}\) 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