Bijection/Examples/n+1 Mapping on Integers

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Example of Bijection

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

$\forall n \in \Z: \map f n = n + 1$

Then $f$ is a bijection.


Let $n_1$ and $n_2$ be integers.


\(\ds \map f {n_1}\) \(=\) \(\ds \map f {n_2}\) by supposition
\(\ds \leadsto \ \ \) \(\ds n_1 + 1\) \(=\) \(\ds n_2 + 1\) Definition of $f$
\(\ds \leadsto \ \ \) \(\ds n_1\) \(=\) \(\ds n_2\)

Hence by definition $f$ is an injection.


Let $m \in \Z$.

Let $n = m - 1$.

We have that:

$n \in \Z$


$\map f n = m$

Hence by definition $f$ is a surjection.


Thus $f$ is both an injection and a surjection, and so a bijection