Bijection/Examples/n+1 Mapping on Integers

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.

Then:

$\ds \map f {n_1}$

$=$

$\ds \map f {n_2}$

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.

$\Box$

Let $m \in \Z$.

Let $n = m - 1$.

We have that:

$n \in \Z$

and:

$\map f n = m$

Hence by definition $f$ is a surjection.

$\Box$

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

$\blacksquare$

1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections