Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be defined by
$f(n)=\left\{\begin{array}{l}n+1, \text { if } n \text { is odd } \\ n-1, \text { if } n \text { is even }\end{array}\right.$
Show that f is a bijection. [CBSE 2012, NCERT]
We have,
$f(n)= \begin{cases}n+1, & \text { if } n \text { is odd } \\ n-1, & \text { if } n \text { is even }\end{cases}$
Injection test:
Case I: If $n$ is odd,
Let $x, y \in \mathbf{N}$ such that $f(x)=f(y)$
As, $f(x)=f(y)$
$\Rightarrow x+1=y+1$
$\Rightarrow x=y$
Case II : If $n$ is even,
Let $x, y \in \mathbf{N}$ such that $f(x)=f(y)$
As, $f(x)=f(y)$
$\Rightarrow x-1=y-1$
$\Rightarrow x=y$
So, $f$ is injective.
Surjection test :
Case I: If $n$ is odd,
As, for every $n \in \mathbf{N}$, there exists $y=n-1$ in $\mathbf{N}$ such that
$\mathrm{f}(y)=f(n-1)=n-1+1=n$
Case II : If $n$ is even,
As, for every $n \in \mathbf{N}$, there exists $y=n+1$ in $\mathbf{N}$ such that
$\mathrm{f}(y)=f(n+1)=n+1-1=n$
So, $f$ is surjective.
So, $f$ is a bijection.