Question

Let $f:N\to N$ be defined by $f\left(n\right)=\left\{\begin{array}{c}\frac{n+1}{2},\text{if n is odd}\\ \frac{n}{2},\text{if n is even}\end{array}\right\}$ for all $n\in N$.

State whether the function $f$ is bijective. Justify your answer.

Answer 1

$f\left(n\right)=\left\{\begin{array}{c}\frac{n+1}{2},\text{if n is odd}\\ \frac{n}{2},\text{if n is even}\end{array}\right\}$ for all $n\in N$
$f:N\to N$ is defined as

It can be observed that:

$f\left(1\right)=\frac{1+1}{2}=1$ and $f\left(2\right)=\frac{2}{2}=1$

$\therefore f\left(1\right)=f\left(2\right)$, where $1\ne 2$

$\therefore f$ is not one-one.

Consider a natural number $\left(n\right)$ in co-domain $N$Case $I$: $n$ is odd
$\therefore n=2r+1$ for some $r\in N$.
Then, there exists $4r+1\in N$ such that $f(4r+1)=\frac{4r+1+1}{2}=2r+1$
Case $II$: $n$ is even
$\therefore n=2r$ for some $r\in N$.
Then, there exists $4r\in N$ such that $f\left(4r\right)=\frac{4r}{2}=2r$
$\therefore f$ is onto.
Hence, $f$ is not a bijective function