Question

Let $f:N\to N$ be defined by $f\left(n\right)=\{\begin{array}{cc}\frac{n+1}{2},& \text{if}n\text{is odd}\\ \frac{n}{2},& \text{if}n\text{is even}\end{array}$.$n\in N$
State whether the function $f$ is bijective. Justify your answer.

Answer 1

For $n$ being odd,

let $n=2K+1$

where $K$ is odd.

now, $\frac{n+1}{2}=K+1$ is even.

And when $K$ is even.

$\frac{n+1}{2}=K+1$ is odd.

Also $n=$even$=2K$

for $K=$even.

$\frac{n}{2}=K+1$ is even.

for $K=$odd.

$\frac{n}{2}=K+1$ is odd.

Thus every value of set $N$ is covered in range as well as domain.

So $f\left(x\right)$ is bijective