Question

Let f : N → N be defined by

\(f(n) = \begin{cases} \frac{n+1}{2},{if}\, n \,is\,odd\\ \frac{n}{2},\,{if}\,n\,is\,even \end{cases} \) for all n ∈ N.

find whether the function f is bijective.

Answer 1

Given,

f : N → N defined such that

Let x, y ∈ N and let they are odd then,

If x, y ∈ N are both even then also,

If x, y ∈ N are such that x is odd and y is even then,

Thus, 

x ≠ y for f(x) = f(y)

Let x = 6 and y = 5

We get,

∴ f(x) = f(y) but x ≠ y

So, f(x) is not One - one.

Hence, f(x) is not bijective.