Question

Let f: N → N be defined by

f(n) = { (n+1) / 2, if n is odd
n / 2 , if n is even } for all n ∈ N

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

Answer 1

Given,
$f(n)=\left\{\begin{array}{ll}\frac{n+1}{2}&\text{If n is odd}\\\frac{n}{2}&\text{if n is even}\end{array}\right.$
now take n = 1 { it is odd}
so, f(1) = (1 + 1)/2 = 1
take n = 2 { it is even }
so, f(2) = 2/2 = 1
here, we can see f(1) = f(2) = 1 but 1 ≠ 2
so, f is not one - one function.

$\bf{case-I}$: When n is odd.
and assume n = 2r +1 for some r ϵ N.
⇒ there exist, 4r + 1 ϵ N such that f(4r+1) =(4r + 1 + 1)/2 = 2r + 1

$\bf{case-II}$: when n is even
and we assume n = 2r for some r ϵ N.
⇒ there exist 4r ϵ N such that f(4r) = (4r)/2 = 2r
hence, co-domain = range
so, f is onto function.
finally we get f is not one -one but it is onto.
so,f is not bijective function.