Let f : N → N be defined as 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

Solution:

A bijective function shows both one one and onto behaviour.

f : N → N be defined as f (n) = {(n + 1)/2,

if n is odd; n/2, if n is even} for all n ∈ N

f (1) = (1 + 1)/2 = 1

and f (2) = 2/2 = 1

f (1) = f (2), where 1 ≠ 2

⇒ f is not one-one.

Consider a natural number n in codomain N.

Case I:

n is odd

⇒ n = 2r + 1

for some r Î N there exists 4r + 1 ∈ N such that

f (4r + 1) = (4r + 1 + 1)/2 = 2r + 1

Case II:

n is even

⇒ n = 2r for some r ∈ N there exists 4r ∈ N such that

f (4r ) = 4r = 2r

⇒ f is onto.

f is not a bijective function

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 9

Let f : N → N be defined as 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.

Summary:

For the function f : N → N be defined as f (n) = {(n + 1)/2 if n is odd; n/2, if n is even} for all n ∈ N, we have proved that the given function is not a bijective function