Q13 of 45 Page 1

Let f:N → N is defined by for all n ∈ N.

Find whether the function f is objective.

Given: f (n) = (n + 1)/ 2, if n is odd


n/ 2, if n is even


Case 1: When n is odd,


When n = 1,


When n = 3,


When n = 2k – 1, … (1)


Case 2: When n is even,


When n = 2,


When n = 4,


When n = 2k, … (2)


When n = 2n, … (3)


From (1) and (2),


⇒ f (2k – 1) = f (2k)


∴ f is not one – one.


From (3),


⇒ f (2n) = n


∴ f is onto.


∴ f is onto but not one – one.


Hence f is not bijective.


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