Q. 94.2( 36 Votes )

# Let f : N → N be defined by

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

Answer :

It is given that

f : N → N be defined by

We can observed that:

(by using the definition of f)

Thus, f(1) = f(2), where 1 ≠ 2.

Therefore, f is not one-one.

Now, let us consider a natural number (n) in co domain N.

**Case I:** When n is odd.

Then, n = 2r +1 for some r ϵ N.

⇒ there exist 4r + 1 ϵ N such that f(4r+1) =

**Case II:** When n is even.

Then, n = 2r for some r ϵ N.

⇒ there exist 4r ϵ N such that f(4r) =

Therefore, f is onto.

⇒ Function f is not one-one but it is onto.

Thus, Function f is not bijective function.

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