One – One Function: – A function f: A → B is said to be a one – one functions or an injection if different elements of A have different images in B.

So, f: A → B is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all a, b ∈ A

⇔ f(a) = f(b)

⇒ a = b for all a, b ∈ A

Onto Function: – A function f: A → B is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, f: A → B is Surjection iff for each b ∈ B, there exists a ∈ B such that f(a) = b

Bijection Function: – A function f: A → B is said to be a bijection function if it is one – one as well as onto function.

Now, f : N → N given by f(x) = x³

Check for Injectivity:

Let x,y be elements belongs to N i.e x, y ∈ N such that

⇒ f(x) = f(y)

⇒ x³ = y³

⇒ x³ – y³ = 0

⇒ (x – y)(x² + y² + xy) = 0

As x, y ∈ N therefore x² + y² + xy >0

⇒ x – y = 0

⇒ x = y

Hence f is One – One function

Check for Surjectivity:

Let y be element belongs to N i.e y ∈ N be arbitrary, then

⇒ f(x) = y

⇒ x³ = y

⇒ x = 3√y

⇒ 3√y not belongs to N for non–perfect cube value of y.

Since f attain only cubic number like 1,8,27….,

Therefore no non – perfect cubic values of y in N (co – domain) has a pre–image in domain N.

Hence, f is not onto function

Thus, Not Bijective also