One – One Function: – A function f:A → B a  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

Now, Let, f:N → N given by f(x) = x²

Check for Injectivity:

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

So, from definition

⇒ f(x) = f(y)

⇒ x² = y²

⇒ x² – y² = 0

⇒ (x – y)(x + y) = 0

As x, y ∈ N therefore x + y>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

⇒ x² = y

⇒ x = √y

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

Therefore no non – perfect square value of y has a pre image in domain N.

Hence, f:N → N given by f(x) = x² is One – One but not onto.