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.