Let N be the set of natural numbers and f : N → N be a function given by f (x) = x + 1 ∀ x ∈ N. Which one of the following is correct?

Concept:

One -One Function / Injective Function:

A function f : A → B is said to be an one - one function, if different elements in A have different f - images in B.

i.e if f (x₁) = f (x₂) ⇒ x₁ = x₂, ∀ x₁, x₂ ∈ A

Onto Function / Surjective Function:

A function f : A → B is said to be an onto function if each element in B has at least one pre-image in A.

Bijective Function:

A function f : A → B is said to be a bijective function if it is both one -one and onto function.

Calculation:

Given: f : N → N and is defined as f (x) = x + 1

One - One:

Let f (x₁) = f (x₂)

⇒ x₁ + 1 = x₂ + 1

⇒ x₁ = x₂

Hence, the given function f is one - one.

Onto:

Let y = f(x) = x + 1

⇒ x = y -1

But If y = 1 ∈ N then x = 0 ∉ N.

Hence, all the elements of co - domain i.e N does not has a pre - image in the domain i.e N.