Question

Consider the following statements:

1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.

2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.

Which of the above statements is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 3 : Both 1 and 2

Concept:

Onto Functions - 

A function f from X to Y is onto (or surjective), if and only if for every element y ∈ Y there is an element x ∈ X with f(x) = y.

In words: “Each element in the co-domain of f has a pre-image” 

Mathematical Description: f : X →Y is onto ⇔ \(\rm \forall \) y \(\rm \exists \)x, f(x) = y 

One-to-one Correspondence

A function f is a one-to-one correspondence (or bijection), if and only if it is both one-to-one and onto

In words: “No element in the co-domain of f has two (or more) pre images” (one-to-one) and “Each element in the co-domain of f has a pre-image” (onto).

Calculation:

1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.

f(x) = x + 1,

calculate f(x1) :

 f(x1) = x1 + 1

calculate f(x2) :

 f(x2) = x2 + 1

Now, f(x1) = f(x2) 

⇒ x1 + 1 =  x2 + 1

⇒ x1 =  x2 

So, f is one-one function.

Consider f(x) = y

y = x + 1

x = y - 1

f(y-1) = y - 1 + 1 = y

f is onto.

2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.

f(x) = x + 1,

calculate f(x₁) :

 f(x1) = x₁ + 1

calculate f(x₂) :

 f(x₂) = x₂ + 1

Now, f(x1) = f(x2) 

⇒ x1 + 1 =  x2 + 1

⇒ x1 =  x2 

So, f is one-one function.

Clearly, f(x) = x + 1 ≥ 2 for all x ∈ N

So, f(x) does not assume values 1.

 f is not an onto function.

So, Both 1 and 2 are correct.