Give an example of a map

which is one-one but not onto

Let f : N → N, be a function given by f(x) = 2x.

In order to prove that f is one-one, it is sufficient to prove that f(x₁)=f(x₂) ⇒ x₁=x₂∀ x₁, x₂ ∈ N

Now, let f(x₁) = f(x₂)

⇒ 2x₁= 2x₂

⇒ x₁= x₂

⇒ f is one-one.

f is not onto, as for 1 ∈ N, there does not exist any x in N such that f(x) = 2x = 1.

Thus, f : N → N, be a function given by f(x) = 2x, which is one-one but not onto.