Question

Let X = {x² : x ∈ N} and the function f : N `rightarrow` X is defined by f(x) = x², x ∈ N. Then this function is ______.

Options

• injective only

• not bijective

• surjective only

• bijective

Answer 1

Let X = {x² : x ∈ N} and the function f : N `rightarrow` X is defined by f(x) = x², x ∈ N. Then this function is injective only.

Explanation:

Let x₁, x₂ ∈ N

f(x₁) = f(x₂)

`\implies x_1^2 = x_2^2`

`\implies x_1^2 - x_2^2 = 0`

`\implies` (x₁ + x₂) (x₁ – x₂) = 0

`\implies` x₁ = x₂    ...{x₁ + x₁ ≠ 0 as x₁, x₂ ∈ N}

Hence, f(x) is injective.

Also, the elements like 2 and 3 have no pre-image in N.

Thus, f(x) is not surjective.