Question

Let A = {−1, 0, 1} and f = {(x, x²) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

Answer 1

Given that A = {−1, 0, 1} and f = {(x, x²): x ∈ A}

Also given that, f(x) = x²

Let us prove that given function neither one-one or nor onto.

Injectivity:

Let x = 1

Therefore f(1) = 1² = 1 and

f(-1) = (-1)² = 1

⇒ 1 and -1 have the same images.

Therefore, f is not one-one.

Surjectivity:

Co-domain of f = {-1, 0, 1}

f(1) = 1² = 1,

f(-1) = (-1)² = 1 and

f(0) = 0

⇒ Range of f  = {0, 1}

Therefore, both are not same.

Hence, f is not onto