Let f AB and g BC be functions a Assume that gf is injective

Let f: AB and g: BC be functions.

(a) Assume that gf is injective. Does this imply that both f and g are injective?

(b) Assume that gf is surjective. Does this imply that both f and g are surjective?

Solution

Solution: (a). If gf is injective then only f is injective. We will prove this fact as follow.

Suppose that gf is injective; we show that f is injective. let x₁, x₂ A and suppose that f(x₁) = f(x₂). Then (gf)(x₁) = g(f(x₁)) = g(f(x₂)) = (gf)(x₂). But since g f is injective, this implies that x₁ = x₂. Thus, f is injective.

(b).If gf is surjective, then only g is surjective.We will prove this fact as follow.

Suppose that g f is surjective. Let z C. Then since g f is surjective, there exists x A such that (g f)(x) = g(f(x)) = z. Therefore if we let y = f(x) B, then g(y) = z. Thus, g is surjective.