Real Square Function is not Bijective

Theorem

Let $f: \R \to \R$ be the real square function:

$\forall x \in \R: \map f x = x^2$

Then $f$ is not a bijection.

Proof

From Real Square Function is not Injective, $f$ is not an injection.

From Real Square Function is not Surjective, $f$ is not a surjection.

The result follows by definition of bijection.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $3$