Real Square Function is neither Injective nor Surjective

Example of Mapping which is Not an Injection nor a Surjection

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

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

Then $f$ is neither an injection nor a surjection.

Proof

Consider the real square function:

From Real Square Function is not Injective:

$f$ is not an injection.

From Real Square Function is not Surjective:

$f$ is not a surjection.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $6$