Real Square Function is not Surjective
(Redirected from Surjection/Examples/Non-Surjection/Square Function)
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Example of Mapping which is Not a Surjection
Let $f: \R \to \R$ be the real square function:
- $\forall x \in \R: \map f x = x^2$
Then $f$ is not a surjection.
Proof
For $f$ to be a surjection, it would be necessary that:
- $\forall y \in \R: \exists x \in \R: \map f x = y$
However from Square of Real Number is Non-Negative:
- $\forall y \in \R_{< 0}: \nexists x \in \R: \map f x = y$
Hence $f$ is not a surjection.
$\blacksquare$