Surjection/Examples/Real Square Function to Non-Negative Reals

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Example of Surjection

Let $f: \R \to \R_{\ge 0}$ be the real square function whose codomain is the set of non-negative reals:

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

Then $f$ is a surjection.


Let $y \in \R_{\ge 0}$.

Let $x = +\sqrt y$.

From Existence of Square Roots of Positive Real Number, there exists such a $y$.


$x^2 = y$

That is:

$y = \map f x$


$\forall y \in \R_{\ge 0}: \exists x \in \R: y = \map f x$

and the result follows by definition of surjection.