Surjection/Examples/Real Square Function to Non-Negative Reals
Jump to navigation
Jump to search
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.
Proof
Let $y \in \R_{\ge 0}$.
Let $x = +\sqrt y$.
From Existence of Square Roots of Positive Real Number, there exists such a $y$.
Then:
- $x^2 = y$
That is:
- $y = \map f x$
Hence:
- $\forall y \in \R_{\ge 0}: \exists x \in \R: y = \map f x$
and the result follows by definition of surjection.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions