Surjection/Examples/Doubling Function on Reals

Example of Surjection

Let $f: \R \to \R$ be the real function defined as:

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

Then $f$ is a surjection.

Proof

For $f$ to be a surjection, it is necessary that:

$\forall y \in \R: \exists x \in \R: y = 2 x$

This is the case.

Thus $f$ is a surjection by definition.

$\blacksquare$

Sources

• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions