# Exponential on Real Numbers is Injection

## Theorem

Let $\exp: \R \to \R$ be the exponential function:

- $\map \exp x = e^x$

Then $\exp$ is an injection.

## Proof

From Exponential is Strictly Increasing:

- $\exp$ is strictly increasing on $\R$.

From Strictly Monotone Mapping with Totally Ordered Domain is Injective:

- $\exp$ is an injection.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $48$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.5$