Exponential on Real Numbers is Injection
Jump to navigation
Jump to search
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 $2.5$