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 $2.5$
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