Real Sine Function is neither Injective nor Surjective
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Theorem
The real sine function is neither an injection nor a surjection.
Proof
This is immediately apparent from the graph of the sine function:
For example:
- $\map \sin 0 = \map \sin \pi = 0$
and so the real sine function is not an injection.
Then, for example:
- $\nexists x \in \R: \map \sin x = 2$
and so the real sine function is not a surjection.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.5$