Real Sine Function is neither Injective nor Surjective

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 $2.5$