Is $f$ surjective function?

Question

Let $f:S \to S$ be a function which is injective. Assume that $S$ is finite. Must $f$ be surjective?

Since $f$ is injective then any single point of $S$ is mapped by only a single point of $S$ . Again , since $S$ is bounded then no other condition holds.Then $S$ can be surjective......am I right?

Answer 1

If by bounded you mean finite, with your hypothesis, $f$ is a permutation of elements of $S$ and it is therefore surjective.

Answer 2

Too simple, a bug?

$S:=(0,1)$ is bounded .

$f: S\rightarrow S , f(x)=(1/2)x$,

$f$ is injective, but not surjective.