Injective functions also surjective? Is it true that for each set $M$ a given injective function $f: M \rightarrow M$ is surjective, too?
Can someone explain why it is true or not and give an example?

For finite sets, consider the two point set $\{a,b\}$ .  If you have an injective function, $f(a)\neq f(b)$, so one has to be $a$ and one has to be $b$, so the function is surjective.  The same idea works for sets of any finite size.  If the size is $n$ and it is injective, then $n$ distinct elements are in the range, which is all of $M$, so it is surjective.
