[Math] Is every injection between sets of the same cardinality also a bijection

Question

I have to decide if the following statement is true or not:

For all sets $A$ and $B$ of the same cardinality, if a function $f: A \to B$ is injective, then it is also bijective.

I would say that this statement is true, but in my answer sheet it is written that this statement is False and I really do not know why?

Can you guys give me some hint please? Is it because the statement is false when the sets $A$ and $B$ are empty?

Answer 1

For example, $f : \mathbb{N} \rightarrow \mathbb{N}$ defined by $f(n) = 2n$ is injective but not surjective.