Functions between subsets of a set over which $f$ is a bijection

Question

Is it true that for a bijective map $f:X \rightarrow X$, that for any non-empty subset $A$ of $X$ we have that $f:A \rightarrow A$ is also a bijection?

I assume the answer in general is no, in which case does there exist a statement along similar lines to the one written above?

Answer 1

$f$ need not map $A$ into $A$: Example: $X=\{1,2\}, f(1)=2,f(2)=1$ and $A=\{1\}$. If you know that $f(A)=A$ then $f$ restricted to $A$ is a bijection on $A$ because it is automatically an injection.