Subset and bijection implies cardinal equality?

Question

I hope I have not hereby created a duplicate, please perdon me if I did, but I had this question for a while now:

Let $A \& B $ be two sets such that $A \subseteq B$. Suppose there exist a one to one (bijective) function $f : A \to B $. Then have we got $|A| = |B|$?

I know that if these sets are finite, it works, but what about the infinite case

Thank you

T. D

Answer 1

Let $A = 2\mathbb{Z}$ (ie the even integers) and $B = \mathbb{Z}$. Then, $A \subseteq B$, and there is a bijection between them, (namely $f:B \to A$ defined by $f(x) = 2x$) but the sets are not equal