Showing that $\sum_{a\in A} 4^{-a}\neq \sum_{b\in B}4^{-b}$ where $A\neq B \subseteq \mathbb{N}$

Let $A,B\in P(\mathbb{N})$ with $A\neq B$. Note that $A, B$ can be finite or infinite. Show that$$\sum_{a\in A} 4^{-a}\neq \sum_{b\in B}4^{-b}$$

So I'm not entirely sure this claim is true, but I think it is (or might be true with some modification). I think I know how to prove it assuming that $A, B$ has a finite number of different elements using induction. However, in the more general case, I'm stuck. The bigger problem I'm facing is to show that$$f:P(\mathbb{N})\to\mathbb{R}\\A\mapsto\sum_{a\in A}4^{-a}$$is injective, so solutions that tackle this directly are welcome.