The question is to prove the set $S$ of all the functions $f:\mathbb{N}\to \{0,1\}$, for which $f^{-1}(\{1\})$ is finite, is countable.
After considering this for a while I do understand what it means, but I have no idea how to solve it. How do I even make a function from neutral numbers to functions and how can I prove such function is bijective?
I'm not even sure how to start this, so I'll be happy with any push in the right direction.
edit: Thanks you guys for your answers. from them I realized I'm missing something since I didn't understand half of what you said. Though I'm a bit surprised since I only missed a 1-hour lecture once and I don't remember discussing most of what written here. The exercise itself is due in a bit less then a week. I'll go study for a bit and come back to this soon. Will leave the question open in the meantime.
Edit 2: OK, after asking around for a bit and reading some stuff, and then sitting for 15 minutes just thining about all the pieces, I think I finally understand this. I haven't written the proof yet, but I feel like I know how to do this, so I'll be closing the question. Thanks again everyone.