Q:

Proving a set of functions from $\mathbb{N}$ to $\{1,0\}$ is countable

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.

A:

You shouldn’t try to find a bijection: that gets very messy very quickly. You should look for a more indirect argument. 
Here’s one possible approach. There’s an easy bijection between your set of functions and the set of finite subsets of $\Bbb N$: pair a function $f$ with $\{k\in\Bbb N:f(k)=1\}$. For each $n\in\Bbb N$ let $[\Bbb N]^n$ be the set of subsets of $\Bbb N$ having exactly $n$ elements. If you can show that each $[\Bbb N]^n$ is countable, you can then use the fact that the union of a countable family of countable sets is countable.
To show that $[\Bbb N]^n$ is countable, find an injection (one-to-one mapping) of $[\Bbb N]^n$ into $\Bbb N^n$, the set of $n$-tuples of natural numbers. (I’m assuming here that you’ve already shown that $\Bbb N^n$ is countable.) HINT: You can list any set of natural numbers in increasing order.

