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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.

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Thanks for your edit/update: It seems you want to consider all/only functions that map a finite number of $n\in \mathbb{N}$ to $1$, and maps all other (countably many) $n^{\prime} \in \mathbb{N}$ to $0$? Am I correct? –  amWhy Nov 6 '12 at 15:32
Yep, these are the functions I try to prove there are countably infinite number of. –  Nescio Nov 6 '12 at 15:44
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4 Answers

up vote 4 down vote accepted

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.

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actually we didn't talk at all about of $\mathbb{N}^n$ like that. unless I missed a lot more then I can remember –  Nescio Nov 7 '12 at 9:14
@Nescio: In that case you should read about the pairing function, which shows that $\Bbb N^2$ is countable; you can then use induction to get countability of $\Bbb N^n$ for $n>2$. –  Brian M. Scott Nov 7 '12 at 9:20
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Hint: Any natural number can be written in binary. Prepend an infinite sequence of zeros the the front of such a binary representation, and then reverse it.

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Any such function is uniquely determined by $f^{-1}(\{1\})$, you can construct a bijection between the set of such functions and the set of all finite subsets of $\mathbb N$.

Thus, we need to prove that the set of all finite subsets of $\mathbb N$ is countable..

Hint Can you prove that for every $n$, the set of subsets of $\mathbb N$ with $n$ elements is countable?

Can you get the problem from here?

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Hint: first try to prove that the set of all functions for which $|f^{-1}(\{1\})|=1$ is countable. Then the same for the set of all functions for which $|f^{-1}(\{1\})|=2$. Then generalize.

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I just touched up typos in the formatting...Check to make sure I got what you intended! –  amWhy Nov 6 '12 at 15:06
Yes, this is it. Thanks! –  Dan Shved Nov 6 '12 at 15:09
I tried looking at the private case of $|f^{-1}(\{1\})|=1$, But I'm not exactly sure how to write it. I understand that exists a function $g$ that maps every $n\in\mathbb{N}$ to the function for which $f(n)=1$, but how do I write it formally? –  Nescio Nov 7 '12 at 9:09
Hmm, I don't think that you really need to write it any more formally than you just did... I guess you could use indicator functions, so then you can say that $g \colon n \to \mathbf{1}_{\{n\}}$ is a bijection from $\mathbb{N}$ to the set of all functions for which $|f^{-1}(\{1\})|=1$. But again, in my opinion, what you already said in your comment seems like enough to me. –  Dan Shved Nov 7 '12 at 9:25
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