Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For school, I have to prove that every finite subset of $\mathbb N$ is countable. Wikipedia tells me, that "By definition a set $S$ is countable if there exists an injective function $f$ from $S$ to the natural numbers.". I'm probably missing something obvious here, but why is this true?

share|cite|improve this question
That means you can tag (or name, or count) each element in $S$ via an element in $\mathbb{N}$. That's actually the definition of counting. Which part exactly makes you uncomfortable in that sentence? – user13838 Oct 8 '11 at 8:41
@SrivatsanNarayanan Yes, but I didn't know ho to add the N... – fabian789 Oct 8 '11 at 8:47
@percusse mm, so you can say that $s_1$ is Element 1, $s_2$ Element 2? – fabian789 Oct 8 '11 at 8:49
Yes, exactly. If it is a finite element set, you can exhaust its elements by simply assigning elements from $\mathbb{N}$. That's what you are doing when counting anyway. :) Then, of course, there are infinite countable sets (odd numbers set etc.) and uncountable sets. – user13838 Oct 8 '11 at 8:56
If you write your comment as an answer I can accept it... – fabian789 Oct 8 '11 at 9:35
up vote 2 down vote accepted

If it is a set with finite number of elements, you can exhaust its elements by simply assigning elements from $\mathbb{N}$. This is a special case of the definition : A set $S$ is countable if and only if there exists a one-to-one correspondence with a subset of natural numbers.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.