(a) Prove that $N \times N$ is a countable set

(b) Let T be the set of two element subsets of N. Prove that T is countable.

This is a question in my exam review package. I missed the lesson on countable sets and such, proving them seems simple enough (I have what seems to be a decent solution package provided to me by peers), however this question is just all over the place in terms of an explanation.

I'm just wondering if someone could guide me through the explanation of the solution.

Homework question.

  • 2
    $\begingroup$ (1) Don't treat the title as part of the question body. Make the question body self contained. (2) Search the site before posting questions that have been asked more than twice or thrice. Show some effort, that's the least you can do to respect those people that put their effort writing the answers. $\endgroup$ – Asaf Karagila May 6 '14 at 22:23
  • $\begingroup$ math.stackexchange.com/questions/linked/71850 contains no less than 15 links within this site discussing these sort of questions. There are probably more. And I'm not even getting to the second question, which can be deduced from the first using a myriad of other questions and answers on this site; or directly if you prefer. $\endgroup$ – Asaf Karagila May 6 '14 at 22:28

Here's a simple proof. Define $f : N \times N \rightarrow N$ by

$$f(a,b) = 2^a * 3^b$$

By the Fundamental Theorem of Arithmetic, f is an injection. Then if you believe that a set that injects into a countable set must be countable, you're done. Of course if you don't believe that, you have to prove it :-)

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