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In this proof can I show independently that positive rationals and negative rationals are countable by using Cantor's zigzag and then say that the union of two countable sets is countable so therefore

positive and negative rationals are countable.

Is this correct ?

user602774
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1 Answers1

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You said "irrationals" in your last sentence by mistake - but "rationals" in the body of the problem. Yes, the argument is correct. Here's an even more direct method (without splitting into two cases)

Order everything in the usual positive matrix, and put the negative number immediately after the corresponding positive number:

$ \\ \frac{1}{1}, -\frac{1}{1}, \frac{1}{2}, -\frac{1}{2}, \ldots \\ \frac{2}{1}, -\frac{2}{1}, \frac{2}{2}, -\frac{2}{2}, \ldots \\ \vdots \\ \frac{n}{1}, -\frac{n}{1}, \frac{n}{2}, -\frac{n}{2}, \ldots \\ \vdots $

or use your favourite matrix (this one is missing $0$, by the way).

GLG
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