Is set of points (p,q) countable?

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I want to show that the set of points (p,q) on the plane with rational coordinates p and q is countable. I proved set of rational numbers is countable by drawing table and I find (http://web01.shu.edu/projects/reals/infinity/proofs/combctbl.html ) Combining Countable Sets. However, I cannot put these together in table.
Could it be proven by matematically instead of table drawing?
 
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If you know how to prove that there are only countably many rationals, the proof is exactly the same.
 
For that matter, since you have proven that there are only countably many rationals, any subset of the rationals is either finite or countable.
 
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