I'm new to this community and I was wondering if this is a valid way to solve the following problem:
I have to prove that the set of all pairs of integers in a cartesian plane are countable. So would it be valid to prove this in the following manner:
- Prove that $\mathbb N$ and $\mathbb Z$ are countable (this should be easily mapped)
- Prove that the Cartesian Product of two countable sets are countable
- Then prove that the countable union of sets are countable
so... ex: In the cartesian plane, all the positive pairs would be $\mathbb N \times\mathbb N$
Also, I was referred to the Cantor pairing function. I read up on the Wikipedia article but I wasn't exactly sure how they applied to my case.