How do I prove the set Z x Z is countable?

Question

Can anybody help me with this problem?

Thanks

I'm thinking that say for A1 = a1,1 a1,2 a1,3 and so one, A1 is a infinite sequences with elements (1, Z), then A2 is a sequence with elements (2, Z). So Ai for i is an element of Z, I have an infinite collection of infinitely countable set and by a corollary stated in my text book, the collection of Ai is countable.

I'm not sure if you can understand me. But if you could, can you tell me if this proof is good enough?

Answer 1

Nice thinking! Indeed, you can "decompose" the cartesian product Z x Z in disjoint sets:
An: = { (n, z) ; where z runs over Z} , for each and every n in Z.

Clearly, every An is equipolent (=has the same cardinality) as Z and is, thus, countable: just define (n,z) --> z from An to Z. This is clearly a bijection.

Now, Z x Z = UNION (An) , and this is a countable union because it is over n in Z, and Z is countable.
Now use that rather known theorem in your book that says thata countable union of countable sets yields a countable set and we're wrapped up...way to go!

Regards
Tonio

Answer 2

yes, a countable union of countable sets is countable itself.
you're right
but not any infinite union of countable sets is countable
you see the difference?

Answer 3

z (the set of integers) is countable. So, z*z is again z only. Thus, it is proved.

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