Proove that following are countably infinite.?

Question

PLEASE SHOW COMPLETE PROOF
Here, Z is integer, N is Natural, and Q is rational numbers.

1. Prove Z x N is countably infinite.
2. Prove Z x Z is Countably infinite.
3. Prove Q is countably infinite using 1 and 2 (or however you can).


One more thing i am an international student from France at University of Arizona. i am an engineering major, so i don't have to take this class, but because of my advisor i am stuck with this class and i don't know anything. I am looking for a tutor but they are too expensive so if someone knows about this subject and can help me out, i'll appreciate it. email me at [e-mail] if interested. Thank you in advance

NO SENTENCES

USE MATHEMETICAL NOTATIONS

Answer 1

A set is countably infinite if its elements can be enumerated in a sequence (in other words, its elements can be put into one-to-one correspondence with the natural numbers).

2.) ZxZ consists of ordered pairs of integers. They can be represented by grid points in a coordinate system. Drawing one might help you see the pattern and catch the idea of this:
Start the enumeration by (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1), (2,-1), (2,0) etc. (along a spiral). This way, you can enumerate the entire set ZxZ, so it is countably infinite.

1.) ZxN is a subset of ZxZ, which is a countably infinite set, so it is either finite or also countably infinite. However, it can not be finite, since it contains the pairs (0,1), (0,2), (0,3), etc. which correspond to the natural numbers, which are countably infinite.

3.) The members of Q can be represented by fractions p/q, where p and q are integers and relative primes. These could also be represented as pairs (p,q), where p and q are relative prime integers. However, this is a subset of ZxZ, which is countably infinite, and clearly, it cannot be finite, so it is countably infinite, as well.

Answer 2

I have read a little in this field, but can't really do the problems. You have to associate each element of the set with an integer for it to be countable. You might find some help here:
http://en.wikipedia.org/wiki/Denumerable
And check the references also.

Answer 3

i'm still a primary 4 student.