# Cardinality and Counting

I need help with this math problem:

Show that the set of rational numbers, Q, is countable.

and

Show that the set of irrational numbers is uncountable.

What do you know about cardinality already? Are there any theorems you've seen that might be useful?

well as far as the first one goes I know that the set of integers Z is a subset of Q and that Z is countable. now I make B as the set of all non-integral rational numbers.

if B is countable, then Q is countable because a union of two countable sets is countable.
thats my approach, but i'm not much of an example person.

and for the second one i'm thinking that I can show the union of the reals IR with the irrational, thus making them uncountable as well?

lanedance
Homework Helper
1) try and set up a 1:1 map between Q and Z
2) show that no such map exists for R-Q