Set of rational numbers is countable.
We are unable to prove that the set of irrational numbers is countable. And so, we conclude that it is uncountable.
The set of real numbers is union of the above two, of which one is uncountable. Therefore, set of real numbers is uncountable.
If we are able to define a function(one-one) whose domain is the set of positive integers and range is set S then, set S is countable. It means that every element of set S can be enumerated and thus the elements in the set can be exhausted sooner or later in counting.
Suppose we write the set of positive rational numbers along with zero as a set as below:
{0/1 0/2 0/3 0/4 0/5 0/6 0/7 ................
1/1 1/2 1/3 1/4 1/5 1/6 1/7 ................
2/1 2/2 2/3 2/4 2/5 2/6 2/7 ................
3/1 3/2 3/3 3/4 3/5 3/6 3/7 ................
..........................................................................
..........................................................................
..........................................................................}
(See how these numbers exhaust all numbers of the form p/q.)
These elements can be mapped on to the set of positive integers and written down in the form of a sequence as below.
0/1, 1/1, 0/2, 2/1, 1/2, 0/3,3/1, 2/2, 1/3, 0/4,.................
(Look at the set and see how the terms form a pattern)
Thus, set of positive rational numbers along with zero is countable. Similarly, set of negative rational numbers is also countable. Their union is also countable.
And hence set of rational numbers is countable.