# Prove the set of irrational numbers is uncountable.

1. Sep 24, 2009

### gotmilk04

1. The problem statement, all variables and given/known data
Prove the set of irrational numbers is uncountable.

2. Relevant equations

3. The attempt at a solution
We proved that the set [0,1] is uncountable, but I'm not sure how to do it for the irrational numbers.

2. Sep 24, 2009

### rasmhop

You have probably shown:
1) The set $\mathbb{Q}$ of rational numbers is countable.
2) The set $\mathbb{R}$ of real numbers is uncountable.
3) The union of two countable sets is countable.
Now if both the set of rational numbers and the set of irrational numbers were countable would you be able to get a contradiction using fact 2 and 3? You should be able to use this contradiction to show that the set of irrational numbers must be uncountable.