# Prove the set of irrational numbers is uncountable.

## Homework Statement

Prove the set of irrational numbers is uncountable.

## 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.

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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.