You have probably shown:
1) The set Q of rational numbers is countable.
2) The set 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.
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