Corollary 7.4.6

Claim

The set of irrational numbers is uncountable.

Proof

We will do a proof by contradiction. Assume the set of irrationals is countable. The union between the set of rationals and the set of irrationals is equal to $\mathbb {R}$. By Exercise 7.3.9, we know that the union of two countable sets is also countable. This is our contradiction because $\mathbb {R}$ is uncountable. Showing the set of irrational numbers is uncountable.

$\square$

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Hints

Dana Ernst 03 Dec 2009 14:36

Proceed by contradiction. Recall that $\mathbb{R}$ is equal to the union of the rationals and irrationals. If the irrationals are countable, what does Exercise 7.3.9 tell us?