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# Show that the set of irrational numbers is an uncountable set.

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To proof: Set of irrational numbers $\bold{R}-\bold{Q}$ is uncountable.

$\textbf{PROOF BY CONTRADICTION}$

The set of real numbers $\bold{R}$ is uncountable.

The set of rational numbers $\bold{Q}$ is uncountable.

The set of irrational numbers $\bold{R}-\bold{Q}$ is the set of real numbers without all rational numbers. The set of real numbers is then the union of the irrational and rational numbers:

$\bold{R}=(\bold{R}-\bold{Q})\cup \bold{Q}$

Let us assume that $\bold{R}-\bold{Q}$ is countable. Since the union is two countable sets is also countable, we then obtain that $\bold{R}$ is countable. However $\bold{R}$ is known to be uncountable and thus we obtained a contradiction. Thus are assumption that $\bold{R}-\bold{Q}$ is countable is incorrect and thus $\bold{R}-\bold{Q}$ is uncountable.

$\square$

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