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Chapter 2: Q32SE (page 187)

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

Short Answer

Expert verified

Set of irrational numbers is uncountable.

Step by step solution

01

Step: 1

We know that rational number is countable and set of all real numbers is uncountable.

02

Step: 2

R is union of rational and irrational number and now if irrational number is countable then as rational number is countable so union of two countable set is also countable. But union of rational numbers and irrational numbers is R which is clearly uncountable so a contradiction.

Therefore set of irrational numbers is uncountable.

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