Use the result of exercise 29 to prove that the set of all irrational numbers is uncountable.
The set of all irrational numbers is uncountable using that, “union of any two countably infinite sets is countably infinite”.
The set of rational numbers is infinitely countable and the set of all real numbers is uncountable.
Union of any two countably infinite sets is countably infinite.
Now, let us assume that the set of irrational numbers is also countable.
Now, from the concept of number, we can say mat the set of rational and the set of irrational numbers together form the set of real numbers.
Thus, we can say that real numbers is the union of rational numbers and the irrational numbers
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