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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Use the result of exercise 29 to prove that the set of all irrational numbers is uncountable.

To determine

To prove:

The set of all irrational numbers is uncountable using that, “union of any two countably infinite sets is countably infinite”.

Explanation

Given information:

The set of rational numbers is infinitely countable and the set of all real numbers (R) is uncountable.

Concept used:

Union of any two countably infinite sets is countably infinite.

Proof:

Now, let us assume that the set of irrational numbers (R/Q) 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 (R) is the union of rational numbers (Q) and the irrational numbers (R/Q)

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