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1.90 Show that the set I of all irrational numbers must be uncountable. (Hint: Use Exercise 1.87.)

1.90 Show that the set I of all irrational numbers must be uncountable. (Hint: Use Exercise 1.87.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 64RE
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1.90 Show that the set I of all irrational numbers must be uncountable. (Hint: Use Exercise 1.87.) For Reference 1.87 t a) If A and B are each countable sets, show that A UB is countable. (Hint: For each set, consider a sequence of all elements, and show how to splice the sequences together to make one sequence. Remember that the sets need not be disjoint.) b) Prove that the union of countably many finite sets is either countable or finite.
Transcribed Image Text:1.90 Show that the set I of all irrational numbers must be uncountable. (Hint: Use Exercise 1.87.) For Reference 1.87 t a) If A and B are each countable sets, show that A UB is countable. (Hint: For each set, consider a sequence of all elements, and show how to splice the sequences together to make one sequence. Remember that the sets need not be disjoint.) b) Prove that the union of countably many finite sets is either countable or finite.
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