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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.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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