Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.
Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.1: Sets
Problem 26E
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Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.
Expert Solution
Step 1
First of all we have to prove that,
The union of countable family of countable sets is countable.
PROOF:
Without loss of generality, we can denote a countable family of sets by
Suppose is an enumeration for . Then,
To the element we assign a natural number so that there corresponds at most distinct elements of A.
Therefore, A is countable by Countable Lemma.
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