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3. (a) Given a family {X; | i E N} of sets where each X; is countable, what is the cardinality of UX: iEN (you may quote a result from class to justify your answer). (b) Let X be the set of all subsets of N of cardinality at most two, that is, X%3 {{п, m} | п,т€ N}. Prove that X is countable. (Hint: Try to use (a) - express X as a union of sets. Consider all subsets of N of car- dinality at most two containing 0, these are {0}, {0, 1}, {0, 2}, {0, 3}, etc. Now consider all subsets of size 2 containing 1, these are {0, 1}, {1}, {1,2}, {1,3}, etc. )

3. (a) Given a family {X; | i E N} of sets where each X; is countable, what is the cardinality of UX: iEN (you may quote a result from class to justify your answer). (b) Let X be the set of all subsets of N of cardinality at most two, that is, X%3 {{п, m} | п,т€ N}. Prove that X is countable. (Hint: Try to use (a) - express X as a union of sets. Consider all subsets of N of car- dinality at most two containing 0, these are {0}, {0, 1}, {0, 2}, {0, 3}, etc. Now consider all subsets of size 2 containing 1, these are {0, 1}, {1}, {1,2}, {1,3}, etc. )

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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3. (a) Given a family {X; | i E N} of sets where each X; is countable, what is the cardinality of UX: iEN (you may quote a result from class to justify your answer). (b) Let X be the set of all subsets of N of cardinality at most two, that is, X — {{п, т} | п,т E N}. Prove that X is countable. (Hint: Try to use (a) - express X as a union of sets. Consider all subsets of N of car- dinality at most two containing 0, these are {0}, {0, 1}, {0, 2}, {0, 3}, etc. Now consider all subsets of size 2 containing 1, these are {0,1}, {1}, {1,2}, {1,3}, etc. )
Transcribed Image Text:3. (a) Given a family {X; | i E N} of sets where each X; is countable, what is the cardinality of UX: iEN (you may quote a result from class to justify your answer). (b) Let X be the set of all subsets of N of cardinality at most two, that is, X — {{п, т} | п,т E N}. Prove that X is countable. (Hint: Try to use (a) - express X as a union of sets. Consider all subsets of N of car- dinality at most two containing 0, these are {0}, {0, 1}, {0, 2}, {0, 3}, etc. Now consider all subsets of size 2 containing 1, these are {0,1}, {1}, {1,2}, {1,3}, etc. )
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