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In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn of the nth number r, E (0, 1) in a purported fixed list of all numbers r E (0, 1) by changing dnn to dn = 1 if dnn + 1 and letting dn = 2 otherwise. Then we form a number r* 0.d,d2 ... and claim that because r* is not in the purported list, therefore |(0, 1)| > |N]. State the function used in the proof clearly and, using the definition of two sets being equal in cardinality |A| = |B|, explain why the proof shows that the cardinality of (0, 1) is strictly bigger than the cardinality of N.

In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn of the nth number r, E (0, 1) in a purported fixed list of all numbers r E (0, 1) by changing dnn to dn = 1 if dnn + 1 and letting dn = 2 otherwise. Then we form a number r* 0.d,d2 ... and claim that because r* is not in the purported list, therefore |(0, 1)| > |N]. State the function used in the proof clearly and, using the definition of two sets being equal in cardinality |A| = |B|, explain why the proof shows that the cardinality of (0, 1) is strictly bigger than the cardinality of N.

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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10. In the proof that |(0, 1)| > |N, we use Cantor's Diagonal Method, where we change the nth digit dnn of the nth number rn E (0, 1) in a purported fixed list of all numbers r e (0, 1) by changing dnn to dm 1 if dnn + 1 and letting dnn 2 otherwise. Then we form a number r* 0.d d, ... and claim that because r* is not in the purported list, therefore |(0, 1)| > |N|. State the function used in the proof clearly and, using the definition of two sets being equal in cardinality |A| = |B|, explain why the proof shows that the cardinality of (0, 1) is strictly bigger than the cardinality of N.
Transcribed Image Text:10. In the proof that |(0, 1)| > |N, we use Cantor's Diagonal Method, where we change the nth digit dnn of the nth number rn E (0, 1) in a purported fixed list of all numbers r e (0, 1) by changing dnn to dm 1 if dnn + 1 and letting dnn 2 otherwise. Then we form a number r* 0.d d, ... and claim that because r* is not in the purported list, therefore |(0, 1)| > |N|. State the function used in the proof clearly and, using the definition of two sets being equal in cardinality |A| = |B|, explain why the proof shows that the cardinality of (0, 1) is strictly bigger than the cardinality of N.
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