Nines. Would Cantor's argument work if we used 2 and 9 instead of the digits? That is, for each digit we ask whether the digit is a 2. If make the analogous digit of Ma 9, and otherwise we make the digit this method, we are not guaranteed that the number M we construc our list. Provide a scenario in which the constructed number is on the Remember from Chapter 2 that 0.1999999. .. = 0.2. The number 9 is %3D
Nines. Would Cantor's argument work if we used 2 and 9 instead of the digits? That is, for each digit we ask whether the digit is a 2. If make the analogous digit of Ma 9, and otherwise we make the digit this method, we are not guaranteed that the number M we construc our list. Provide a scenario in which the constructed number is on the Remember from Chapter 2 that 0.1999999. .. = 0.2. The number 9 is %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Question 21
Transcribed Image Text:Ic, a Nó Vacancy sign (or actually a Not Enough
Uign) might actually be necessary. Explain why it is not possible to
give each person from your group a room.
20. Just guess. This is just a "guessing question" Do you think there are sets
whose cardinality is actually larger than that of the set of real numbers? Or,
do you think the infinity of reals is the largest infinity? Just make a guess and
informally explain it.
Further Challenges
21. Nines. Would Cantor's argument work if we used 2 and 9 instead of 2 and 4 as
the digits? That is, for each digit we ask whether the digit is a 2. If it is 2, we
make the analogous digit of M a 9, and otherwise we make the digit a 2. Using
this method, we are not guaranteed that the number M we construct is not on
our list. Provide a scenario in which the constructed number is on the list! (Hint:
Remember from Chapter 2 that 0.1999999... = 0.2. The number 9 is key here.)
22. Missing irrational. Could you modify the diagonalization procedure so that
the missing real you produce is a rational number? How could you modify
the diagonalization argument so that the missing real number you produce is
an irrational number? (Hint: Using the construction in this section, each digit
of M is a 2 or a 4. Modify the construction so the 10th place, the 100th place,
the 1000th place, and so on, are either a 3 or a 5, whereas the rest are 2's and
4's. Why will such a number be irrational?)
23. Personal perspectives. Write a short essay describing the most interesting or
surprising discovery you made in exploring the material in this section. If any
In Your Own Words
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