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Consider the set of real numbers between 0 and 1 having a decimal expansion that can be written using only the digits 2, 3, and 5. True or false: this set is countable. True False

Consider the set of real numbers between 0 and 1 having a decimal expansion that can be written using only the digits 2, 3, and 5. True or false: this set is countable. True False

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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Consider the set of real numbers between 0 and 1 having a decimal expansion that can be written using only the digits 2, 3, and 5. True or false: this set is countable. True False
Transcribed Image Text:Consider the set of real numbers between 0 and 1 having a decimal expansion that can be written using only the digits 2, 3, and 5. True or false: this set is countable. True False
Expert Solution
Step 1

If possible, let the set of numbers between 0 and 1 with decimal expansion that can be written using only the digits 2, 3 and 5 be denoted by S and let this set be countable. 

Then, the set S can be written as

     S = {x1 , x2 , x3 , . . . . . . . . . . . } 

  where for any i,

        xi = 0.xi1xi2xi3. . . . . . . . . 

              with each xij = 2, 3, or 5. 

 

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