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5. The set S contains some real numbers, according to the following three rules. (i) is in S. (ii) If is in S, where is written in lowest terms (that is, a and b have highest common factor 1), then is in S. (iii) If and are in S, where they are written in lowest terms, then is in S. These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S? For example, by (i), is in S. By (ii), sinceisin S, is in S. Since both andare in S, (iii) tells usis in S.

5. The set S contains some real numbers, according to the following three rules. (i) is in S. (ii) If is in S, where is written in lowest terms (that is, a and b have highest common factor 1), then is in S. (iii) If and are in S, where they are written in lowest terms, then is in S. These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S? For example, by (i), is in S. By (ii), sinceisin S, is in S. Since both andare in S, (iii) tells usis in S.

Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.1: Basic Assumptions
Problem 40WE
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Claim: Set S is contained in interval [½, 1] for a/b where 0<a≤b≤2a

Hint: Farey sequences (please do some research)

Unfortunately, I have only found this through observation, and I am unsure how to prove this, I need a rigorous proof

DO NOT COPY AND PASTE ANOTHER ANSWER FROM THIS WEBSITE (I've seen it 3 times, and it is not complete)

 

5. The set 5 contains some real numbers, according to the following three rules. (i) is in S. (ii) If is in S, where is written in lowest terms (that is, a and b have highest common factor 1), then is in S. (iii) If and are in S, where they are written in lowest terms, then is in 5. These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S? For example, by (i), is in S. By (ii), since is in 5, is in S. Since both andare in 5, (iii) tells us is in S.
Transcribed Image Text:5. The set 5 contains some real numbers, according to the following three rules. (i) is in S. (ii) If is in S, where is written in lowest terms (that is, a and b have highest common factor 1), then is in S. (iii) If and are in S, where they are written in lowest terms, then is in 5. These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S? For example, by (i), is in S. By (ii), since is in 5, is in S. Since both andare in 5, (iii) tells us is in S.
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