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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 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), sinceis in S, is in S. Since both andare in S, (iii) tells us is in S.

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 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), sinceis in S, is in S. Since both andare in S, (iii) tells us is in S.

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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Your answer MUST have a RIGOROUS proof (not just explanation, some math that THOROUGHLY demonstrates ALL cases work) AND Farey Sequences

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), since is in 5, is in S. Since both andare in S, (iii) tells usis in S.
Transcribed Image Text: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), since is in 5, is in S. Since both andare in S, (iii) tells usis in S.
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