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Finding the possible numbers in sets

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1

2

$begingroup$

The set S contains some real numbers, according to the following three rules.

(i) $frac{1}{1}$ is in S

(ii) If $frac{a}{b}$ is in S, where $frac{a}{b}$ is written in lowest terms (that is, a and b have highest common factor 1), then $frac{b}{2a}$ is in S.

(iii) If $frac{a}{b}$ and $frac{c}{d}$ are in S, where they are written in lowest terms, then $frac{a+b}{c+d}$ 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?

discrete-mathematics

share|cite|improve this question

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments.
  $endgroup$
  – saulspatz
  Mar 19 at 15:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not able to proceed any further
  $endgroup$
  – Aadhavan Srinivasan
  Mar 19 at 15:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You must have some ideas. Can you list infinitely many numbers in the set? Can you list some real numbers that are definitely not in the set?
  $endgroup$
  – saulspatz
  Mar 19 at 15:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The numbers that are definitely not in the set: 1/3, ¼, 1/5, 1/6, … 2/5, 2/6, 2/7, 2/8,… 3/7, 3/8, 3/9,… 4/9, 4/10, 4/11,… 5/11, 5/12, 5/13…
  $endgroup$
  – Aadhavan Srinivasan
  Mar 19 at 15:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  All the numbers you've listed as definitely not in the set are less than $frac12.$ Are you willing to make a hypothesis? Can you prove it?
  $endgroup$
  – saulspatz
  Mar 19 at 15:38
  

  
  
  
  

 | 
show 2 more comments

1

2

$begingroup$

The set S contains some real numbers, according to the following three rules.

(i) $frac{1}{1}$ is in S

(ii) If $frac{a}{b}$ is in S, where $frac{a}{b}$ is written in lowest terms (that is, a and b have highest common factor 1), then $frac{b}{2a}$ is in S.

(iii) If $frac{a}{b}$ and $frac{c}{d}$ are in S, where they are written in lowest terms, then $frac{a+b}{c+d}$ 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?

discrete-mathematics

share|cite|improve this question

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments.
  $endgroup$
  – saulspatz
  Mar 19 at 15:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not able to proceed any further
  $endgroup$
  – Aadhavan Srinivasan
  Mar 19 at 15:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You must have some ideas. Can you list infinitely many numbers in the set? Can you list some real numbers that are definitely not in the set?
  $endgroup$
  – saulspatz
  Mar 19 at 15:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The numbers that are definitely not in the set: 1/3, ¼, 1/5, 1/6, … 2/5, 2/6, 2/7, 2/8,… 3/7, 3/8, 3/9,… 4/9, 4/10, 4/11,… 5/11, 5/12, 5/13…
  $endgroup$
  – Aadhavan Srinivasan
  Mar 19 at 15:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  All the numbers you've listed as definitely not in the set are less than $frac12.$ Are you willing to make a hypothesis? Can you prove it?
  $endgroup$
  – saulspatz
  Mar 19 at 15:38
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

The set S contains some real numbers, according to the following three rules.

(i) $frac{1}{1}$ is in S

(ii) If $frac{a}{b}$ is in S, where $frac{a}{b}$ is written in lowest terms (that is, a and b have highest common factor 1), then $frac{b}{2a}$ is in S.

(iii) If $frac{a}{b}$ and $frac{c}{d}$ are in S, where they are written in lowest terms, then $frac{a+b}{c+d}$ 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?

discrete-mathematics

share|cite|improve this question

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

$endgroup$

share|cite|improve this question

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

share|cite|improve this question

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

edited Mar 19 at 15:11

Mauro ALLEGRANZA

67.7k449117

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132

asked Mar 19 at 15:09

Aadhavan SrinivasanAadhavan Srinivasan

132