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Thread: Help me find Numbers in S, please!!!

  1. #1
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    poland
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    Help me find Numbers in S, please!!!

    The set S contains some real numbers, according to the following three rules.
    1) 1/1 is in S.
    2) If a/b is in S, where a/b is written in lowest terms (that is, a and b have highest common factor 1), then b/2a is in S.

    3) If a/b and c/d are in S, where they are written in lowest terms, then a+c/b+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?
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  2. #2
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    Re: Help me find Numbers in S, please!!!

    Start playing with it.
    You have 1/1 in S. It is in lowest terms (the greatest common factor of 1 and 1 is 1). So, by (2), you have 1/2 is in S.
    So, you know 1/1 and 1/2 are in S. Both are in lowest terms. Apply (2) to 1/2 and you get 2/2 = 1/1, so that does not add any new numbers.
    Apply (3) to 1/1 and 1/2 to get 2/3. Now you have 3 numbers. 1/1, 1/2, and 2/3.
    Applying (2) to 2/3 gives 3/4, as does applying (3) to 1/1 and 2/3. So, 3/4 is in S. Applying (3) to 1/2 and 2/3 gives 3/5, which is in S.

    So, now you have 1/1, 1/2, 2/3, 3/4, 3/5.
    Let's look for patterns. Applying (3) on 1/1 and 3/4 gives 4/5. Applying (3) on 1/1 and 4/5 gives 5/6. Now, we can use an induction argument to show that this will always be satisfied.
    For any positive integer n, we know the GCD(n,n+1) = 1 (you can prove this yourself if you want, but I am not going to bother with the proof). Therefore, 1/1 and n/(n+1) in S for some positive integer n implies (n+1)/(n+2) is in S by (3). Therefore by the principle of mathematical induction, n/(n+1) is in S for all positive integers n.
    So, now you have 1/1, n/(n+1), and 3/5. Since n/(n+1) is always in lowest terms, we can take any two of them and apply (3):
    a/(a+1) and b/(b+1) are both in S implies (a+b)/(a+b+2) is in S. a+b can be any positive integer no less than 2. So, you have n/(n+2) where $n \ge 2$.
    Now, you have 1/1, n/(n+1) where $n \ge 1$, n/(n+2) where $n \ge 2$.
    Continuing these patterns, you should find that n/(n+k) is in S where $n\ge k$.
    Next, apply (2) and see what you get. Then apply (3), then (2). Keep doing that until you do not uncover any new information. Be careful to make sure you are only applying it to numbers in lowest terms.
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  3. #3
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    Re: Help me find Numbers in S, please!!!

    Thank you very much for your help:-)
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