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Is this well-defined in the rational numbers

  1. Jul 31, 2005 #1
    Need help with proving:

    Show that (a,b) + (c,d) = (a+c, b+d) is not well-defined in the rational numbers.
    [Note: (a,b) + (c,d) = (ad+bc, bd) is well-defined because (a,b) is related to (c,d) when ad = bc.)]
     
  2. jcsd
  3. Jul 31, 2005 #2

    Hurkyl

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    Have you tried at all? Do you know what it means to fail to be well-defined?
     
  4. Jul 31, 2005 #3
    I know what it means. But is it enough with just stating an example?
     
  5. Jul 31, 2005 #4

    HallsofIvy

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    When proving a general statement is not true, yes, it is sufficient to give a "counter-example": one case showing for which the statement is not true, thus proving it is not always true.

    What you need to do is find an example of rational numbers (a,b),(a',b'),(c,d),(c',d') such that (a,b) and (a', b') are equivalent, (c,d) and (c',d') are equivalent, but (a+c,b+d) is not equivalent to (a'+c', b'+d').

    (For those who are not clear on this, this is using a method of defining rational number from the integers by saying that two ordered pairs of integers, (a,b) and (a',b') (with second integer non-zero) are equivalent if and only if ab'= a'b. That is an equivalence relation and so partitions the set of all such pairs into equivalence classes. A rational number is such an equivalence class.)
     
  6. Jul 31, 2005 #5
    That's just what I did, I wasn't just sure if that was enough.. :bugeye:
     
  7. Jul 31, 2005 #6

    HallsofIvy

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    Out of curiosity then, what was your counter-example?
     
  8. Jul 31, 2005 #7
    (1,2) ~ (1,2) and (1,3) ~ (2,6).
     
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