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Proof that there exist such an element in Q

  1. Jan 11, 2009 #1


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    1. The problem statement, all variables and given/known data
    I'm asked to say whether the set Q,+,. is a field.
    To be a field it must respect 8 conditions. And one of them is that there exists a unique element -x in Q such that x+(-x)=0 for all x in Q. I realize I have to prove the existence of -x but also its uniqueness. For the existence I don't know how could I approach it. For the uniqueness I'm sure that I could do it by absurd. That is by suposing that there exist more than one element -x that satisfies the same property and fall into a contradiction.
    Can you get me started or help to get started for showing the existence?
    Thank you!
  2. jcsd
  3. Jan 11, 2009 #2


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    Hi fluidistic! :smile:

    (good sig! :biggrin:)

    If Q is the rationals, then define the inverse of p/q as -p/q. :wink:
  4. Jan 11, 2009 #3
    assume there is an element -x such that x + (-x) = 0

    now assume there is an element y such that x + y =0
    This implies y=-x therefore we conclude that

    -x is the unique element
  5. Jan 11, 2009 #4


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    Ok thank you both, I think I got it. Wasn't that hard it seems!
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