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Another Proof by Induction Question

  1. Apr 14, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove -(-x) = x.

    2. Relevant equations

    A2: x + y = y + x [additive commutativity]
    A5: x + (-x) = 0
    M3: x(yz) = (xy)z [multiplicative associativity]
    M4: x (1) = x
    Lemma: (-1)(-1) = 1
    Theorem c: (-1)x = -x


    3. The attempt at a solution

    -(-x) = (-1)[(-1)x] by Theorem c

    = [(-1)(-1)]x by M3

    = (1)x by lemma

    = x by M4

    Q.E.D.



    This is what my approach was; however, the solution in the back of the book was something like this:

    From A5 we have x + (-x) = 0. Then (-x) + x = 0 by A2. Hence x = - (-x) by the uniqueness of -(-x) in A5.

    Q.E.D.


    Is my approach a viable proof? I thought I was starting to understand this, but when I see the solution it completely throws me off. Additionally, the proof given by the book makes no sense to me. I have no intuition for this.
     
  2. jcsd
  3. Apr 14, 2013 #2
    All they are saying is that since (-x) + x = 0, subtract (-x) from both sides of the equation and the result is immediate.

    This is not a proof by induction, by the way.
     
  4. Apr 14, 2013 #3

    Fredrik

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    Yes, your approach is fine (if you have proved the lemma). It has nothing to do with induction though.
     
  5. Apr 14, 2013 #4

    tiny-tim

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    hi mliuzzolino! :smile:
    your proof is valid …

    but it's far too complicated :redface:

    in maths theorems, you get more marks for simpler proofs

    you've used two theorems to prove it, but it could have been proved using no theorems, and two axioms
    hint: what is the definition of -(-x) ? :wink:
     
  6. Apr 14, 2013 #5

    Fredrik

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    I think it's even more immediate than that. They're saying that by definition of "additive inverse" (see below), we can immediately conclude that the additive inverse of -x (which is denoted by -(-x)) is x.

    Definition of additive inverse: For each real number y there's a real number z such that y+z=0. This z is said to be the additive inverse of y, and is denoted by -y.
     
  7. Apr 14, 2013 #6
    Ah, my apologies. I don't know why I put induction in the title. My brain is going haywire - exam is tomorrow and this material is very hard won for me. Thanks for the clarification everyone!
     
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