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Math Help - 2 ordered field theorems that need proving

  1. #1
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    2 ordered field theorems that need proving

    My textbook has kindly decided to present twotheorems without proof, meanwhile all others have proofs. I don't want to come up with a flawed proof, so I need some help.

    The first of the two involved proving that 0 =/= 1
    I did this one, with the help of mhf.

    The remaining theorem will probably incorporate the latter into its proof...

    Def. a < b <=> a<=b and a =/=b

    Thm. For any ordered field, 0 < 1.
    Proof:

    I'm not really sure where to start, contradiction seems like it's the way to go...

    Suppose that for some ordered field, 1 <= 0
    0 =/= 1, so 1 < 0
    by another theorem that i've proved already, (-0) < (-1)
    but (-0) = 0 < (-1)
    and... i feel like i'm going in circles, please help


    Q.E.D.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Noxide View Post
    My textbook has kindly decided to present twotheorems without proof, meanwhile all others have proofs. I don't want to come up with a flawed proof, so I need some help.

    The first of the two involved proving that 0 =/= 1
    I did this one, with the help of mhf.

    The remaining theorem will probably incorporate the latter into its proof...

    Def. a < b <=> a<=b and a =/=b

    Thm. For any ordered field, 0 < 1.
    Proof:

    I'm not really sure where to start, contradiction seems like it's the way to go...

    Suppose that for some ordered field, 1 <= 0
    0 =/= 1, so 1 < 0
    by another theorem that i've proved already, (-0) < (-1)
    but (-0) = 0 < (-1)
    If a< b and 0< c then ac< bc.
    Use that with a= 0, b= -1 and c= -1.

    and... i feel like i'm going in circles, please help


    Q.E.D.
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  3. #3
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    Joined
    Sep 2009
    Posts
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    Thm. For any ordered field, 0 < 1.
    Proof:

    Suppose that for some ordered field, 1 <= 0
    (-0) <= (-1)
    (-0) = 0 <= (-1)
    Using a<= b and 0 <= c implies ac <= bc, 0 * (-1) = 0 <= 0 = 0*(-1)
    but 0=/= 1, so 0 < 0
    but 0 = 0
    Q.E.D.

    Thanks
    Last edited by Noxide; January 15th 2011 at 06:48 PM.
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