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Math Help - Transitivity and Ordered Fields. How do you prove this? Please help.

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    Transitivity and Ordered Fields. How do you prove this? Please help.

    In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
    prove that only one of the following can be true:
    x<y,x=y,y<x
    Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
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    Quote Originally Posted by amm345 View Post
    In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
    prove that only one of the following can be true:
    x<y,x=y,y<x
    Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
    there are a couple of ways to define an ordered field. which one are you using?
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    To define a total order S.
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    Quote Originally Posted by amm345 View Post
    In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
    prove that only one of the following can be true:
    x<y,x=y,y<x
    Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
    The set P satisfied that given a\in S we exactly one of following: a\in P \text{ or }-a\not \in P\text{ or } a=0. Also given a,b\in P we have a+b\in P \text{ and }ab\in P. Given, x,y\in S consider x-y. If x-y=0 then we are done. Otherwise x-y\in P xor -(x-y)\in P. Thus, x-y\in P xor y-x\in P i.e. x<y xor y<x.

    To prove the other statement, notice that if y-x\in P and z-y\in P then (y-x)+(z-y) = z-x\in P so x<z.
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