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Math Help - How can we prove this?

  1. #1
    Junior Member
    Joined
    Nov 2008
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    53

    How can we prove this?

    Let X and Y be isomorphic ordered sets and X is well ordered.
    How can we show that Y is well ordered as well?

    It is easy to just say A->B bijective and order preserving therefore Y is also well ordered.
    But it is hard to come up with more constructive answer.
    Can anybody help?
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  2. #2
    Senior Member
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    Hi
    (X,\leq_X),\ (Y,\leq_Y) are two isomorphic ordered sets iff there is a bijection f between them such that \forall a,b\in X,\ a\leq_Xb\Leftrightarrow f(a)\leq_Yf(b).

    Let S_Y be a subset of Y, there is a S_X subset of X such that f(S_X)=S_Y, and since f is injective, S_X and S_Y are in bijection.

    \leq_X is well-ordered, so there is a x\in S_X\ \text{s.t.}\ \forall a\in S_X,\ x\leq a and with f properties, we have \forall b\in S_Y,\ f(x)\leq_Y b.

    Therefore \leq_Y is a well-order.
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