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Thread: How can we prove this?

  1. #1
    Junior Member
    Joined
    Nov 2008
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    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
    $\displaystyle (X,\leq_X),\ (Y,\leq_Y)$ are two isomorphic ordered sets iff there is a bijection $\displaystyle f$ between them such that $\displaystyle \forall a,b\in X,\ a\leq_Xb\Leftrightarrow f(a)\leq_Yf(b).$

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

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

    Therefore $\displaystyle \leq_Y$ is a well-order.
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