# How can we prove this?

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• February 16th 2009, 04:09 PM
ninano1205
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?
• February 16th 2009, 11:16 PM
clic-clac
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.