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Math Help - Linear ordering question

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    Linear ordering question

    Show that if (P,<) and (Q, \prec) are isomorphic strictly ordered sets and < is a linear ordering, then \prec is a linear ordering.
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  2. #2
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    we need to show that if a,b\in Q, a \neq b then a\prec b or a\succ b.

    Let \varphi : P\to Q be an order isomorphism.

    As a\neq b then \varphi ^{-1}(a)\neq \varphi ^{-1}(b), so \varphi ^{-1}(a)< \varphi ^{-1}(b) or \varphi ^{-1}(a)> \varphi ^{-1}(b) ( (P,<) is linear).

    Therefore a=\varphi \varphi^{-1}(a)\prec \varphi \varphi^{-1}(b)=b or a=\varphi \varphi^{-1}(a)\succ \varphi \varphi^{-1}(b)=b, hence (Q,\prec ) is linear.
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  3. #3
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    Quote Originally Posted by hammertime84 View Post
    Show that if (P,<) and (Q, \prec) are isomorphic strictly ordered sets and < is a linear ordering, then \prec is a linear ordering.
    What distinction does your text material make between linear ordering and strict ordering?
    Is it a matter of the reflexive property?
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