1. ## Linear ordering question

Show that if (P,<) and (Q,$\displaystyle \prec$) are isomorphic strictly ordered sets and < is a linear ordering, then $\displaystyle \prec$ is a linear ordering.

2. we need to show that if $\displaystyle a,b\in Q, a \neq b$ then $\displaystyle a\prec b$ or $\displaystyle a\succ b$.

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

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

Therefore $\displaystyle a=\varphi \varphi^{-1}(a)\prec \varphi \varphi^{-1}(b)=b$ or $\displaystyle a=\varphi \varphi^{-1}(a)\succ \varphi \varphi^{-1}(b)=b$, hence $\displaystyle (Q,\prec )$ is linear.

3. Originally Posted by hammertime84
Show that if (P,<) and (Q,$\displaystyle \prec$) are isomorphic strictly ordered sets and < is a linear ordering, then $\displaystyle \prec$ 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?