1. ## 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.

2. 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.

3. Originally Posted by hammertime84
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?