Question about axioms of an ordered field.

Question

I’m currently studying Introduction to Analysis by Ross. I wanted to ask if the either - or is an inclusive or exclusive disjunction in property O1 below. I believe this should just expressing the trichotomy of the reals, right?

Answer 1

It's inclusive disjunction, i.e. that there can be $a\leq b$, $b\leq b$ or both. If we define the relation $<$ by the formula $a<b \iff a\leq b$ and $a\neq b$ then from O1 we get trichotomy: $a<b$, $a=b$ or $b<a$, which is easy to prove. Namely, take any $a,b$. Then either $a=b$ or $a\neq b$ (exclusive disjunction). If $a\neq b$ then we see from O1 that either $a\leq b$ or $b\leq a$ (exclusive so far). Since $a\neq b$ then either $a<b$ or $b<a$. Observe that if $a<b$ and $b<a$ then $a=b$, which is a contradiction. Therefore we get trichotomial exclusive disjunction.