Need help with proofs using axioms only

Question

Prove if $p,q∈R$ and $pq>0$ then either $p>0$ and $q>0$, or, $p<0$ and $q<0$ using only the field axioms.

I have no idea how to do this using only the field axioms. Seems pretty straightforward but how would you approach this question using only the field axioms. Remember the premise here is $pq >0$ and we must begin with this to prove the "then either .... or ....". I've seen similar questions to this but the implication is usually the other way around. Eg. we begin with if p>0 and q >0 then pq >0 which is a lot easier to prove. But how do we do it the other way around while only being able to use the field axioms.

Edit: These include distribution axioms, order axioms, ordered field axioms. Edit: I know the question is similar but the answers don't take pq >0 as the premise. I really need help/hints for this question and any help would be appreciated.

$p>0$ doesn't mean anything in fields in general. Perhaps you want the axioms of ordered fields? $\qquad$ — Michael Hardy, Apr 02 '16 at 02:40
You should specify which field axioms you mean because none of the standard field axioms involve an ordering of the elements. — Matt Samuel, Apr 02 '16 at 02:41
I saw that. And like I said, the answers did not start with pq > 0 as the premise. — Mathew, Apr 02 '16 at 02:46
Proofs don't have to start with the premise. The proofs in the other post do all possibilities and note they only ones that meet the premise are the ones to prove. That's a solid and complete proof. — fleablood, Apr 02 '16 at 03:03
You can try to prove 1/p > 0 iff p>0. But this is actually too thorough for to little result. — fleablood, Apr 02 '16 at 03:09
What the heck is the problem with a proof by contadition if p <0 and q>9 then pq <0 thus.... — fleablood, Apr 02 '16 at 03:23

fleablood · Accepted Answer · 2016-04-02T03:19:27.307

0

Prove 1 > 0. Then prove that means 1/p >0 iff p>0. So pq > 0 means if 1/p > 0 q >0 and if 1/p <0 then q <0.

But I don't advise this. It repeats way too many similar results that should be more versatile. Almost to the point of being circular.

I'd advise not worrying about a direct premise. Prove more vetsitile propositions and have this particular result come out as consequence.

edited Apr 02 '16 at 03:19

answered Apr 02 '16 at 03:12

fleablood

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Need help with proofs using axioms only

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