[Math] Prove that for every three non-zero integers, a,b and c, at least one of the three products ab,ac,bc is positive

Question

The question is:

Prove that for every three non-zero integers, a,b and c, at least one of the three products ab,ac,bc is positive. Use proof by contradiction.

My general approach to doing contradiction is as follow:

I always like to turn the statements into propositional logic, with an implication. In this case it would be like:

[For all a,b,c in the domain of non-zero integer If a,b and c are three non-zero integers], then at least one of the three products ab,ac,bc is positive.

Then I take the negation of it: (P ^ ~Q)

Assume to the contrary,there exist a,b and c that are non-zero integers and none of the three products ab,ac and bc are positive.

Now, I pick a = 1, b = -1 , c = 2

a.b = -1
a.c = 2
b.c = -2

Since a.c is positive, our assumption is false and we have a contradiction. Hence, the original statement is true.

Please feel free to share any other alternatives…(contradiction ones)

Answer 1

The product of three negative numbers is negative. So if $ab$, $ac$, and $bc$ are all negative, then $(ab)(ac)(bc)\lt0$. But $(ab)(ac)(bc)=a^2b^2c^2$ is the product of three squares, which are all positive.