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)