I have to prove that for every non-zero integer $x,y,z$ at least one of the products $xy,xz,yz$ must be positive.
What I try to do is proof by contradiction. Assume $xy,xz,yz$ are all negative
$$xy = -a$$ $$xz = -b$$ $$yz = -c$$
$$(xy)(yz) = (-a)(-c) = ac$$
Is it possible to use transitivity for product of numbers to say that $$(xy)(yz) = xz$$ the equation above doesn't make sense yes I know. How else can I approach this problem. It's relatively easy but I just can't seem to prove that the assumption is wrong to get a contradiction..
Or can I just show a simple counter example to disprove the assumption.. so let
x be positive
y be negative and
z be negative.
From this, I can show y.z = negative . negative = positive. Hence, the assumption is false and we have a contradict.