Shouldn't this be transitive relation also?

Question

$ R = \{ (a,b) ∈\Bbb R^2 ; 1 + ab > 0 \} $

It is clearly reflexive and symmetrical but I feel that it is transitive also because the relation R can be stated as $ R = \{(0,0), (0,1), (1,2)...\}$ and as per the rules of transitive $ x,y ; y,z ; x,z = true $, and in the relation R it is true also, $ (0,1) $. Then why is not transitive? Are there some specific rules regarding $(0,0)$?

Answer 1

No it is not transitive consider $(-1, 1/2)$ and $(1/2,100)$.

Answer 2

Hint: Let $a=0.1$, $b=2$, and $c=-3$.