prove a set relation R is transitive?

Question

I have been thinking this problems all the evening, please help

Let R be a relation on A. Prove that if Dom(R)∩ Range(R) = Ø, then R is transitive.

Oh my god, how to prove this???

Answer 1

Transitivity means if $x R y$ and $y R z$, then $x R z$.

Since $\text{dom}(R) \cap \text{rang}(R) = \emptyset$, there are no $x,y,z$ such that $x R y$ and $y R z$ since otherwise $y \in \text{dom}(R) \cap \text{rang}(R)$. So the statement of transitivity is vacuously true.