why this is transitive relation?

Question

$\rho\subseteq \mathbb{N}\times \mathbb{N},\rho=\{(x,y):y=x+5,x<4\}$ is the relation, so $\rho=\{(1,6),(2,7),(3,8)\}$ in my book it is written that $\rho$ is an transitive relation, but why? I know the definition as if $(a,b)\in \rho, (b,c)\in\rho\Rightarrow (a,c)\in\rho$, then $\rho$ is transitive.

Answer 1

An alternative way to look at the definition of transitivity is as follows:

A relation $\rho$ is transitive unless there exist $a, b, c$ such that $(a, b) \in \rho$ and $(b, c) \in \rho$, but $(a, c)\notin \rho$.

What you're noting is the fact that the antecedent of your definition is not fulfilled (hence is false); there are no $a, b, c$ such that both $(a, b)\in \rho$ and $(b, c) \in \rho$, so the relation is vacuously transitive.