Let $R = \{(x, y) : y = x + 5\ and\ x \lt 4\}$ be a relation in $\Bbb N$. Is it transitive?

Question

Clearly $R = \{(1, 6), (2, 7), (3, 8)\}$.

From this it follows that for any $R(a, b), \ R(b, c)$ does not exist. Does this imply that the relation is transitive?

Edit: Since, there are no examples to prove transitivity shouldn't it just be intransitive?

Answer 1

There are no counterexamples to transitivity, so the relation is vacuously transitive.

Phrased differently; in order for a relation, $\sim$, to fail to be transitive, there needs to be a triple $\{a,b,c\}$ with $a\sim b, b\sim c$ and $a\not \sim c$. As there is no such triple in the case of your relation, it is transitive.