Why is $R=\{(1,6),\,(2,7),\,(3,8)\}$ a transitive relation?

Question

Can someone please clear me transitive relations. books too have confused some say this is not as there are no pair to look for transitivity .While the true answer is it is but i couldn't understand why????

Answer 1

As is often the case, clarity arises by recourse to the definitions.

A relation is said to be transitive if $(a,b)\in R$ and $(b,c)\in R$ implies $(a,c)\in R$ for all $a,b,c$.

In the present relation, it is never the case that $(a,b)\in R$ and $(b,c)\in R$; hence the condition is vacuously true, and the relation is transitive.

Answer 2

A binary relation $R \subseteq S\times S$ is transitive if

$(1.) \quad \text{For all }\; a,b,c \in S,\; \left[(a,b) \in R \; \text{ and } \; (b,c) \in R \right]$ implies $(a,c) \in R$.

In your case

$$(a,b)\in R\;\text{ and }\;(b,c)\in R$$

will always be FALSE. So, according to the rules of logic, condition (1.) will be TRUE.