Based the definition that a relation $R$ on a set $A$ is called transitive: $∀_a ∀_b ∀_c (((a,b)∈R∧(b,c)∈R)→(a,c)∈R)$
I thought the relation ${(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}$ would be transitive due to the case: $$((1,3)\in R \land (3,4)\in R)\to (1,4)\in R$$
If it's not would adding $(1,1)$ to the relation make it transitive?
It's not enough to check transitivity in one, or a couple of cases, it has to hold in all cases for which it makes sense. Providing a single counterexample is, however, enough to disprove transitivity.
Here we have $2\sim_R 3\;$ and $3\sim_R 1\;$ but we do not have $2\sim_R 1$, so transitivity fails.