Why is this relation not transitive but R = {(3,4)} is ?

Question

While studying relations, I came across a strange question.

Set $A=\{1,2,3,4\}$ on which the relation $R=\{(2,4),(4,3),(2,3),(4,1)\}$ is defined. It is said in the answer that the relation is not transitive. I am not able to find out why is it so.

Let $R=\{(3,4)\}$. This is said to be transitive in the answer.

Can anyone give proper reason for it?

Answer 1

Recall that $R$ is transitive if $(a,b),(b,c)\in R\Rightarrow (a,c)\in R$.

Here, you have that $(2,4),(4,1)\in R$, but $(2,1)\not\in R$, so $R$ is not transitive.

Answer 2

$(2,4)$ and $(4, 1)$ are in the relation, but $(2, 1)$ is not, and so the relation cannot be transitive.

Answer 3

$R$ is a transitive relation if for every $a,b,c\in A$ if $(a,b), (b,c)\in R$ then $(a,c)\in R$.

As others have said, in the first relation $(2,4),(4,1)\in R$ but $(2,1)\notin R$. This proves that $R$ is not transitive.

This is because in order to prove $R$ is not transitive we need to find three elements $(a,b),(b,c)\in R$ such that $(a,c)\notin R$.

However in the case $R=\{(3,4)\}$ there is only one pair of elements. There is no possible counterexample to transitivity, and so it holds vacuously.