Transitive relations on sets

Question

So I'm having a bit of an issue understanding transitive relation property. I feel like I understand the rule well enough.

On: the set $\{1, 2, 3, 4\}$

on this relation $\{(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)\}$

I understand that $(2,3)$ matches the $(a,b)$ condition then $(3,2)\to (b,c)$ and finally $(2,2)$. But I thought I had to also consider $(2,4)$ and $(3,4)$?

And a second one here where the relation is not transitive but I'm not sure why. On set: $\{0, 1, 2, 3\}$

$\{(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)\}$

Answer 1

Regarding your first point. To evaluate transitivity you have to look at two couples $(a,b),(b,c)$ in your relation set and verify that $(a,c)$ also belongs to it. Hence, you don't need to evaluate $(2,4),(3,4)$.

Regarding your second point. Name $R=\{(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)\}$.

You have $(0,2) \in R$ and $(2,3) \in R$. If $R$ was transitive, you should have $(0,3) \in R$ which is not the case.

Answer 2

I believe the above by mathcounterexamples.net takes care of the second question you ask about quite nicely. As for the first question, I think this is where you are misunderstanding transitivity.

Transitive is $(a,b)$ AND $(b,c)$ implies $(a,c)$. The relation need not be symmetric, so the order matters. It is the second coordinate of the first and the first coordinate of the second that must be the same. When you are checking $(2,3)$, you need to check (1) things of the form $(3,x)$ and be sure that $(2,x)$ is in there and (2) things of the form $(y,2)$ to be sure $(y,3)$ is in there.