I'm having a bit of a problem in understanding when a relation is transitive.
So let's say we have $A=\{a,b,c,d\}$. Do these following constitute a transitive relation? $\{(a,b), (b,c), (c,a)\}$?
I'm just confused since what I know is that for a set to have a transitive relation, $xRy$ and $yRz$ -> $xRz$. So it should be $(a,c)$ instead of $(c,a)$? Or is it the same thing?
As stated in the comments, $(a,c)$ and $(c,a)$ are different things. Transitivity would mean that $(a,b),(b,c) \in R$ (a relation on $A$) implies $(a,c) \in R$. Similarly, it would imply $(c,b),(b,a) \in R \implies (c,a) \in R$. These are ordered pairs, after all, and thus each component must be equal for two ordered pairs to be equal.
Of course, if we had symmetry, it would be a different matter. But that isn't always the case.
Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.