Given the relation on the set $\{1,2,3,4\}$ why is $\{(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)\}$ considered to be transitive?
As I go through it I can see that we have $(2,2)$ and $(2,3)$ so we'd need $(3,2)$ as well. But then I see $(3,3)$ and $(3,4)$ but not $(4,3)$? Obviously my processing of this is flawed, so I am asking this:
What is the process one would go through to determine if a relation set is transitive? Does the sequence/ordering of the set matter? I feel there's a key I don't understand yet.
You've misunderstood transitivity. Transitivity says that if $(x,y)$ and $(y,z)$ are in the set, then $(x,z)$ must also be; transitivity tells us nothing about $(z,x),(z,y),$ or any other ordered pair made from these elements.
With transitivity, $(3,3)$ and $(3,4)$ together only imply that $(3,4)$ and $(3,3)$ must be part of the set, which they clearly are. The situation is exactly the same for $(2,2)$ and $(2,3)$; $(3,2)$ may be in the set, but we could exclude it without violating transitivity.
But we do have things like $(2,3)$ and $(3,2)$ in our set, and it would not be transitive if we didn't have $(2,2)$ in as well. Likewise, if we notice that we have $(3,4)$ and $(2,3)$, we'd better have $(2,4)$, which we do.