I am just now starting to grasp the concepts of transitivity in relations and I have the following question:
In my textbook, it is noted that $R = \{(1, 1), (1, 2), (2, 1)\}$ is not transitive.
How can that be right?
There are only two elements in this relation and there aren't any "steps" that aren't covered.
It was explained to us that a relation is not transitive when you can not perform a two-step action with one step. However there isn't any possible action in the $R$ above that requires more than one step.
To give an example, this is transitive: $\{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)\}$
But this is not: $\{(1,1),(1,2),(2,1),(2,2),(3,4),(4,4)(4,1)\}$
But that is obviously because there isn't a $(3, 1)$
I can not see a fundamental difference between the two from the first example.
Thank you for your help
EDIT:
Apparently, this also isn't transitive and I have no idea why:
$R_5 = \{(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)\}$
Transitivity simply means if $(x,y) \in R, (y,z) \in R$, then $(x,z)\in R$. In the first case, $(2,1), (1,2) \in R$, but $(2,2) \notin R$.
The last one is not transitive because $(2,1), (1,4) \in R$, but $(2,4)\notin R$