I was reading my notes when i came across some transitive proofing which i can't understand. For example
A = {0, 1, 2} and let R be a relation on given by
R = {(0,0), (1, 1), (2, 2), (0, 1), (1, 0)}
So this is how my notes do some transitive proofing which is rather weird
(0, 0), (0, 1) ∈ R, so (0, 1) should be in R, which it is
(1, 1), (1, 0) ∈ R, so (1, 0) should be in R, which it is
(0, 1), (1, 1) ∈ R, so (0, 1) should be in R, which it is
(0, 1), (1, 0) ∈ R, so (0, 0) should be in R, which it is
(1, 0), (0, 1) ∈ R, so (1, 1) should be in R, which it is
(1, 0), (0, 0) ∈ R, so (1, 0) should be in R, which it is
Therefore R is a transitive
Here is the question, how do they exactly derive the information that produce that 6 statements? What about the other elements in the relation such as (1, 1) and (2, 2)?
The reason you are interested in these six statements in particular is that transitivity says "if $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$". Note here that $y$ is a common, connecting variable between the two pairs. Thus, we only need to consider those pairs that have such a common variable. This means that it makes sense to consider $(1,0)$ and $(0,1)$ together, because they are connected through $0$, but there is no reason to consider $(1,1)$ and $(2,2)$ together, for example, because they are not connected through $1$ and $2$. If you want to be extremely pedantic (but please don't), then you would also have to consider that if $(1,1) \in R$ and $(1,1) \in R$, then $(1,1) \in R$, but this is too obvious to merit mention.