I have a doubt trying to determine wether a relation is transitive or not.
Given this sets A={1,2,3,4} and B={1,2,3,4} and relation AxB = {(1,1),(3,4),(2,2),(3,3)} we can determine that is transitive given the next definition: if (a,b) in R, and (b,c) in R there must be (a,c) to be transitive.
We take (3,3) as (a,b), then we take (3,4) as (b,c) so (3,4) is (a,c), which is already in the set, therefore this is a transitive relation.
Your argument is correct but it is not written well.
First, the relation $\{(1,1),(3,4),(2,2),(3,3)\}$ should not be called "$A \times B$". It is a particular subset of $A \times B$. Give it its own name, perhaps "$C$".
Second, before giving your correct argument about $(3,3)$ and $(3,4)$ you should say explicitly that they are the only pair of elements of $C$ where the second element of the first matches the first element of the second, so that is the only pair you need to check. I think that was in the back of your mind when you wrote the proof but you didn't write that down.
PS When posting on this site, use mathjax.