Can a relation be transitive vacuously if no pairs share a middle element?

[matticus]

in the book I'm reading it gives a set S={0,1,2,3}, and it says that the relation R where (m,n) $$\in$$ R if m + n = 3, m,n $$\in$$ S.

it says that this relation isn't transitive, but couldn't you give a vacuous argument for transitivity.

more specifically there are no x,y,z s.t. (x,y) and (y,z) are elements of the S, therefore the statement
if (x,y) and (y,z) are in S then (x,z) is in S should be true, right?

 

[masnevets]

(0,3) and (3,0) are both in R, but (0,0) is not.

 

[matticus]

thanks, i don't know how i missed that. i must have had myself fooled that the 3rd number had to be unique from the first, when clearly it doesn't.