Consider the following relation on the set of all people.
a R b <--> a and b have visited the same cities.
is R transitive i.e. (aRb and bRc -->aRc) ?
a has visited Rome, paris, london
b has visited rome, paris, New York
c has visited paris, New York, Honolulu
then aRb since they have visited the same cities (Rome, Paris)
bRc (paris, New York)
but a is not related to c since they have only one city in common not "cities" as required.
"have visited the same cities" means that the set of cities visited by a and b is the same. obviously trnasitive.
Basically the question reduces to wether there is an invisible "all" sitting in the phrase "have visitied (all) the same cities."
I think in math the convention is that there is never an implicit all, and in English in this case, there is an implicit "all."
what do you (all) think?