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) ?

1) No.

suppose

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.

2) Yes.

"have visited the same cities" means that the set of cities visited by a and b is the same. obviously trnasitive.

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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?