For transitivity, suppose that ab > 0 and bc > 0. Note that ac = (ab)(bc) / bē.

Alternatively, it is easy to find a function f from S to {0, 1} such that

x ~ y iff f(x) = f(y) (*)

(hint: the name of f starts with s and ends with n). The property (*) says, "x and y are related if they have the same ... ." Every relation ~ that satisfies (*) for some function f is an equivalence relation.