John - sorry, but I don't see that "the inverse image of a nonempty set is always nonempty." Am I missing something? The part \\(S\\) in \\(Y\\) may simply have nothing mapped onto it by \\(f\\) - then \\(f^{\ast}(S)\\) will be empty and \\(f^{\ast}(P)\\) will have an empty part. I know this is rather pedantic, because if you define the partition \\(f^\ast(P)\\) in terms of an equivalence relation, then empty parts make no difference. To be clear, if you change your example and map \\(X_{14}\\) to \\(Y_{22}\\), then \\(f^{\ast}({Y_{13}})\\) will be empty. I suppose I have been trained never to treat an empty set as "nothing"!