> \\(f_*(A)\\) does not just give you "all the buckets that only contain balls from A", but also all the buckets that never get any ball from \\(X\\).
This is a nice example of how ordinary language treats "all" in a different way than mathematics. Suppose I'm unmarried and I say "all my wives are millionaires". Is this true?
A mathematician would say yes, it's true. It's "vacuously true". In mathematics, "All my wives are millionaires" is equivalent to "I have no wife that is not a millionaire". Go through my wives and check this. Yes it's true, because there are no wives to check!
\\(f_*(A)\\) gives you the buckets all of whose balls are from \\(A\\). If a bucket has no balls in it, it's vacuously true that all the balls in this bucket are from \\(A\\).