One way of picturing about this is in terms of the natural \\(Y\\)-indexed partition \\(f\\) induces on \\(X\\). Colour each partition _white_ if none of its members is in \\(S\\), _black_ if all of its members are in \\(S\\), _grey_ if it's "on the borderline". Then \\(f_* (S)\\) is the \\(y\\) indices corresponding to grey or black partitions, and \\(f_!(S)\\) is the \\(y\\) indices of just the black ones. The double complements thing that Owen talks about is basically a matter of swapping black and white.