**Puzzle 22**. I think that \\(X\\) is "not \\(Y\\)" if \\(X \wedge Y = \bigvee \emptyset\\) (they have no elements in common) and \\(X \vee Y = \bigwedge \emptyset\\) (their union is the entire set).

Instead of referring explicitly to other elements of the poset like "the empty subset" or "the full subset" I have used the properties we derived earlier that the join of nothing is the bottom element of the poset and the meet of nothing is the top element of the poset.