For those who like pictures, here's an upset in the power set \\(P\\{1,2,3,4\\}\\), as drawn by

You can see that if anything is marked in green, so is everything above it: that's what makes it an "upset". It's a "principal upset", because it consists of all \\(S \subseteq \\{1,2,3,4\\} \\) with \\(\\{1\\} \subseteq S \\).

**Puzzle.** If \\(X\\) is some set, can there be upsets of \\(P(X)\\) that aren't principal upsets?