I added a new puzzle to the above post, which is very useful:

Here's another way to tell if one partition is finer than another:

**Puzzle 34.** Given two partitions \\(P\\) and \\(Q\\) of a set \\(X\\), show that \\(P \le Q\\) if and only if every part of \\(P\\) is contained in a part of \\(Q\\).