Re **puzzle 32**:

The big difference between your definition and the one in Seven Sketches is that in your definition every element of \\(P\\) _is_ a subset of \\(X\\), whereas in Seven Sketches every element of \\(P\\) _labels_ a subset of \\(X\\).

The Seven Sketches definition allows you to permute the labels and still get essentially the same partition (ie the same equivalence relationship). Or even switch the set of labels for another set entirely, providing you do this sensibly (ie distinct old labels get replaced with distinct new ones). So it's no longer one-to-one exactly.

The big difference between your definition and the one in Seven Sketches is that in your definition every element of \\(P\\) _is_ a subset of \\(X\\), whereas in Seven Sketches every element of \\(P\\) _labels_ a subset of \\(X\\).

The Seven Sketches definition allows you to permute the labels and still get essentially the same partition (ie the same equivalence relationship). Or even switch the set of labels for another set entirely, providing you do this sensibly (ie distinct old labels get replaced with distinct new ones). So it's no longer one-to-one exactly.