Artur - thanks for catching that typo in #11. I fixed it.
And yes, you solve Puzzle 32. Does everyone understand what Artur said? With Fong and Spivak's definition of partition, _infinitely many different partitions_ correspond to the _same_ equivalence relation! The reason is that for them, different label sets \\(P\\) give different partitions, and there are always infinitely many choices of label set.
So, I needed to adjust their definition by eliminating the label set. This is a well-known Jedi math trick: _get rid of labels by letting a set label itself!_ For me, \\(P\\) is not an arbitrary set labeling the parts of the partition. It _is_ the set of parts of the partition.