If you start with a finite family of subsets of an arbitrary nonempty universe set and close that family under an arbitrary collection of operations that includes set complementation, you'll always end up with an even number (assuming it's finite) of subsets total, because no subset equals its own complement.
For some reason the authors of an otherwise impressive math research paper published several years ago wrote a convoluted eight-sentence proof to get this same job done (the closed family has even cardinality when finite). Am I missing something? My argument looks correct to me. Put another way, the number has to always be even for the same reason that Noah's Ark carried an even number of passengers. (Wait - Noah wasn't single was he? Silly me - of course he wasn't single - we wouldn't be here!)