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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!)

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    @Phira, I stand corrected. Genesis 7:13, "On that same day Noach entered the ark with Shem, Ham and Yefet the sons of Noach, Noach’s wife and the three wives of his sons accompanying them...." There is still the question of who counts as a passenger. Do you count members of the crew?2012-06-17

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The following is contributed by an anonymous user who attempted to edit the OP's post with this addition:

Unbeknownst to Shango at the time this question was posted, it turns out that the authors use a subresult contained in their longer-than-necessary proof a little later in the paper. This helps explain their choice. It's likely they were aware of the shorter proof, but preferred the longer one given its residual usefulness later.