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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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The argument is correct, but I do not see the conection to Noah's Ark? –  Michael Greinecker Jun 4 '12 at 6:28
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Noah was the captain, not sure if he counts as a passenger. Anyway, he had a wife and three sons (if I remember right), so if you coount him as a passenger, that would make an odd number. More to the point, your argument sounds good, but I'd like to see this paper you refer to, to see what's in it. –  Gerry Myerson Jun 4 '12 at 6:41
    
@GerryMyerson The three sons also had wives. –  Phira Jun 4 '12 at 12:48
    
@Shango It sounds like you are right, but this is not really a big issue. The authors probably filled in the first proof that came to their mind and eight sentences seemed reasonably short, so they didn't think about it further. If you referee a paper like that, sure, point it out, but otherwise, it doesn't really matter. –  Phira Jun 4 '12 at 12:52
    
The following comment was proposed as an edit by an anonymous user. It is better suited as a comment rather than an edit of the OP, so I place it here instead. –  Ragib Zaman Jun 17 '12 at 17:15
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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.

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