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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?2012-06-04
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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.2012-06-04
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    @GerryMyerson The three sons also had wives.2012-06-04
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    @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.2012-06-04
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    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.2012-06-17
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    "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."2012-06-17
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    @Ragib : Why not post your comment as an answer?2012-06-17
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    @MichaelHardy Well it was not my comment, it was that of an anonymous user who attempted to edit the original post with that addition. I hadn't read this question in any detail, only performing an administrative action, and didn't realize the comment answered the question. Now that I see it does, I'll place it as an answer.2012-06-17
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    @GerryMyerson : The way I remember Genesis, his three son's wives were also aboard.2012-06-17
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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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