The "opposite" of being finite is having a finite complement.
The "opposite" of a vector is a linear functional.
If a set is a small collection of elements... then what is the "opposite" of a set?
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Errata:
2:53 - Some of the equations are incorrect; thanks @sstadnicki!
6:18 - The preimage function g^{-1} is pointing the wrong way; it should be mapping P(X) to P(Y)... like every other instance of g^{-1} in this video. Thanks @ДмитрийГнатюк-к4ц !
14:23 - This example is not complete! The algebra here does not have infinite joins... sorry everyone! This only provides an example of a Boolean algebra with no atoms. For a *complete* Boolean algebra with no atoms, you can take a similar-looking (but much bigger example) of Lebesgue measurable subsets of the real numbers, where two subsets are viewed as "essentially the same" if they differ by a set of Lebesgue measure zero. Thank you @SlipperyTeeth for pointing out the error with the infinite joins and meets, and thank you also to @yuvalpaz3752 for demonstrating that A doesn't have infinite joins at all.
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Timestamps:
00:00 - Introduction
01:18 - Cosets, as the name implies
02:02 - Formal duality
03:51 - Undoing functions
04:53 - Preimage
06:30 - Towards an algebraic definition of cosets
08:22 - The structure of a coset
09:35 - Function reconstruction
12:41 - Cosets are a kind of algebra
13:22 - John Dalton would be very upset with this example
15:55 - Coset building blocks
16:40 - Thx 4 watching!