#Can there not be less than 2^k subsets? I don't get it what he means by this?

30 messages · Page 1 of 1 (latest)

empty bolt
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This is from a book of Terence Tao for Measure theory and he is defining the Lebesgue integral. What I don't understand is this part. If we use a Veen diagram and let's say we have the picture below we only have 5 (at most 6 if we include the empty set as the intersection of B and C). What does he mean by 2^k?

image.png image.png
toxic pewterBOT
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empty bolt
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or if ppl are scared of links just search: "Terence Tao measure theory" and it should be the first link

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Oh I think I see it now

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The other sets, C intersect B, C intersect A, C intersect A intersect B are counted separate even though all of them are the null set and still disjoint ( their interesection is still the empty set)

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But isn't this a bit of a strech since we also say that it can partitioned into 2^k + 17 other empty sets

flat whale
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per the definition of partitions, the sets arent empty

flat whale
empty bolt
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Then how do you explain what he said

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or maybe I am misunderstanding something

flat whale
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he means if you list out the partition

empty bolt
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also why would there be a problem with including empty sets in your partition

flat whale
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there are 2^k sets

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cause then there's no equivalence relation that generates it

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and it's known a collection of sets is a partition iff it's a set of equivalence classes

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Like in the 2 set example (k=2), you have your partition as ${A\setminus(A\cap B),A\cap B, B\setminus(A\cap B),U\setminus(A\cup B)}$ for universal set $U$

charred sequoiaBOT
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Omegabet_

186109587366215691.png
flat whale
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which is 4=2^2 sets

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it's basically the same proof that {1,2,...,k} has 2^k subsets

empty bolt
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What if the sets A and B were disjoint

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The partition would be of only 3 non-sets

flat whale
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this is what to do if they are not disjoint

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if they are disjoint, then they already partition

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if they arent disjoint, then this process 'repurposes' it into a partition

empty bolt
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Ok

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I see

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+close