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Mathematics 56 Online
OpenStudy (anonymous):

What's the power set of the power set of {1}? I know the power set of {1} is the set{null set, {1}}, correct? So the power set of that would be {null set, {1}, {null set, {1}}}?? or do I need the last one, because doesn't {1} contain the null set by implication? Also, would it contain {null set} as well as null set?

OpenStudy (amistre64):

Power set is a set of all subsets right?

OpenStudy (amistre64):

all the subsets of {1} are then 0s { } 1s {1} thats all of them sooo, { { } , {1}}

OpenStudy (anonymous):

yeah, power set is the set of all subsets

OpenStudy (amistre64):

lets say the set was: {a,b} 0s { } 1s {a},{b} 2s {a,b} and its done power set of [a,b] is: {{ },{a},{b},{a,b}}

OpenStudy (anonymous):

so if i want the power set of the powerset of {1}, that would be the powerset of {a,b} where a={} and b={1}?

OpenStudy (anonymous):

so it would be {{},{{}},{1},{{},{1}}}

OpenStudy (amistre64):

{a,b} was just an example with more elements to play with.

OpenStudy (anonymous):

sure, but it works in this case because the first powerset has two elements, right? Sorry, I'm just being thrown off by the null set I think

OpenStudy (amistre64):

a set with a single element only has null and itslef as subsets

OpenStudy (amistre64):

its best to work thru it systematically, define the nothings: { } define all the subsets of a single element define all the subsets of two elements etc ...

OpenStudy (anonymous):

I guess the question is would P{P{1}} where P{} is the power set be P{1} or P{ {} ,{1} } or are they the same thing?

OpenStudy (amistre64):

the power set, of the power set, of the set {1} ?? is that the question

OpenStudy (anonymous):

yep, sorry to be unclear

OpenStudy (amistre64):

work it from inside to outside ... P{1} = { {}, {1} } P{P{1}}: 0s { } 1s { }, {1} 2s {{ },{1}} and were done since we dont need to list the same element twice ... { { }, {1}, {{ },{1}} } looks like its right

OpenStudy (anonymous):

see, power set is the set of all sets that can be formed with various combinations of the given set. as it is the maximum set that can be formed (we call it as span in linear algebra). powerset of it shall be itself. i.e. powerset of a power set is the same set

OpenStudy (anonymous):

@amistre64, {{},{1}} is no different from {1}

OpenStudy (anonymous):

ahh, got it. so {} is the same as {{}}.

OpenStudy (anonymous):

yes

OpenStudy (amistre64):

simplifying after the fact is always good too

OpenStudy (anonymous):

so @amistre64, @vamgadu would it be { { }, {1}, {{ },{1}} } or {{},{1}}?

OpenStudy (amistre64):

my gut would go; {{},{1}} but then this interactions got me wondering lol

OpenStudy (anonymous):

ok, good to know, I'll revisit my notes again as well. Thanks for the patient help!

OpenStudy (amistre64):

i recall doing this in discrete math, but the specifics get fuzzy after a few terms :) good luck with it

OpenStudy (anonymous):

yep, taking discrete right now. Thanks again!

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