Finding the Power Set of a Power SetDate: 02/17/2005 at 04:45:48 From: Richard Subject: "The second power set", or the power set of a power set What is "the second power set", or the power set of the power set of a set, say set <1,2,3>? As we know, the power set of set <1,2,3> is << >,<1>,<2>,<3>,<1,2>,<1,3>,<2,3>,<1,2,3>>. But I do not know the power set of that set. Date: 02/22/2005 at 09:47:35 From: Doctor Ian Subject: Re: Hi Richard, Suppose our original set is A = {a,b} The power set of this is the set of all subsets: P(A) = { {}, No elements {a}, {b}, One element {a,b} Two elements } To make subsequent expansion less of a notational nightmare, I'm going to use variables to represent these subsets, i.e., P(A) = { W, X, Y, Z } Now we can take the power set of the power set, which is the set of all subsets of P(A): P(P(A)) = { {}, No elements {W}, One element {X}, {Y}, {Z}, {W,X}, Two elements {W,Y}, {W,Z}, {X,Y}, {X,Z}, {Y,Z}, {W,X,Y}, Three elements {W,X,Z}, {W,Y,Z}, {X,Y,Z}, {W,X,Y,Z} Four elements } This is 2^4 = 16 elements, as we would expect. To express them in terms of the original elements, we can substitute as needed: W -> {} X -> {a} Y -> {b} Z -> {a,b} P(P(A)) = { {}, {{}}, {{a}}, {{b}}, {{a,b}}, {{},{a}}, {{},{b}}, {{},{a,b}}, {{a},{b}}, {{a},{a,b}}, {{b},{a,b}}, {{},{a},{b}}, {{},{a},{a,b}}, {{},{b},{a,b}}, {{a},{b},{a,b}}, {{},{a},{b},{a,b}} } If I wanted to find P(P(P(A))), I would use the same trick again, i.e., assign a variable to each element of P(P(A)), expand using those variables, and substitute. Otherwise, it's too easy to get lost in a blizzard of brackets. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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