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### Finding the Power Set of a Power Set

```Date: 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/
```
Associated Topics:
College Logic
High School Logic

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