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Finding Power Set of Power Set of an Empty Set?

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M. Richard

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Feb 22, 2010, 1:05:27 PM2/22/10

Hello!

I am confused about what is the answer of P(P(P(Ø)))?

I have tried to solve it as follows

P(Ø) = ?

Ø has no elements therefore P(Ø) will contain 2^0=1 element, that is
P(Ø) = {Ø}

What will be the power set of the above set?

P(P(Ø)) = ?

P(Ø) has one element therefore P(P(Ø)) will contain 2^1=2 elements, that is
P(P(Ø)) = {Ø, {Ø}}

What will be the power set of the above set?

P(P(P(Ø))) = ?

P(P(Ø)) has two elements therefore P(P(P(Ø))) will contain 2^2=4 elements, that is
P(P(P(Ø))) = {Ø, {Ø}, {{Ø}}, {Ø, {Ø}}}

.....

Please tell me is the last power set correct? How?

Thanks!

Arturo Magidin

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Feb 22, 2010, 1:19:14 PM2/22/10

On Feb 22, 12:05 pm, "M. Richard" <muaa...@gmail.com> wrote:
> Hello!
>
> I am confused about what is the answer of P(P(P(Ø)))?
>
> I have tried to solve it as follows
>
> P(Ø) = ?
>
> Ø has no elements therefore P(Ø) will contain 2^0=1 element, that is
> P(Ø) = {Ø}

Yes.

>
> What will be the power set of the above set?
>
> P(P(Ø)) = ?
>
> P(Ø) has one element therefore P(P(Ø)) will contain 2^1=2 elements, that is
> P(P(Ø)) = {Ø, {Ø}}

Yes.

>
> What will be the power set of the above set?
>
> P(P(P(Ø))) = ?
>
> P(P(Ø)) has two elements therefore P(P(P(Ø))) will contain 2^2=4 elements, that is
> P(P(P(Ø))) = {Ø, {Ø}, {{Ø}}, {Ø, {Ø}}}

Yes.

>
> .....
>
> Please tell me is the last power set correct? How?

"How" what?

Look, take your previous set, P(P(emptyset)). Call it A. A has two
elements; call them a and b (in "reality", "a' is emptyset, and "b" is
{emptyset}).

If A = {a,b}, then P(A) = { emptyset, {a}, {b}, {a,b} }. No problem,
right?

Well, now substitute the "real" values of A, a, and b to convince
yourself you got the correct answer.

Doesn't matter what you call the elements, after all.

--
Arturo Magidin

Aatu Koskensilta

unread,

Feb 22, 2010, 1:21:01 PM2/22/10

"M. Richard" <mua...@gmail.com> writes:

> Please tell me is the last power set correct? How?

One way to approach the problem of computing the powerset of a finite
set A is to first list the zero-element subsets (i.e. the empty set),
then the one-element subsets (singletons of elements of A), then the
two-element subsets, ... finally collecting them all together and
removing duplicates.

--
Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus