Power Set of a Power set with the same element

Question

I am working with Power Sets and I am stumped conceptually on a problem that looks as such: A = {a, {a}}.

Find P(P(A)).

I am under the impression the Power set of A would be: {∅ , a}

and then...

P(P(A)) would be: {{∅ }, {a}, {{∅ }} {a, ∅ }}

Which would resulting in a cardinality of 4.

Answer 1

Hint:

Try to instead look at finding $\mathcal{P}(\mathcal{P}(\{a,b\}))$ and after you are done, replace $b$ with $\{a\}$.

Additional hint:

So, you should be well familiar with that $\mathcal{P}(\{a,b\}) = \{\emptyset, \{a\},\{b\},\{a,b\}\}$. If you still struggle with finding the power set of that result, then try doing something similar as before... naming these something else. You should be able to find $\mathcal{P}(\{u,v,x,y\})$ and then after you are done, replace $u$ with $\emptyset$, replace $v$ with $\{a\}$, replace $x$ with $\{b\}$, and replace $y$ with $\{a,b\}$ (and then finally replace $b$ with $\{a\}$ at the end)