Cardinality of power sets

Question

How would I find the cardinality of these two sets?

$\mathcal{P} (\{a,b,\{a,b\}\})$

$\mathcal{P} (\mathcal{P} (\emptyset))$

I believe the first set has a cardinality of 5 but I am not sure how the inner set works with power sets.

Answer 1

For a finite set $S$, the following formula holds:

$$|P(S)|=2^{|S|}$$

So $|P(\{a,b,\{a,b\}\})|=2^3=8$. Your second problem can be solved similarly:

$$|P(P(\emptyset))|=2^{|P(\emptyset)|}=2^{2^0}=2$$