How do you find the power set within a power set?

Question

I'm trying to find P(P(A)), where A = {0, 1, 2, 4, 7, 9}. Any ideas?

Answer 1

Mostly as said in the comments - there are no clever tricks here, you could try to compute $\mathcal{P}(\mathcal{P}(A))$ manually. However, this is inadvisable, because for a set of cardinality $n$, the power set has cardinality $2^n$. That is to say, $\mathcal{P}(\mathcal{P}(A))$ will have $2^{64}$ elements, which is a very big number (though most numbers are bigger). So don't waste your time.

However, if you're doing this with some purpose in mind - such as attempting to find whether a certain element is in this set - then we can be of more help, because such problems are more easily solvable.

Surely your question is a stepping stone in a larger problem?