Solution 0.3.4.1.

\(\pow(A)\) is a set of sets, all of which are subsets of \(A\text{.}\) So

\begin{equation*} \pow(A) = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1, 3\}, \{2,3\}, \{1,2,3\}\}\text{.} \end{equation*}

Notice that while \(2 \in A\text{,}\) it is wrong to write \(2 \in \pow(A)\) since none of the elements in \(\pow(A)\) are numbers! On the other hand, we do have \(\{2\} \in \pow(A)\) because \(\{2\} \subseteq A\text{.}\)

What does a subset of \(\pow(A)\) look like? Notice that \(\{2\} \not\subseteq \pow(A)\) because not everything in \(\{2\}\) is in \(\pow(A)\text{.}\) But we do have \(\{ \{2\} \} \subseteq \pow(A)\text{.}\) The only element of \(\{\{2\}\}\) is the set \(\{2\}\) which is also an element of \(\pow(A)\text{.}\) We could take the collection of all subsets of \(\pow(A)\) and call that \(\pow(\pow(A))\text{.}\) Or even the power set of that set of sets of sets.

in-context