Q:

(Sets) Find the relationship between A and 2^D in terms of ∈ and ⊆.

The task is to find the various relationships between some sets (listed):
A = {0, ∅, {∅}}
B = A
C = {A}
D = {A, {A}}
For the most part, it is quite simple, with questions like "B and C", but the question in the title "A and 2^D" is giving me grief. I don't understand where the 2 is coming from and what the superscript D means. I know superscript C means complement, but I doubt that it is a typo.
Any help would be greatly appreciated.

A:

$2^D$ can be only an (ugly) way to write the power set  of  $D$, i.e. the set of all subsets of $D$ usually denoted as $\mathscr{P}(D)$
as $D=\{A,\{A\}\}$
can be quite confusing writing the power set. First of all if $X$ has $n$ elements, then $\mathscr{P}(X)$ has $2^n$ elements
So $\mathscr{P}(D)$ has $2^2=4$ elements
Furthermore a set has by default two subsets: itself and the empty set
The other two are the so called singleton sets of only one element, namely 
$\{A\}$ and $\{\{A\}\}$
Finally we have 
$$ \mathscr{P}(D)=\left\{\emptyset,\;D,\;\{A\},\;\{\{A\}\}\right\}$$
Thus $A$ is neither an element nor a subset of $\mathscr{P}(D)$
Hope this is useful

