If $x$ is in $A$ then is $\{x\}$ an element of the power set $A$ or a subset of the power set of $A$? If $x \in A$ then is $\{x\} \in \wp{A}$ or is $\{x\} \subseteq \wp{A}$?
I know that since $x \in A$ then $\{x\} \subseteq A$ but what does that make $\{x\}$ in relation to $\wp{A}$?
 A: By definition the elements of $\wp(A)$ are the subsets of $A$. If $x\in A$, then $\{x\}\subseteq A$, so $\{x\}$ is a subset of $A$ and therefore an element of $\wp(A)$. In short, $\{x\}\in\wp(A)$.
For a little more practice:
Let $X=\{x\}$. Then $X\in\wp(A)$, so $\{X\}\subseteq\wp(A)$, and therefore $\{X\}\in\wp(\wp(A))$. In other words, $\{\{x\}\}\subseteq\wp(A)$, so $\{\{x\}\}\in\wp(\wp(A))$. In fact,
$$\underbrace{\{\{\{\dots\{}_nx\underbrace{\}\dots\}\}\}}_n\in\underbrace{\wp(\wp(\wp(\dots\wp(}_nA\underbrace{)\dots)))}_n\;.$$
A: Let $A=\left\{1,2,3\right\}$. Then, $2\in A$. Now,
$$
\wp(A)=\left\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\right\}.
$$
This implies that $\{2\}\in\wp(A)$.
A: Saying that $B \subset A$ is the same as saying that $B \in \wp{A}$. In your case it means that $\left\{x\right\} \in \wp{A}$.
A: The previous answers are excellent and give the right picture, but I'd like to point out that we can have both $\{x\}\in\mathcal{P}(A)$ and $\{x\}\subseteq\mathcal{P}(A)$ for $x\in A$. As an example, consider $A=\{\emptyset,\{\emptyset\}\}$ and $x=\emptyset$. We now have
$\mathcal{P}(A)=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\},\{\emptyset,\{\emptyset\}\}\}$
and both $\{\emptyset\}\in \mathcal{P}(A)$ and $\{\emptyset\}\subseteq \mathcal{P}(A)$.
In fact, there are sets $A$ such that both of your options hold for every element $x\in A$; such sets $A$ are called transitive.
