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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 $\wp{A}$?

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The power set is one whose elements are sets themselves. –  Josué Molina Oct 30 '12 at 17:02
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Indeed, $\{x\} \subseteq A$, while $\{x\} \in \wp{A}$ –  amWhy Oct 30 '12 at 17:07
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4 Answers

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,

$$\underbrace{\{\{\{\dots\{}_nx\underbrace{\}\dots\}\}\}}_n\in\underbrace{\wp(\wp(\wp(\dots\wp(}_nA\underbrace{)\dots)))}_n\;.$$

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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 $\{2\}\in\wp(A)$.

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+1. I think an example is clearest here. It should also be noted that we could write $\{2\} \subseteq A$ and $\{\{2\}\} \subseteq \mathcal{P}(A)$. –  JavaMan Oct 30 '12 at 18:29
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Saying that $B \subset A$ is the same as saying that $B \in \wp{A}$. In your case it means that $\left\{x\right\} \in \wp{A}$.

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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.

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