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Thread: Subset of a Power set

  1. #1
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    Subset of a Power set

    I am a little confused by this question: Suppose $\displaystyle A \subseteq \mathcal{P}(A)$, Prove that $\displaystyle \mathcal{P}(A) \subseteq \mathcal{P}(\mathcal{P}A)$.

    By definition, I know that $\displaystyle A\in \mathcal{P}(A)$ but not $\displaystyle A \subseteq \mathcal{P}(A)$. I believe only when $\displaystyle A=\varnothing$ that it can be a subset of $\displaystyle \mathcal{P}(A)$. I am right?
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  2. #2
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    Quote Originally Posted by novice View Post
    By definition, I know that $\displaystyle A\in \mathcal{P}(A)$ but not $\displaystyle A \subseteq \mathcal{P}(A)$.
    In general, it is not the case that $\displaystyle A \subseteq \mathcal{P}(A)$; however, it is possible. Such sets are called transitive. Suppose $\displaystyle A\subseteq\mathcal{P}(A)$. Then for every $\displaystyle x\in A$, $\displaystyle x\in\mathcal{P}(A)$, i.e., $\displaystyle x\subseteq A$. This in turn means that for every $\displaystyle y\in x$, $\displaystyle y\in A$, which explains the name.

    Quote Originally Posted by novice View Post
    I believe only when $\displaystyle A=\varnothing$ that it can be a subset of $\displaystyle \mathcal{P}(A)$. I am right?
    Other examples include $\displaystyle \{\emptyset,\{\emptyset\}\}$, $\displaystyle \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\ }\}\}$ and other von Neumann ordinals.
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