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Math Help - 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 A \subseteq \mathcal{P}(A), Prove that \mathcal{P}(A) \subseteq \mathcal{P}(\mathcal{P}A).

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

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