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I'm trying to review some set theory. The question I'm encountering is "How many elements are in a power set of a set?" I know the answer if my interpretation of the question is correct. If the original set A has n elements, the power set of A will have 2^n new sets within it, but are these sets considered "elements" of the power set, or am I misinterpreting the question? If these sets are not considered elements of the power set, then is the question asking for the total number of elements of all the sets of the power set? I'm not entirely sure.

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    $\begingroup$ Each set is an element. In set-builder, $\mathcal{P}(S) = \{ X : \ X \subset S \}$ where $\mathcal{P}$ is the power set "function." $\endgroup$ – Sean Roberson Aug 24 '18 at 1:17
  • $\begingroup$ Should the subset symbol have an underline for possible equality? $\endgroup$ – Oscar Lanzi Aug 24 '18 at 2:00
  • $\begingroup$ @OscarLanzi likely a notation difference. $\endgroup$ – George V. Williams Aug 24 '18 at 5:34
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Short Answer: Yes

Long Answer:

I know why this is confusing, but always think of it this way: a set can contain any kind of objects, but the term elements exclusively refers to the objects that are members of the set. For example, if I say $x$ is an element of $y$, then $x\in y$, regardless of what $x$ is, even if it is another set.

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    $\begingroup$ Also in ZFC (for example), everything is a set anyways $\endgroup$ – Ashwin Iyengar Aug 24 '18 at 1:55

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