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If $s=\{1,2\}$, then we say that $P(s) = \{\{\},\{1\},\{2\},\{1,2\}\}$.

But the power set is the set of all subsets, of which $\{\}$ is one of them. So why doesn't the power set also include sets such as $\{\{\},1\}$, $\{\{\},2\}$, and $\{\{\},1,2\}$?

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  • $\begingroup$ Because being a member of and being a subset of are two different properties. The common usage using A in S to mean A subset S is why so many modern students are having difficulty with confusing 'in' and 'subset'. Don't use A in S for A subset S! You will have clarity of mind. $\endgroup$ – William Elliot Oct 23 '17 at 1:36
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The empty set is not a member of the set in question. In much the same way, the set is not a member of the set, so the powerset doesn't include $\{2,\{1,2\}\}$

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  • $\begingroup$ I didn't even think of that. Thanks! $\endgroup$ – Michael Kolber Oct 22 '17 at 23:21
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    $\begingroup$ I think another way to put it would be that {{}} isn’t a subset of the set either. $\endgroup$ – Michael Kolber Oct 23 '17 at 2:16

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