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Is the set $\{1\}$ a member, and not a subset of the power set of $\{1, 2\}$?

I just want to be sure I am not making a mistake here, my reasoning:

A is a subset of B if for every $x \in A$, $x \in B$

However, the element in $\{1\}$ is simply $1$. But the power set of $\{1, 2\}$ is the set $\{\{\},\{1\},\{2\},\{1,2\}\}$ and this power set has no such element, it does however have an element that is the set containing the element $1$. So $\{1\}$ is an element of this power set, but NOT a subset, since the only element in $\{1\}$ does not appear in this power set.

Is my reasoning correct?

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up vote 4 down vote accepted

Yes, you are totally correct.

$\{1\}$ is a member of the power set; $\{\{1\}\}$ is a subset of the power set.

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Thank you for confirming, some of the question get a bit sticky. – Leonardo Jan 15 '13 at 6:13
Yes, I have witnessed my fair share of confusion surrounding, in particular, the empty set, the power set of a nonempty set, and membership vs. being a subset. But writing out your reasoning as you did is a good idea. – Benjamin Dickman Jan 15 '13 at 6:17

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