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from my understanding,every set has at least two subsets; the null set and the original set itself.

My question is, what is the power set of the null set? Shouldn't it be just itself?

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The empty set? There's nothing to it... –  JDH Jan 30 '11 at 12:22

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up vote 15 down vote accepted

Every nonempty set has at least two distinct subsets, namely the empty set and the set itself.

However, the empty set has only one subset: itself.

Thus, the power set of the empty set has one element, namely the empty set. That is, $\mathcal{P}(\emptyset) = \{\emptyset\}$.

Notice that the set whose only element is the empty set, $\{\emptyset\}$, is not empty: a bag that has an empty bag inside is not, itself, empty. So the power set of the empty set is not the empty set.

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Pointing out that an empty bag is better than no bag at all reminded me of one of my favorite Futurama quotes: "Yes, all that is and ever shall be is in that box! And the box itself is probably worth something too." Knowing how many Futurama writers have math degrees, there may in fact be a set theory joke in there somewhere... –  Pete L. Clark Jan 30 '11 at 6:33
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I find that for those new to the subject, the emptyset notation $\emptyset$ can be confusing. Or rather, also using $\{\}$ helps. SO then you can always say: $\cal{P}(\{\}) = \{\{\}\}$, so that you see the empty set as an element. (Yes of course you see it your way, but it's hard to realize that.) –  Mitch Jan 31 '11 at 1:00
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@Mitch: Interesting; I find exactly the opposite: multiple brackets just confuses some people. (I have a hard time getting my students to balance the parenthesis, brackets, and curly brackets appropriately, after all; it's hardly a surprise that they seem to have trouble parsing $\{\{\}\}$; perhaps $\Bigl\{\{\}\Bigr\}$ might be somewhat better, but even with variable size I still have a hard time getting them to balance their parentheses). –  Arturo Magidin Jan 31 '11 at 1:05
    
hmmm...maybe you're right (you have more experience). I think I must be projecting my own internal feelings when first learning it, that I felt I understood it when I translated the single $\emptyset$ sign to curly braces. –  Mitch Jan 31 '11 at 18:31

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