How do I understand the null set

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?

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

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.
• 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
• @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