TITLE: Is the empty set a power set?
QUESTION [17 upvotes]: I have some home work with problems such as...
Determine whether each of these sets is the power set of a set:
$ 1. \{\phi, \{a\}\}$
$ 2. \phi$
So yes $1$ is the power set of $\{a\}$.
But what about $2?$ Since power sets have to have $2^k$ members then no it can't be a power set. It would have to be $P(\phi) = \{\phi, \{\phi\}\}$ correct? Since it must contain $2$ members.  However, when I try to check myself with Wolfram Alpha, it says $P(\{\}) = \{\{\}\}...$ why? I thought it must have $2^k$ members!

REPLY [9 votes]: Every powerset must contain at least one member, namely $\emptyset$.  The set $\emptyset$ itself contains no members, so it cannot be the powerset of any set.
The powerset of $\emptyset$ is $\{\emptyset\}$, which has exactly $1 = 2^0$ members.  It is the smallest powerset, in the strict sense that it is a proper subset of every other powerset, all of which contain at least two members.  (Specifically, the powerset of any set $S \ne \emptyset$ has at least $\emptyset$ and $S$ as members.)