TITLE: Is a power set null because it contains an empty set?
QUESTION [7 upvotes]: All power sets contain an empty set, so does this make the power set itself an empty set? Or am I mistaken?

REPLY [3 votes]: An empty set contains nothing. A power set of some set $A$, $\mathscr{P}(A)$  always contains something, at least $A$ itself, thus $\mathscr{P}(A)$ is not empty. 
The reason I decided to add my two cents is to point out to the frequent source of confusion which is associating empty set with zero, and then associating zero with "nothing". Then, as the (faulty) reasoning goes, the set that contains only the empty set contains nothing, and therefore it is the empty set. The major cause of such confusion is mostly due to notation: designating the empty set as $\emptyset$, which looks almost like $0$. 
My rule of thumb: if in doubt, replace $\emptyset$ with $\{ \, \}$ and then think of $\{ \, \}$ as a container that contains nothing. Empty container. Clearly, the empty set. But, say, we have some other set $A$ that contains only the empty set. Again, $A$ is a container, but now it contains another container that contains nothing: $\{ \ \{ \, \} \ \}$. Clearly, a set $A$ contains something, so it's not empty. However writing the same thing as $\{ \ \emptyset \ \}$ may cause some confusion to those exploring the set theory: is that zero the set  $A$ contains? Looks like zero, anyway! But zero is nothing! So the set contains nothing. So it is the empty set.