# Subsets of the empty set

Having read Velleman's 'How to prove it' I came across a question I am not sure I can answer. He states that the power set of the empty set is equal to a set consisting only of the empty set: $\mathscr P (\emptyset) = \{\emptyset \}.$ That is clear. He then asks what the power set of $\{\emptyset \}$ is.

What is $\mathscr P (\{\emptyset \})$ equal to?

• You simply need to list the subsets of $\{\varnothing\}$. One of them is $\varnothing$, since that’s a subset of every set. Can you find any others? – Brian M. Scott Jul 19 '15 at 5:39
Empty set is a subset of every set. So, $\emptyset\in\mathscr P (\{\emptyset\})$. And for any set $a\in \mathscr P(a)$. But there are no elements of $\{\emptyset\}$ except $\emptyset$. So, $\mathscr P(\{\emptyset\})=\{\emptyset, \{\emptyset\}\}$.