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?

Thanks in advance.

  • 4
    $\begingroup$ 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? $\endgroup$ – Brian M. Scott Jul 19 '15 at 5:39
  • $\begingroup$ Related: Cardinality of power set of empty set and [Is a power set null because it contains an empty set?](Is a power set null because it contains an empty set?) $\endgroup$ – Martin Sleziak Jul 19 '15 at 7:27

This is just Brian's words.

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, $\mathscr P(\{\emptyset\})=\{\emptyset, \{\emptyset\}\}$.


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