Question

Determine whether each of these sets is the power set of a set, where a and b are distinct elements.

Answer 1

A. $\varnothing$ cannot be the power set of any set. Consult Cantor's theorem, which says that the cardinality of the power set of any set (even the empty set) is strictly greater than the cardinality of the set.

(No part b?)

c. Also not the power set of any set, because any power set must have $2^n$ elements, where $n$ is the cardinality of the original set. The cardinality of this set is 3, but there is no integer $n$ such that $2^n=3$. This set would be a power set if $\{\varnothing\}$ (that is, the set containing the empty set) were a member of it.