Power Set/Examples/Nested Sets of Empty Sets

Example of Power Set

Let $\O$ denote the empty set.

Let $S$ be the set defined as:

$S = \set {\O, \set \O, \set {\set \O} }$

Then the power set of $S$ is:

$\powerset S = \set {\O, \set \O, \set {\set \O}, \set {\set {\set \O} }, \set {\O, \set \O}, \set {\O, \set {\set \O} }, \set {\set \O, \set {\set \O} }, \set {\O, \set \O, \set {\set \O} } }$

and so has $2^3 = 8$ elements.

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 2$. Sets of sets: Exercise $2$