$\begingroup$What is the size of a set of sets of the empty set {{}, {{}}, {{{}}}}?
I am not sure if the empty set, in this case, can be considered as a set and make the size 3 or if it is 2 or 0.
thanks
$\endgroup$53 Answers
$\begingroup$Your set can be written as $\{a,b,c\}$ where $a$ happens to be the emptyset, $b$ happens to be the set containing the empty set, and $c$ happens to be the set containing the set containing the emptyset.
You should not have any trouble seeing that $\{a,b,c\}$ is of size three so long as they are all distinct. The only possible source of confusion in this is in recognizing that each of $\emptyset, \{\emptyset\}$ and $\{\{\emptyset\}\}$ are not only valid possible elements of a set, but are distinct and different than one another. Indeed, they are all valid possible elements of sets and are all different than one another. Remember that the "depth" of each is relevant and is part of what makes these distinct.
$\endgroup$9$\begingroup$As you are only interested in the number of elements in the set and not in the elements themself, you basically just have to count the commas, so your set has three elements, so three is its size.
$\endgroup$1$\begingroup$- The empty set is indeed an element of this set.
