I am wondering why the answer is not "Two":
Formulating the problem as per my understanding
$\rightarrow count(subsets(P(Ø)))$
$∵$ "The power set of the empty set is
the set containing only the empty set"
$\rightarrow count(subsets(\{Ø\}))$
$∵$ "[… $|S| = n$] the number of all the subsets of $S$ is
$|P(S)| = 2^n$"
$\rightarrow 2^1$
($n=1$, not zero, because the set $\{Ø\}$ contains a single element, namely, $Ø$)
Basic exponentiation
$\rightarrow 2$
$\rightarrow$ "Two"
$2^{2^0}$ would have been a much concise mode of conveying the method I used to obtain the answer above, but I considered emphasizing the steps in that procedure as it may increase the chances of finding a mistake, if it exists — a quick googling revealed many answer keys pointing to the answer "One":
I have cited all these sources (inside Markdown) so as to satisfy the relevant licensing obligations to the best of my knowledge.
One of such explanations particularly caught my attention:
Upon scrutinizing that justification, I am not sure whether or not to conclude that the author has misinterpreted "has" as in "comprise" because if the objective was explicitly not to find the number of subsets of the resultant powerset as a characteristic possession but to obtain the count of sets comprising the powerset, then the question might have been something appropriately phrased like (in increasing order of precision based on my limited interpretation of the intended question):
- Powerset of an empty set has exactly _____ set(s)
- How many subsets does the powerset of an empty set contain?
- How many sets does the powerset of an empty set contain?
Source: Google Dictionary
It could perhaps be the case that I am misinterpreting or overlooking words (e.g. "exactly" hinting at proper subsets?) or missing something within the context of Set Theory since I am not more formally qualified than a student pursuing these only as a complementary subject.
The intent of my post is not to assault their answer keys (also why I did not noticeably render hyperlinks to any of those sites) but to achieve a comprehensive analysis of both the answers through the Math SE community, even if either/all of it is incorrect, in which case, indicating the mistake(s) would be more helpful to arrive at the correct answer, within and/or outside the restricted options.
So, in general, how to fill in that blank technically?
Best Answer
The question is badly (!terribly!) worded.
It is asking how many elements $P(\emptyset)$ has. And $P(\emptyset)$ has $2^0 =1$ element.
As every element of $P(A)$ is a subset of $A$ then the question thinks it is asking. "Consider $P(A)$; how many subsets of $A$ are there that are elements of $P(A)$". And the answer to that is $2^{|A|}$.
But by saying "subset" is not clear whether it is asking how many subset of $A$ are elements of $P(A)$ (which would be $2^{|A|}$) or whether it is asking how many subset of $P(A)$ are there (which would not be elements of $P(A)$ but elements of $P(P(A))$. If interpreted this way the answer is $2^{|P(A)|} = 2^{2^{|A|}}$.
$\emptyset$ has exactly one subset (of $\emptyset$). It is $\emptyset$.
So $P(\emptyset) = \{\emptyset\}$. And $P(\emptyset)$ has one element. Its one element is a subset of the emptyset. So $P(\emptyset)$ has one subset of the emptyset as an element.
But $P(\emptyset) = \{\emptyset\}$ has two subsets of itself. They are $\emptyset \subset \{\emptyset\}= P(\emptyset)$; and $P(\emptyset) = \{\emptyset\}\subset \{\emptyset\} = P(\emptyset)$.
......
So to my mind I would think the correct answer to "The power set of the empty set has exactly ------- subsets"; I'd assume the question is asking how many subsets of itself exist. And the answer to that is clearly: $2$.
But that is not what the question intended. The question intended "The power set of the empty set has exactly ------ subsets (of the empty set) as elements". And the answer to that is $1$.
In my opinion the fault is entirely on the authors for writing a !terribly! worded question.
......
(Or possibly the authors of the answer key were not the people who wrote the questions. And the authors of the answer key didn't know the answer and they misread the question and gave the wrong answer.)
Anyway....
if a set $A$ has $|A|$ elements then it has $2^{|A|}$ subsets. So $P(A)$ which is, by definition, is the set of all subsets, will have $2^{|A|}$ elements.
So $\emptyset$ has $0$ elements so it has $2^0 =1$ subset.
So $P(\emptyset)$ has $1$ element so it has $2^1 = 2$ subsets.
I think you've got that (thoroughly by this point I hope) so I'll go and bury this dead horse.