Note: Actually there's no error in the book and the manual. I actually misread it. The answer is of a different question :
True or False: {0} ⊂ {0}
This question is from Discrete Math Book by Rosen.
{{∅}} ⊂ {∅,{∅}}
Answer in the manual is:
This is false. For one set to be a proper subset of another, the two sets cannot be equal.
How is {{∅}} is equal to the set {∅,{∅}} ? I know the two sets have different cardinal numbers. So they cannot be equal.
Best Answer
Is $\{ \{ \emptyset \} \}$ a subset of $\{ \emptyset, \{ \emptyset \} \}$?
Well, it's equivalent to asking whether $\{ \emptyset \} \in \{ \emptyset, \{ \emptyset \} \}$.
And it obviously is.
The answer in the manual is extremely wrong, if you've quoted it correctly. $\{ \{ \emptyset \} \}$ is not equal to $\{ \emptyset, \{ \emptyset \} \}$. (More easily read, $\{ 5 \}$ is not equal to $\{ \emptyset, 5 \}$.) It is possible there is a typographical error in the book, or that you've mis-read it, or misunderstood it.
Notice that $\{ \{ \emptyset \} \}$ is equal to $\{ \{ \emptyset \}, \{ \emptyset \} \}$, because sets must have distinct elements and we discard duplicates: $\{1, 1 \} = \{ 1 \}$. This is one possible way you might have misread the book, or that the book might have been printed incorrectly.
Note: But make sure what the definition of $\subset$ is! Some use $A \subset B$ to mean any subset (i.e., include $A = B$); others use $A \subseteq B$ for this, in which case they use $A \subset B$ if $A \subseteq B$ but $A \ne B$.