How the answer of this question is 4? I'm learning Sets and at current about Power Set. I'm confused in a example given in my book. Please have a look at the image. 

I didn't get the last two lines of the solution. Please help. Thankyou in advance.
Probably the clustering of empty set symbols and nested set braces is the most confusing here.


The empty set has only one subset: $\emptyset$ itself, hence $P(\emptyset)$ is a one-element set with $\emptyset$ as only element.
For any one-element set $X=\{a\}$ we have $P(X)=\{\emptyset,X\}$ as the empty set and the full set itself are the only subsets of $X$. Hence $P(P(\emptyset))$ is a two-element set.
For any two-element set $X=\{a,b\}$ (note that $a\ne b$), there are precisely four subsets of $X$, namely $\emptyset$ and $\{a\}$ and $\{b\}$ and $\{a,b\}$. Hence $P(P(P(\emptyset)))$ is a four-element set.
If you recall that $a=\emptyset$ and $b=\{\emptyset\}$ we can expand the enumaration of $P(P(P(\emptyset)))$:
$$ P(P(P(\emptyset)))=\{\emptyset,\{a\},\{b\},\{a,b\}\}=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\},\{\emptyset,\{\emptyset\}\}\}$$




If you want so experience an even more confusing notation, use $\{\}$ for $\emptyset$: 
$$ P(P(P(\{\})))=\{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\}\}$$
think $a=\phi ,b=\{\phi \}$, then $P[\{a,b\}]=\{\phi,\{a\},\{b\},\{a,b\}\}$