let $\displaystyle P={\{\emptyset,1\}}$ be the given set.
What are elements present in the set $\displaystyle P \cup S$, where $\displaystyle s$ is super set of P
$\displaystyle S$ is superset of $\displaystyle P$ iff $\displaystyle S\supset{P}$ so, $\displaystyle P\cup S=S$
Fernando Revilla
I am guessing that there is a translation error here.
Can it be that super set should be power set, $\displaystyle \mathcal{P}(P)$.
If that is correct then $\displaystyle S=\mathcal{P}(P)= \left\{ {\emptyset ,\{ \emptyset \} ,\{ 1\} ,P} \right\}$.
So, I think the question asks for $\displaystyle P\cup \mathcal{P}(P) $ which is a standard question is a basic set theory class.
$\displaystyle {\emptyset} $ is the emptyset itself while $\displaystyle {\{\emptyset\}}$ is the set containing the empty set ... I'm not sure why it lists both seperately however even my texts do the same ... I've learned to just remember it when it comes to the power set.
It is very important to understand that the above two are different. Think about how will you "define" $\displaystyle {\emptyset}$ ?
Your definition should be precise (basically you cannot say that it contanins nothing)
Once you do that you will understand the difference betweem the above.
Alright -
So now can you see the difference between
$\displaystyle {\{\emptyset\}}$
and
$\displaystyle {\emptyset$
How many elements does each have?
PS: Though I am no expert on this subject but I feel $\displaystyle {\emptyset$ has deep rooted significance - for e.g. you can question what do you mean by a set with 0 element? Maybe someone with more relevant knowledge can comment. But the two sets you mentioned are definately very different and it would be a blunder to consider they are same.
A note on why this topic is included in basic set theory.
The set $\displaystyle P=\{\{a\},\{a,b\}\}$ contains two elements.
Both of the elements are sets. Neither a nor b is an element of $\displaystyle P$.
Because $\displaystyle a\ne \{a\}~\&~ b\ne \{a,b\} $
But we use the set $\displaystyle P$ to define the ordered pair $\displaystyle (a,b)$.