Let P(S) denotes the power set of set S. Which of the following is always true?
a) P(P(S)) = P(S)
b) P(S) ∩ P(P(S)) = { Ø }
c) P(S) ∩ S = P(S)
d) S ∉ P(S)
My approach:-
S = {1},
P(S) = { {}, {1}},
P(P(S)) = { {} , {{}} , {{1}} , {{},{1}} }
So with this approach I can eliminate option a and c.Option d is also false as set always an element of its power set.I think b is also false ,but i am getting confused in b option.
Generally we say that intersection of two sets is a set.And intersection of two disjoint sets is Ø (empty set).
Now P(S) ∩ P(P(S)) has {} in common ,which is same as Ø. But what is confusing me is we say intersection is a set having common elements,Now as my common elements are Ø so my intersection result is a set having Ø i.e {Ø}.These are not disjoint sets as they have something in common that is Ø .So is the representation of { Ø } is correct to represent the intersection as the Ø is common or will it be Ø.
Best Answer
The assertion of $(b)$ is also false, unless some restrictions are imposed on $S$. If not, we can take $S = \{a, \{a\}\}$; then, \begin{eqnarray} \mathcal{P}(S) &&=&& \{\emptyset, \{a\}, \{\{a\}\}, \{a, \{a\}\}\} \\ \mathcal{P}(\mathcal{P}(S)) &&=&& \{\emptyset, \{\{a\}\}, {}\dots \} \end{eqnarray}
and so $\mathcal{P}(S) \cap \mathcal{P}(\mathcal{P}(S)) \supset \{\emptyset, \{\{a\}\}\}$.
After looking over the question you showed us, it seems to me either that the answer is "none," or else that the question as written is ill-posed.