Given A={1,2,3,4}, B = {1,2,3} And C={2,4}
Now, P(A)={  \(\phi\), {1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}}  ......(i)
P(B) ={\(\phi\){1},{2},{3},{1,2},{1,3},{1,3},{2,3},{1,2,3}}    ....(ii)
and P(c) ={\(\phi\)),{2},{4},{2,4}}  ...(iii)
Given conditions is X \(\subseteq\) B and X\(\neq\) B and X \(\nsubseteq\) C..
\(\Rightarrow\) X \(\in\) P(B), X \(\neq\) B and X \(\in\) P(c)
\(\therefore\) X ={1},{3},{1,2},{1,3},{2,3}
[using eqs (ii) and (iii)]

Given A={1,2,3,4}, B = {1,2,3} And C={2,4}
Now, P(A)={ \(\phi\), {1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}}  ......(i)
P(B) ={\(\phi\){1},{2},{3},{1,2},{1,3},{1,3},{2,3},{1,2,3}}    ....(ii)
and P(c) ={\(\phi\)),{2},{4},{2,4}}     ...(iii)
Given condion is X \(\subseteq\) P(B) and X  \(\nsubseteq\) C.
\(\Rightarrow\)  X \(\in\) P(B) and X \(\in\) P(c)
\(\therefore\) X= {1},{3},{1,2},{1,3},{2,3}
[using the equation (ii) and (iii)]

here, A ={1,2,3,4} ,B={1,2}
Now,P(A:B)= set of all subsets of A which contain B
={{1,2},{1,2,3},{1,2,4},{1,2,3,4}

We have, P ( \(\phi\) )={\(\phi\) }
\(\therefore\)  P (P(\(\phi\)))={\(\phi\) ,{\(\phi\) }}
\(\Rightarrow\) P[P(P\(\phi\) ))]={\(\phi\) },{\(\phi\) },{{\(\phi\)}},{\(\phi\) ,{\(\phi\) }}}
Hence, number of elements in P [P(P (\(\phi\) ))] is 4
i.e. n {P [P(P (\(\phi\)))]}=4

The value of the function for every rational number is 1 and for every irrational number is -1.
\(\therefore \quad \frac { 1 }{ 2 } \in Q\Rightarrow f\left( \frac { 1 }{ 2 } \right) =1\)
\(and \quad \pi \notin Q\Rightarrow f\left( \pi \right) =-1\)