Given that \(R = (P \cap Q')'\) , list the elements of R .

For a set S , let \({S^ * }\) denote the set of all subsets of S ,

(i)     find \({P^ * }\) ;

(ii)     find \(n({R^ * })\) .

\(P = \{ 1,{\text{ }}2,{\text{ }}3\} \)

\(Q' = \{ 1,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9\} \)

so \(P \cap Q' = \{ 1,{\text{ }}3\} \)     (M1)(A1)

so \((P \cap Q')' = \{ 2,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9,{\text{ }}10\} \)     A1

[3 marks]

(i)     \({P^ * } = \left\{ {\{ 1\} ,{\text{ }}\{ 2\} ,{\text{ }}\{ 3\} ,{\text{ }}\{ 1,{\text{ }}2\} ,{\text{ }}\{ 2,{\text{ }}3\} ,{\text{ }}\{ 3,{\text{ }}1\} ,{\text{ }}\{ 1,{\text{ }}2,{\text{ }}3),{\text{ }}\emptyset } \right\}\)     A2 

Note: Award A1 if one error, A0 for two or more.

(ii)     \({R^ * }\) contains: the empty set (1 element); sets containing one element (8 elements); sets containing two elements     (M1)

\( = \left( {\begin{array}{*{20}{c}}
  8 \\
  0
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
  8 \\
  1
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
  8 \\
  2
\end{array}} \right) + ...\left( {\begin{array}{*{20}{c}}
  8 \\
  8
\end{array}} \right)\)     (A1)

\( = {2^8}{\text{ }}( = 256)\)     A1 

Note: FT in (ii) applies if no empty set included in (i) and (ii).

[5 marks]

This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.

This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.