Question 11: A shopping mall has 150 employees lablled 1 to 150, representing the Universal set U. The empolyees fall into the following categories:
* Set A: 40 empolyees with a salary range of 30k-45k, labelled from 50 to 89.
* Set B: 50 employees with a salary range of 50k-80k, labelled from 101 to 150.
* Set C: 60 empolyees with a salary range of 100k-150k, labelled from 1 to 49 and 90 to 100.
\((b)\) \(\;\) Find \(n\{A\cap (B^{c}\cap C^{c})\}\)
Solution:
\(U=\{1,2,3,\ldots,150\}\)
\(A=\{50,51,52,\ldots,89\}\)
\(B=\{101,102,103,\ldots,150\}\)
\(C=\{1,2,3,\ldots,49,90,91,92,\ldots,100\}\)
\({B}’=U-B\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots,150\}\)\(\ -\ \)\(\{101,102,103,\ldots,150\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots,100\}\)
\({C}’=U-C\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots,150\}\)\(\ -\ \)\(\{1,2,3,\ldots,49,90,91,92,\ldots,100\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{50,51,52,\ldots,89,101,102,103,\ldots,150\}\)
\({B}’\cap {C}’=\ \) \(\{1,2,3,\ldots,100\}\)\(\cap\)\(\{50,51,52,\ldots,89,101,102,103,\ldots,150\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{50,51,52,\ldots,89\}\)
\(A\cap ({B}’\cap {C}’)=\ \) \(\{50,51,52,\ldots,89\}\)\(\cap\)\(\{50,51,52,\ldots,89\}\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{50,51,52,\ldots,89\}\)
\(n(A\cap ({B}’\cap {C}’))=40\)