Discussion
Let A and B denote the sets of students in a school who play cricket and football respectively. If \(n(A) = 45, n(B) = 35\) and \(n(A \cap B = 13)\), then \(n((A \cap B)' \cap (A \cup B))\) =
Answer: E
The answer is 54.
\((A \cap B)' \cap (A \cup B)\) is the set of students who play at least one game but not both, so it is the union minus the intersection.
\[ \begin{aligned} n(A \cup B) &= 45 + 35 - 13 = 67 \\ n &= 67 - 13 = 54 \end{aligned} \]
Check: 45 − 13 = 32 play only cricket and 35 − 13 = 22 play only football, and 32 + 22 = 54. 67 is the whole union, which is not offered, and none of the other values matches either count.
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