First slide

Operations on Sets

Question

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese
and apples, then a value of x can be

Moderate

A

37
B

67
C

45
D

None of these

Solution

Let A denote the set of Americans who like cheese and let B denote the set of Americans who like apples.
Let population of America be 100.
$\begin{array}{l} Then n (A) = 63, n (B) = 76 \\ Now, n (A \cup B) = n (A) + n (B) - n (A \cap B) \\ = 63 + 76 - n (A \cap B) \\ \Rightarrow n (A \cap B) = 139 - n (A \cup B) \\ But n (A \cup B) \leq 100 \end{array}$
$\begin{array}{l} ∴ 139 - n (A \cup B) \geq 139 - 100 = 39 \\ ∴ n (A \cap B) \geq 39 i.e., 39 \leq n (A \cap B) \\ Again, A \cap B \subseteq A, A \cap B \subseteq B \\ ∴ n (A \cap B) \leq n (A) = 63 and n (A \cap B) \leq n (B) = 76 \\ ∴ n (A \cap B) \leq 63 \\ Then, 39 \leq n (A \cap B) \leq n (B) \Rightarrow 39 \leq x \leq 63 \end{array}$

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