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

Easy

A

$x = 39$
B

$x = 63$
C

$39 \leq x \leq 63$
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 American be 100.
Then $n (A) = 63, n (B) = 76$
Now, $n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$= 63 + 76 - n (A \cap B)$
$∴ n (A \cup B) + n (A \cap B) = 139$
$\Rightarrow n (A \cap B) = 139 - n (A \cup B)$
But $n (A \cup B) \leq 100$
$∴ - n (A \cup B) \geq - 100$
$∴ 139 - n (A \cup B) \geq 139 - 100 = 39$
$∴ n (A \cap B) \geq 39 i . e ., 39 \leq n (A \cap B) . . . (i)$
Again, $A \cap B \subseteq A, A \cap B \subseteq B$
$∴ n (A \cap B) \leq n (A) = 63 and n (A \cap B) \leq n (B) = 76$
$∴ n (A \cap B) \leq 63 . . . (ii)$
Then, $39 \leq n (A \cap B) \leq 63 \Rightarrow 39 \leq x \leq 63 .$

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