First slide

Operations on Sets

Question

In a town of 10,000 families, it was found that 40% families buy newspaper A, 20% buy newspaper B and 10% buy newspaper C. Also,5o families buy newspapers A and B, 3% buy newspapers B and C and 4% buy newspapers A and C. lf 2% families buy all the three newspapers, then number of families which buy newspaper A only is

Moderate

A

3100
B

3300
C

2900
D

1400

Solution

$\begin{aligned} n (A) = 40 % of 10, 000 = 4, 000 \\ n (B) = 20 % of 10, 000 = 2, 000 \\ n (C) = 10 % of 10, 000 = 1, 000 \end{aligned}$
$\begin{array}{l} n (A \cap B) = 5 % of 10, 000 = 500 \\ n (B \cap C) = 3 % of 10, 000 = 300 \\ n (C \cap A) = 4 % of 10, 000 = 400 \\ n (A \cap B \cap C) = 2 % of 10, 000 = 200 \end{array}$
Number of families which buy newspaper A only
$\begin{array}{l} = n (A \cap B^{c} \cap C^{c}) \\ = n [A \cap (B \cup C)^{c}] \\ = n (A) - n [A \cap (B \cup C)] \end{array}$
$\begin{array}{l} = n (A) - n [(A \cap B) \cup (A \cap C)] \\ = n (A) - [n (A \cap B) + n (A \cap C) - n (A \cap B \cap C)] \\ = 4000 - [500 + 400 - 200] \end{array}$
$\begin{array}{l} = 4000 - 700 \\ = 3300 \end{array}$

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