Sets Relations And Functions Question 105
Question: A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product $ P_1 $ and 1450 consumers like product $ P_2. $ What is the least number that must have liked both the products?
Options:
A) 1150
B) 2000
C) 1170
D) 2500
Show Answer
Answer:
Correct Answer: C
Solution:
- [c]Let U be the set of all consumers who were questioned, A be the set of consumers who liked product $ P_1 $ and B be the set of consumers who liked product $ P_2 $ .
It is given that $ n(U)=2000,n(A)=1720,n(B) $
$ =1450,n(A\cup B)=n(A)+n(B)-n(A\cap B) $
$ =1720+1450-n(A\cap B) $
$ =3170-n(A\cap B) $
Since, $ A\cup B\subseteq U\therefore n(A\cup B)\le n(U) $
$ \Rightarrow 3170-n(A\cap B)\le 2000 $
$ \Rightarrow 3170-2000\le n(A\cap B) $
$ \Rightarrow n(A\cap B)\ge 1170 $
Thus, the least value of $ n(A\cap B) $ is 1170.
Hence, the least number of consumers who liked both the products is 1170.
BETA
