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Example 32 - A market research reported that 720 consumers like

Example 32 - Chapter 1 Class 11 Sets - Part 2


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Example 32 A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B, what is the least number that must have liked both products? Let set of consumers who liked the product A & product B be A & B respectively Number of consumers who liked product A = n(A) = 720, Number of consumers who liked product B = n(B) = 450 We know that n (A ∪ B) = n (A) + n (B) – n (A ∩ B) n (A ∪ B) = 720 + 450 – n (A ∩ B) n (A ∪ B) = 1170 – n(A ∩ B) n (A ∩ B) = 1170 – n(A ∪ B) Therefore, n (A ∩ B) is least when n (A ∪ B) is maximum Number of people who liked product A or B (i.e. n (A ∪ B)) is maximum when it is equal to number of people surveyed So, maximum value of n (A ∪ B) is 1000. Thus, the least value of n (A ∩ B ) is n ( A ∩ B ) = 1170 – n(A ∪ B) = 1170 – 1000 = 170 Hence, the least number of consumers who liked both products is 170. Example 33 Out of 500 car owners investigated, 400 owned car A and 200 owned car B, 50 owned both A and B cars. Is this data correct? Let the set of owners of car A & car B be A & B respectively Number of car A owned = n(A) = 400 Number of car B owned = n(B) = 200 Number of car A and car B owned = n(A ∩ B) = 50 We know that n (A ∪ B) = n (A) + n (B) – n (A ∩ B) = 200 + 400 – 50 = 550 So, number of people who owned either car A or car B = 550 But total number of car owners is only 500 (given in question) Number of people who own car A or car B should be less than total number of car owners investigated But, here 550 > 500 This is a contradiction. So, the given data is incorrect.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.