Question

Example: A company studies the product preferences of 20,000 consume. I was found
that each of the products A, B, C was liked by 7020, 6230 and 5980 respectively and all the
products were liked by 1500; Products A and B were liked by 2580, products A and C were
liked by 1200 and products B and C were liked by 1950. Prove that the study results are not
correct.​

Answer 1

To check whether the study results are correct or not, we will use the concept of set theory,

We will take,

A = set of people who like product A

B = set of people who like product B

C = set of people who like product C

Now, the following information can be concluded from the question,

$n(A) = 7020\\n(B) = 6230\\n(C) = 5980$

n(A∩B∩C) = 1500

n(A∩B) = 2580

n(A∩C) = 1200

n(B∩C) = 1950

Now, to prove that study results is correct or not, we will check whether n(A∪B∪C) = 20,000 or not.

So,

=> n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)

=> n(A∪B∪C) = $7020 + 6230 + 5980 - 2580 - 1200 - 1950 + 1500$

=> n(A∪B∪C) = $20730 - 5730$

=> n(A∪B∪C) = $15000$

Since, n(A∪B∪C) ≠ 20,000

Hence, the results of the study are not correct.

Answer 2

Given,

A company studies the product preferences of 20,000 consume. It was found that each of the products A, B, C was liked by 7020, 6230 and 5980 respectively and all the products were liked by 1500; Products A and B were liked by 2580, products A and C were liked by 1200 and products B and C were liked by 1950.

To find,

Prove that the study results are not correct.​

Solution,

To check whether the study results are correct or not, we will use the concept of set theory,

We will take,

A = set of people who like product A

B = set of people who like product B

C = set of people who like product C

Now, the following information can be concluded from the question,

n(A) = 7020

n(B) = 6230

n(C) = 5980

n(A∩B∩C) = 1500

n(A∩B) = 2580

n(A∩C) = 1200

n(B∩C) = 1950

Now, to prove that study results is correct or not, we will check whether n(A∪B∪C) = 20,000 or not.

So,

=> n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)

=> n(A∪B∪C) = 7020 + 6230 + 5980 - 2580 - 1200 - 1950 + 1500

=> n(A∪B∪C) = 19230 - 5730 + 1500

=> n(A∪B∪C) = 15000

Since, n(A∪B∪C) ≠ 20,000

Hence, the results of the study are not correct.