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Question on Set Theory











































The problem statement is: "It was found that out of 100 consumers, foll. are the details of preference for 3 brands A,B and C.
(a) Brand A - 50 consumers
(b) Brand B - 40 consumers
(c) Brand C - 30 consumers
(d) Both brands A and B - 20 consumers
(e) Both brands B and C - 15 consumers
(f) Both brands A and C - 10 consumers
Find how many used all the 3 brands A, B and C?"

I used the formula (x = intersection):
n(AUBUC) = n(A) + n(B) + n(C) - n(AxB) - n(AxC) - n(BxC) + n(AxBxC)
=> 100 = 50+40+30-20-15-10+n(AxBxC)
=> n(AxBxC) = 25

Somehow, I am not convinced that the answer is correct, one of the reasons being n(AxBxC) is > n(AxB), n(AxC), n(BxC).

Can you pl. clarify/confirm if my approach and answer are correct?

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Feb 06, 2011
Question on Set Theory
by: Staff


The question:

The problem statement is: "It was found that out of 100 consumers, foll. are the details of preference for 3 brands A,B and C.

(a) Brand A - 50 consumers
(b) Brand B - 40 consumers
(c) Brand C - 30 consumers
(d) Both brands A and B - 20 consumers
(e) Both brands B and C - 15 consumers
(f) Both brands A and C - 10 consumers
Find how many used all the 3 brands A, B and C?"

I used the formula (x = intersection):
n(AUBUC) = n(A) + n(B) + n(C) - n(AxB) - n(AxC) - n(BxC) + n(AxBxC)

=> 100 = 50+40+30-20-15-10+n(AxBxC)

=> n(AxBxC) = 25

Somehow, I am not convinced that the answer is correct, one of the reasons being n(AxBxC) is > n(AxB), n(AxC), n(BxC).

Can you pl. clarify/confirm if my approach and answer are correct?

The answer:

I got the same answer you did when I used 100 consumers. I also agree that this answer makes no sense.


n(A∪B∪C) = nA + nB + nC – n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)

100 = 50 + 40 + 30 – 20 – 10 – 15 + n(A∩B∩C)

100 = 75 + n(A∩B∩C)

25 = n(A∩B∩C)

I think there were less than 100 consumers who choose A, B, or C.

The statement of the problem does not say all 100 consumers were required to choose at least one of the brands: A, B, or C.

Some of the consumers may have chosen nothing.

If this is the case, n(A∪B∪C) < 100


I think there were less than 100 consumers who choose A, B, or C.



If the number of consumers who choose A, B, or C was actually 75, then n(A∪B∪C) = 75. In that case n(A∩B∩C) = 0.

If the number of consumers who choose A, B, or C was actually 85, then n(A∪B∪C) = 85. In that case n(A∩B∩C) = 10.


It looks as if anything outside this range (75 to 85) will not work.


Thanks for writing.


Staff
www.solving-math-problems.com



Apr 26, 2011
Keep up this good work
by: Anonymous

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