Looking at sets of numbers to find things like n(A), n(B), n(C).

Question

In a study of factors that influence brand switching by a certain demographic, the following factors were identified as being significant as judged by the responses of the 439 participants in the survey.

1:	Better price
2:	Friend's recommendation
3:	Seeing others use it
4:	Interesting packaging
5:	Buzz—people talking about the brand
6:	An advertisement
7:	Press stories
8:	Entertainer/sports celebrity endorsement

The results of the survey follow.

Factor	1	2	3	4	5	6	7	8
Respondents (%)	68.8	60.2	22.8	20.8	16.6	16.6	14.9	6.7

Let A denote the set of factors that had a response greater than 20%, let B denote the set of factors that had a response between 15% and 25%, and let C denote the set of factors that had a response less than 25%. Find the following.

(a)    n(A), n(B), n(C)

n(A) =
n(B) =
n(C) =

(b)    n(A ∩ B)=

(c)    n(A^c ∩ C)=

(d)    n(A ∩ B^c)=

Answer 1

(a)    A includes the factors 1,2,3 and 4. So n(A)=4,

B includes the factors 3,4, 5 and 6. So n(B) = 4

C includes the factors 3, 4, 5, 6, 7 and 8. So n(C)= 6

(b)    A ∩ B includes the common members of A and B: 3, 4 So n(A ∩ B) = 2

(c)    A^c  includes 5, 6, 7 and 8; A^c ∩ C ={ 5, 6, 7, 8} n(A^c ∩ C)=4

(d)   B^c  = { 1, 2, 7, 8} A ∩ B^c={ 7 ,8} So n(A ∩ B^c)= 2