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Math Help - Venn Diagram question.

  1. #1
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    Venn Diagram question.

    Question 2

    A group of 100 students is polled to see how many watched three TV shows, Angie, Bobo
    and Conway. The results showed that 55 watched Angie, 53 watched Bobo, 55 watched
    Conway, 35 watched Angie and Bobo, 28 watched Angie and Conway, 23 watched Bobo and
    Conway, and 10 did not watch any of the three. Let A denote the set of students who
    watched Angie, and similarly define sets B and C.


    (a) Calculate the number of students in each of the eight subsets shown in the Venn diagram. Copy the Venn diagram and enter the number of students in each subset.

    (b) Hence find how many students watched:

    (i) Angie and Conway, but not Bobo;
    (ii) Bobo only;
    (iii) only two of the three shows;
    (iv) at least two of the shows.
    Last edited by mr fantastic; November 20th 2011 at 07:14 PM. Reason: Title.
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  2. #2
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    Re: Venn Diagram question Help! pls

    what we need to do is find the cardinality of the following 8 sets:

    1.A - (BUC) the people who watched "only A"
    2.B - (AUC) (only B)
    3.C - (AUB) (only C)
    4.A∩B - C, the people who watched A and B, but not C.
    5.A∩C - B (A and C, but not B)
    6.B∩C - A (B and C, but not A)
    7.A∩B∩C, people who watched all 3 shows
    8.T - AUBUC, people who didn't watch ANY show (where T is our "total set").

    we are given the following:

    |A| = 55, |B| = 53, |C| = 55, |A∩B| = 35, |A∩C| = 28, |B∩C| = 23, |T| = 100, |T - AUBUC| = 10.

    so we get subset 8, as a "freebie", there are 10 people who didn't watch any of the 3 shows.

    this means that |AUBUC| = 90, 90 people watched at least one of the three shows.

    now |AUBUC| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|. we know all of these numbers except |A∩B∩C|, so we can calculate it:

    90 = 55 + 53 + 55 - 35 - 28 - 23 + |A∩B∩C|
    90 = 163 - 86 + |A∩B∩C|
    90 = 77 + |A∩B∩C|
    13 = |A∩B∩C|, which is subset 7.

    now |A∩B| = |A∩B - C| + |A∩B∩C|, and again, we know 2 of these numbers, so we can solve for the 3rd:

    35 = |A∩B - C| + 13
    22 = |A∩B - C|, which is subset 4. similarly,

    |A∩C| = |A∩C - B| + |A∩B∩C| ---> |A∩C - B| = 15, which is subset 5, and
    |B∩C| = |B∩C - A| + |A∩B∩C| ---> |B∩C - A| = 10, which is subset 6.

    finally |A| = |A - BUC| + |A∩B - C| + |A∩C - B| + |A∩B∩C|, so

    55 = |A - BUC| + 22 + 15 + 13 ---> |A - BUC| = 5, which is subset 1. in a similar fashion:

    |B| = |B - AUC| + |A∩B - C| + |B∩C - A| + |A∩B∩C| ---> |B - AUC| = 8, which is subset 2,
    |C| = |C - AUB| + |A∩C - B| + |B∩C - A| + |A∩B∩C| ---> |C - AUB| = 17, which is subset 3.

    this should enable you to finish the remainder of the equation.
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