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Problem Involving Counting of Elements in Three Sets

  1. Oct 9, 2012 #1
    -The Union of set A, set B and set C has 104 elements.
    -The Union of Set A and B has 51 elements
    -The Union of Set A and C has 84 elements
    -The Union of Set B and C has 97 elements
    -The Intersection of Set A and the Union of Set B and C has 17 elements.
    -Set C has twice as many elements as set B and three times as many elements as Set A.

    How many elements does A have?

    Hint: Take C is 6x and solve for x.

    This is what I did after drawing a Venn diagram:

    A minus B[itex]\cup[/itex]C = 104-97=7
    B minus A[itex]\cup[/itex]C = 104-84=20
    C minus A[itex]\cup[/itex]B = 104-51=53

    If you add up these numbers and subtract them from 104, you'll get that all the intersections of these sets together have 24 elements.

    It is given that A[itex]\cap[/itex](B[itex]\cup[/itex]C) = 17.

    Therefore, the intersection of B and C without whatever is in A, should be 7.

    From this fact the elements of A can be calculated from the fact that A[itex]\cup[/itex]C has 84 elements. To calculate the elements of A take 84 and subtract the elements of C that are not in A, 84 -53 -7= 24

    I didn't use the hint nor did I use the size relationship between the sets, so I am not sure if I did this problem right.

    How could I solve this problem with the hint (solve as an equation of x) and the given relationship between A, B and C?
    Last edited: Oct 9, 2012
  2. jcsd
  3. Oct 9, 2012 #2
  4. Oct 9, 2012 #3
    Sorry, typo.

    Set C is twice as large as B and three times as large as A.
  5. Oct 9, 2012 #4
    You only need to use three of the above 7 conditions to determine the size of A, so hopefully there are some interesting follow-up questions to your problem. Given that unhelpful hint, that does not seem too likely.
  6. Oct 9, 2012 #5
    The hint and the size relationship confused me. I don't see how to solve this problem as an equation of x and given the fact that C is twice as large as B and three times as A, but I guess it is possible. If someone sees it, I am curious how to proceed.

    There is no follow up question to this problem.

    The teacher of this course is a little bit nuts though, on another test he asked to prove that every even integer greater than 4 can be written as the sum of two primes for extra credit.
    Last edited: Oct 9, 2012
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