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Math Help - Finite set problem

  1. #1
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    Finite set problem

    80 participants in total, each can belong to any number of a,b,c (it is not stated but I assume they can belong to none of the sets as well).

    |a|=50, |b|=40, |c|=30

    Either one of these two following scenarios is correct, we have to figure out which one:

    (1) |a\cap b|=30, |a \cap c|=20, |b \cap c|=20, |a \cap b \cap c|=10

    (2) |a\cap b|=10, |a \cap c|=10, |b \cap c|=20, |a \cap b \cap c|=5

    I reason that it is number 1 that is correct since |a \cup b \cup c| \leqslant 80 must be true, but this can only be said if they can belong to none as if they can't |a \cup b \cup c|=80 must be true. If they can't belong to none then I don't see this as solvable as both are wrong. I would like for someone to please confirm/contradict my thoughts on this? Have I considered every reasonable way of looking at the scenario? If it is the first, does this mean that 20 people belong to none of the groups?
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  2. #2
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    Your reasoning follows using inclusion-exclusion. I would say that the wording of the problem statement ("each can belong to any number of a, b, c") implies that that a participant can belong to none of them, since zero is a number.
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  3. #3
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    |aUbUc| = |a| + |b| + |c| - |a∩b| - |a∩c| - |b∩c| + |a∩b∩c|.

    (1) implies 80 ≥ 50 + 40 + 30 - 30 - 20 - 20 + 10 = 60, which is true.

    (2) implies 80 ≥ 50 + 40 + 30 - 10 - 10 - 20 + 5 = 85, which is false.

    yes, 20 people belong to none of the 3 sets.
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  4. #4
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    Quote Originally Posted by Deveno View Post
    |aUbUc| = |a| + |b| + |c| - |a∩b| - |a∩c| - |b∩c| + |a∩b∩c|.
    Yes, this is what I was using. Thanks.
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