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What am I doing wrong on this simple cardinality of unions/intersections problem?

  1. Jun 22, 2009 #1
    If I know:

    [itex]|A \cup B \cup C|[/itex] = 1000
    [itex]|A|[/itex] = 344
    [itex]|B|[/itex] = 572
    [itex]|C|[/itex] = 296
    [itex]|A \cap B|[/itex] = 301
    [itex]|B \cap C|[/itex] = 252
    [itex]|A \cap C|[/itex] = 213

    and I use the standard formula to compute [itex]|A \cap B \cap C|[/itex], I get 554, which is absurd. Can someone tell me what's wrong here? Is there something inconsistent in the initial data we're given? If so, I can't find it...
  2. jcsd
  3. Jun 22, 2009 #2


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    How did you get 554?
  4. Jun 22, 2009 #3
    Using the following formula:

    [tex]|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|[/tex]
  5. Jun 22, 2009 #4


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    So 1000= 344+ 572+ 296- 301- 572- 296+ x.

    Then you appear to have just done the arithmetic wrong. Solving this equation for x does not give anything like 554!
  6. Jun 22, 2009 #5
    ...are you quite sure what you wrote is correct?
  7. Jun 22, 2009 #6


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    Final analysis: The data is wrong. |A|+|B|+|C|=1212. This allows only 212 for any overlap. Since each of the pairwise intersections is more, this is impossible.

    Note: minor error in the 1000= statement, the -296 should be -213, but it doesn't change the analysis.
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