Solving Union/Intersection Problem: 554 is Absurd!

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Discussion Overview

The discussion revolves around a problem involving the union and intersection of three sets, specifically addressing the calculation of the intersection of those sets based on provided cardinalities. Participants are examining the validity of the initial data and the resulting calculations.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a calculation leading to the conclusion that |A ∩ B ∩ C| equals 554, questioning the consistency of the initial data.
  • Another participant asks for clarification on how the figure of 554 was derived, referencing the standard formula for the union of three sets.
  • Several participants point out potential arithmetic errors in the calculations, suggesting that the result of 554 does not align with the expected outcome based on the provided data.
  • One participant asserts that the total of |A|, |B|, and |C| exceeds the total for |A ∪ B ∪ C|, indicating that the data must be incorrect due to the impossibility of the overlaps as stated.
  • A minor correction is noted regarding the arithmetic in the equation, but it is claimed that this does not affect the overall analysis of the data's validity.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the correctness of the calculations and the validity of the initial data provided.

Contextual Notes

There are unresolved issues regarding the assumptions made in the calculations and the definitions of the sets involved, as well as discrepancies in the arithmetic leading to the conclusion of 554.

AxiomOfChoice
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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...
 
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How did you get 554?
 
mathman said:
How did you get 554?

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]
 
AxiomOfChoice said:
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]

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!
 
HallsofIvy said:
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!

...are you quite sure what you wrote is correct?
 
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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