Solving Union/Intersection Problem: 554 is Absurd!

[AxiomOfChoice]

If I know:

$|A \cup B \cup C|$ = 1000
$|A|$ = 344
$|B|$ = 572
$|C|$ = 296
$|A \cap B|$ = 301
$|B \cap C|$ = 252
$|A \cap C|$ = 213

and I use the standard formula to compute $|A \cap B \cap C|$, 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...

 

[mathman]

How did you get 554?

 

[AxiomOfChoice]

How did you get 554?

Using the following formula:

$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$$

 

[HallsofIvy]

Using the following formula:

$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$$

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!

 

[AxiomOfChoice]

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?

 

[mathman]

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.