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

1. Jun 22, 2009

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...

2. Jun 22, 2009

mathman

How did you get 554?

3. Jun 22, 2009

AxiomOfChoice

Using the following formula:

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

4. Jun 22, 2009

HallsofIvy

Staff Emeritus
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!

5. Jun 22, 2009

AxiomOfChoice

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

6. Jun 22, 2009

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.