Solving Union/Intersection Problem: 554 is Absurd!

  • Thread starter AxiomOfChoice
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In summary, the given data is inconsistent as it does not follow the standard formula for computing the intersection of three sets, resulting in a value of 554 which is impossible to obtain. This suggests that there may be an error in the initial data provided.
  • #1
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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  • #2
How did you get 554?
 
  • #3
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]
 
  • #4
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!
 
  • #5
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?
 
  • #6
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.
 

1. What is the union of two sets in mathematics?

The union of two sets in mathematics is a new set that contains all elements from both original sets. It is denoted by the symbol ∪.

2. What is the intersection of two sets in mathematics?

The intersection of two sets in mathematics is a new set that contains only the elements that are common to both original sets. It is denoted by the symbol ∩.

3. How do you solve a union/intersection problem?

To solve a union/intersection problem, you first need to identify the sets involved and their elements. Then, for a union problem, you combine all the elements from both sets and remove any duplicates. For an intersection problem, you only keep the elements that are common to both sets. Finally, you can express the solution using set notation or a Venn diagram.

4. What does it mean for 554 to be absurd in a union/intersection problem?

If the number 554 is mentioned in a union/intersection problem, it likely means that it is not a valid element in the given sets. In other words, it does not belong in either set and therefore does not affect the solution.

5. Why is the number 554 specifically mentioned in this problem?

The number 554 is likely mentioned in this problem as an example or to demonstrate a specific concept related to union/intersection problems. It is not a significant number in itself and can be replaced with any other number without changing the solution.

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