Is there a simpler way to solve this Venn diagram algebra problem?

In summary, the conversation discusses using the Inclusion-Exclusion Principle to solve a problem involving algebra. The identity is n(A ∪ B ∪ C) = n(A) + n(B) + n(C) -n(A ∩ B) -n(A ∩ C) -n(B ∩ C) + n(A ∩ B ∩ C). The person was looking for other simplifications that would have pointed to needing the + n(A ∩ B ∩ C) term and mentions that this could be the point of the problem to teach a new identity. They are then reminded that the correct term is + n(A ∩ B ∩ C), not just + A ∩ (B ∩ C
  • #1
chris2020
9
0

Homework Statement


I solved the problem myself but i have a question about the algebra

Homework Equations


n(A ∪ B ∪ C) = n(A) + n(B) + n(C) -n(A ∩ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)
= n(A ∪ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)

The Attempt at a Solution


I knew i needed n(A ∪ B ∪ C) and that the book had:

n(A ∪ B) = n(A) + n(B) -n(A ∩ B)

you can see that was the only simplification I had made, but was there any other simplifications that would have pointed to needing the + A ∩ (B ∩ C) term? are there some identities here that would have lead to that conclusion without needing to see the diagram and think about it? maybe that was the point of this problem to teach a new identity?
 
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  • #3
andrewkirk said:
Yes. The identity you are searching for is the Inclusion-Exclusion Principle.
That is exactly what I was looking for, thanks andrew!
 
  • #4
chris2020 said:

Homework Statement


I solved the problem myself but i have a question about the algebra

Homework Equations


n(A ∪ B ∪ C) = n(A) + n(B) + n(C) -n(A ∩ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)
= n(A ∪ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C)

The Attempt at a Solution


I knew i needed n(A ∪ B ∪ C) and that the book had:

n(A ∪ B) = n(A) + n(B) -n(A ∩ B)

you can see that was the only simplification I had made, but was there any other simplifications that would have pointed to needing the + A ∩ (B ∩ C) term? are there some identities here that would have lead to that conclusion without needing to see the diagram and think about it? maybe that was the point of this problem to teach a new identity?

Your last term should be ##+n(A \cap B \cap c)##, not just the ##+ A \cap (B \cap C)## that you wrote (which, incidentally, can be written without parentheses as ##A \cap B \cap C##).
 

FAQ: Is there a simpler way to solve this Venn diagram algebra problem?

1. What is a 3 set Venn diagram algebra?

A 3 set Venn diagram algebra is a visual representation of the relationships between three different sets of data. It is used to show the similarities, differences, and intersections between the sets.

2. How do you create a 3 set Venn diagram algebra?

To create a 3 set Venn diagram algebra, you will need three overlapping circles. Each circle represents a different set of data. The overlapping regions between the circles represent the intersections between the sets. You can use different colors or shading to represent each set and its intersections.

3. What is the purpose of using a 3 set Venn diagram algebra?

The purpose of using a 3 set Venn diagram algebra is to visually represent the relationships between three sets of data. It can help to identify commonalities, differences, and intersections between the sets, making it easier to draw conclusions and analyze the data.

4. What is the difference between a 3 set Venn diagram algebra and a traditional Venn diagram?

A 3 set Venn diagram algebra is an extension of a traditional Venn diagram, which only represents the relationships between two sets. A 3 set Venn diagram algebra allows for the comparison of three sets of data, showing more complex relationships and intersections between the sets.

5. How can a 3 set Venn diagram algebra be used in algebraic equations?

In algebra, a 3 set Venn diagram algebra can be used to represent different algebraic equations by assigning each set to a variable. The intersections between the sets then represent the solutions to the equations. This can be helpful in solving complex algebraic problems and visualizing the solutions.

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