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

[chris2020]

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?

 

[andrewkirk]

Yes. The identity you are searching for is the Inclusion-Exclusion Principle.

 

[chris2020]

Yes. The identity you are searching for is the Inclusion-Exclusion Principle.

That is exactly what I was looking for, thanks andrew!

 

[Ray Vickson]

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 $A \cap B \cap C$).