# 3 set Venn diagram algebra

1. Oct 6, 2015

### chris2020

1. The problem statement, all variables and given/known data
I solved the problem myself but i have a question about the algebra

2. Relevant 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)

3. 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?

2. Oct 6, 2015

3. Oct 6, 2015

### chris2020

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

4. Oct 7, 2015

### Ray Vickson

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 $A \cap B \cap C$).