We have:
- $U = 150$
- $|M| = 90$
- $|E| = 75$
- $|F| = 80$
- $|M ∩ E| = 45$
- $|M ∩ F| = 35$
- $|M ∩ F ∩ E| = 10$
We want:
- $|E ∩ F ∩ M^c| = ?$
I know I have to use the exclusion-inclusion principle. I have to take the last statement and transform it so that it is similar to:
- $|A' ∪ B'|$
But I don't know how transform it; that complementary is problematic. I have tried playing with the complementary, but I get this:
$|E ∩ F ∩ M^c| = |(E∪F∪M^c)^c|$
But that is not useful, right?
I think this properties coud be useful too:
- $A∩U =A$
- $A∩∅=∅$
- $A∪U=U$
- $A∪∅=A$
I would appreciate some tips.
Edit:
My Solution:
$U = 150 = |E ∪ F ∪ M|$
1 $|M^c ∩ F ∩ E| = |F ∩ E| - |M ∩ F ∩ E|$
$|E ∪ F ∪ M| = |E| + |F| + |M| - (|E ∩ F| + |E ∩ M| + |F ∩ M|) + |E ∩ F ∩ M|$
$150 = 75 + 80 + 90 - |E ∩ F| - 45 - 35 + 10$
$|E ∩ F| = 25$
So, back to 1:
$|M^c ∩ F ∩ E| = |F ∩ E| - |M ∩ F ∩ E| = 25 - 10 = 15 $