How to transform a set with complementary so that I can apply Inclusion-exclusion principle 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  $
 A: We know that $E \cap F \cap M^{c} = (E \cap F) - (E \cap F \cap M)$. Thus you need to find $E \cap F$. I think a Venn Diagram will help here since you know the size of $U$ (the universe). Or one could proceed by inclusion-exclusion. We know that $$|E \cup F \cup M| = |E|+|F|+|M|- (|E \cap F|+ |E \cap M|+ |F \cap M|) + |E \cap F \cap M|$$
A: Since you know that $M\cup E\cup F = 150$, the Inclusion-Exclusion Formula gives you
\begin{align*}
150 &= |M\cup E\cup F|\\
&= |M|+|E|+|F|-|M\cap E|-|M\cap F|-|E\cap F|+|M\cap E\cap F|\\
&= 90 + 75 + 80 - 45 - 35 - |E\cap F| + 10\\
&= 175 - |E\cap F|,
\end{align*}
so you know that $|E\cap F|= 25$. 
This gives you all the information you need: For example, to divide $M$ into its four disjoint parts ($M\cap E\cap F$, $M\cap E^c\cap F$, $M\cap E\cap F^c$, and $M\cap E^c\cap F^c$), you use inclusion-exclusion: 


*

*You know that $M$ contains $10$ elements that are also in $E$ and in $F$ (that is, $|M\cap E\cap F| = 10$).

*$M$ contains $35$ elements that are also in $E$ but not in $F$ (because $|M\cap E| = |M\cap E\cap F^c| + |M\cap E\cap F|$, and you know that $|M\cap E| = 45$ and $|M\cap E\cap F|=10$).

*$M$ contains $25$ elements that are also in $F$ but not in $E$ (because $|M\cap E^c\cap F| = |M\cap F| - |M\cap E\cap F|$).

*And $M$ contains $20$ elements that are in $M$ but not in either $E$ nor $F$ (because $|M\cap E^c\cap F^c| = |M| - |M\cap E| - |M\cap F| + |M\cap E\cap F|$).


Similarly, $E$ contains $10$ elements that are also in $M$ and $F$; it contains $35$ that are in $M$ but not in $F$; $15$ that are also in $F$ but not in $M$; and therefore $15$ that in neither $M$ nor $F$.
And $F$ contains $10$ elements that are also in $M$ and $F$; $25$ that are in $M$ but not in $E$; and $15$ that are in $E$ but not in $M$; leaving $30$ that are in $F$ but not in $E$ nor $M$. 
That is:
\begin{align*}
|M\cap E\cap F\;| &= 10;\\
|M\cap E\cap F^c\;| &= 35;\\
|M\cap E^c\cap F\;| &= 25;\\
|M\cap E^c\cap F^c\;| &= 20;\\
|M^c\cap E\cap F\;| &= 15;\\
|M^c\cap E\cap F^c\;| &= 15;\\
|M^c\cap E^c\cap F\;| &= 30;\\
|M^c\cap E^c\cap F^c\;| &= 0.
\end{align*}
