Principle of Inclusion and Exclusion Notation

Question

Let $A,B,$ and $C$ be sets such that

$A\subset C, B\subset C$, with $|C|=n$, $|A|=x$, $|B|=y$, and $|A\cap B|=z$.

What is $|C\setminus(A\cup B)|$?

Answer 1

Using Inclusion-Exclusion:

$$ |A\cup B|=|A|+|B|-|A\cap B|=x+y-z. $$

Now, $C\setminus (A\cup B)$ is the set containing all things in $C$ that are not in the set $A\cup B$. Since $C$ contains $n$ elements, and $A\cup B$ contains $x+y-z$ elements, it follows that

$$ |C\setminus (A\cup B)|=n-(x+y-z)=n+z-x-y. $$