Prove the inclusion-exclusion formula We just touched upon the inclusion-exclusion formula and I am confused on how to prove this:
$|A ∪ B ∪ C| =|A| + |B| + |C|
− |A ∩ B| − |A ∩ C| − |B ∩ C|
+ |A ∩ B ∩ C|$
We are given this hint: To do the proof, let’s denote $X = A ∪ B$, then
$|(A ∪ B) ∪ C| = |X ∪ C|$,
and we can apply the usual subtraction rule (you will have to apply it twice).
That just made me even more confused. I was hoping someone can guide me through this, or explain 

$|(A\cup B)\cup C|=|A\cup B|+|C|-|(A\cup B)\cap C|$
Now, $|A\cup B|=|A|+|B|-|A\cap B|$
$$\text{and  }|(A\cup B)\cap C|=|(A\cap C)\cup (B\cap C)|=|(A\cap C)|+|(B\cap C)|-|(A\cap C)\cap (B\cap C)|=|(A\cap C)|+|(B\cap C)|-|A\cap B\cap C|$$
