Given that sets A, B, C are subsets of a finite set S prove that

$\displaystyle \left | A\cup B\cup \bar{C} \right | =\left | S \right | -\left | C \right |+\left | A\cap C \right | +\left | B\cap C \right | -\left | A\cap B\cap C \right |$

So I started of throwing a dummy element into the sets that I will keep track of:

$\displaystyle Let \; x\in A\cup B\cup \bar{C}$

$\displaystyle \Rightarrow x\in A \; or \; x\in B \; or \; x\notin C$

Then I list out all the possible cases:

Since in all the cases, the cardinality are equal to 1, can I say that hence the theorem is proven?