Having trouble understanding adding back the intersection of all elements when trying to get | A U B U C|, A, B, C being sets

Question

According to the principle of inclusion and exclusion, finding the union of 3 sets A,B, and C I first need to add |A| + |B| + |C|, then subtract the intersections of all 3 sets,
(| A intersection B| +| B intersection C| +| A intersection C |), where I am confused is adding back |A intersection B intersection C|, I can't understand why we only add it back once rather than adding it back 3 times. when we subtract all the combinations of intersections we are removing the intersections 3 times, so wouldn't we have to add it back 3 times. I know my understanding is flawed. Please help me understand.

Answer 1

You want the (size of the) intersection $A\cap B\cap C$ to be counted exactly once in total. When you take $\vert A \vert + \vert B \vert + \vert C \vert$, you add it $3$ times. When you subtract $\vert A\cap B \vert + \vert B\cap C \vert + \vert C\cap A \vert$, you subtract it three times, so it's there $0$ times in total, and so you need to add it back $1$ time.