Let's suppose an object X is in A, B, and C.
Since you are counting A, B, and C serparately, you've counted X three times.
But then, you subtract everything in A & B, B & C, and A & C. X is in each of those, so you've subtracted X three times. That means you haven't actually counted X.
So, you need to add X (and every other object in A, B, and C) to get the correct total, because otherwise you aren't counting it.
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