Venn diagram with 3 intersecting sets

[Darkknightreturning]

The formula for 3 intersecting sets is Total=A+B+C−(AUB+AUC+BUC)+AnBnC + Neither
While solving some questions I came across the formula : Total = A + B + C - (sum of 2-group overlaps) - 2*(all three) + Neither
Is this correct? When 2 group overlaps are considered, aren't we substracting the all 3 intersection area already? $$$$

[Jay@ManhattanReview]

The formula for 3 intersecting sets is Total=A+B+C−(AUB+AUC+BUC)+AnBnC + Neither
While solving some questions I came across the formula : Total = A + B + C - (sum of 2-group overlaps) - 2*(all three) + Neither
Is this correct? When 2 group overlaps are considered, aren't we substracting the all 3 intersection area already? $$$$

Both of them are correct. Let's understand each of them.

1. Total = A + B + C − (AUB + AUC + BUC) + AnBnC + Neither

In the above notations, AnBnC represents 'all three', AUB represents A and C, AUB represents A and C, and BUC represents B and C. Note that AnBnC is a part of all AUB, AUC, and BUC; thus, AnBnC is subtracted 3 times; it's count is missing, we must include it once, thus, AnBnC is added.

2. Total = A + B + C - (Sum of 2-group overlaps) - 2*(all three) + Neither

In the above notations, (Sum of 2-group overlaps) represents ONLY 2-groups or does not include 'all three' groups. Since in A, B and C, 'all three' is counted 3 times, we must subtract it 2 times so as to count it only once.

Hope this helps!

-Jay
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