Venn Diagram Calculator: Fill In Every Region for 2 or 3 Sets
Find the count in every region of a 2- or 3-set Venn diagram from set totals and overlaps: only A, exactly two, all three, none, union and complements.
How it works
|A ∪ B| = |A| + |B| − |A ∩ B| · |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C| · only A = |A| − (A and B only) − (A and C only) − (all three) · none = |U| − |A ∪ B ∪ C|
A Venn diagram of two sets has 3 regions inside the circles (only A, only B, both) plus the area outside them; three sets have 7 inside regions plus the outside. Word problems usually give you the set totals and some overlaps, and ask for one of the regions. The calculator fills every region by working from the centre outwards: first the part in all three sets, then the parts in exactly two, then the parts in only one, and finally the part outside every circle. The link between them is the inclusion–exclusion principle, which adds the set totals and subtracts the overlaps that were counted twice. You can enter what your problem gives: for two sets, the overlap, the union, the number in neither, in only one of them, or in exactly one; for three sets, the centre, the union or the number in none. Choose whether your pair overlaps include the people in all three (the usual case) or are “exactly these two”. If a region comes out negative, the numbers cannot describe real sets and the calculator tells you which region fails. Unlike the probability-of-union page, which works with probabilities of two events, this one works with counts (or percentages) for up to three sets and gives every region and complement.
Worked example
In a class of 43 students, 24 studied with flash cards (A), 14 with notes (B) and 27 by reviewing (C); 7 used flash cards and notes, 12 flash cards and review, 9 notes and review, and 5 used all three. Working from the centre: flash cards and notes only = 7 − 5 = 2, flash cards and review only = 12 − 5 = 7, notes and review only = 9 − 5 = 4. Only flash cards = 24 − 2 − 7 − 5 = 10, only notes = 14 − 2 − 4 − 5 = 3, only review = 27 − 7 − 4 − 5 = 11. The union is 24 + 14 + 27 − 7 − 12 − 9 + 5 = 42, so 43 − 42 = 1 student used none of the three, 24 used exactly one method and 13 used exactly two.
Sources: Omni Calculator: Venn diagram calculator (2-set example U = 50, |A| = |B| = 20, |A ∩ B| = 5) · Foundations of Mathematics 12 (TRU Open Press): Set operations with three sets
Frequently asked questions
How do you find the number in neither set?
Find the union first, |A ∪ B| = |A| + |B| − |A ∩ B|, then subtract it from the total. With 50 people, 20 in A, 20 in B and 5 in both, the union is 35 and 50 − 35 = 15 are in neither.
How do you fill in a 3-circle Venn diagram?
Start in the centre with the number in all three sets. Subtract it from each pairwise overlap to get the “exactly two” regions. Subtract the centre and the two adjacent “exactly two” regions from each set total to get the “only” regions. Finally, subtract everything inside the circles from the total to get the outside region.
What if I know the union but not the centre?
Rearrange inclusion–exclusion: all three = |A ∪ B ∪ C| − |A| − |B| − |C| + |A ∩ B| + |A ∩ C| + |B ∩ C|. Choose “In at least one” above and enter the union; the calculator solves for the centre.
What does “only A” mean in a Venn diagram?
The members of A that are in none of the other sets: the part of circle A that does not overlap any other circle. It is written A ∩ B′ for two sets or A ∩ B′ ∩ C′ for three.
Why do I get a negative region?
The numbers you entered are inconsistent: for example, an overlap larger than one of its sets, or pair overlaps entered as “exactly two” when they actually include the centre. Recheck the problem’s wording and the pair-overlap setting.
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This calculator is for educational purposes. Double-check important results before acting on them.