Two circles, four regions
A Venn diagram draws each set as a circle inside a box, the universal set. Two overlapping circles cut the box into four regions: inside A only, inside both, inside B only, and outside both. The regions never overlap, so every element of the universal set lies in exactly one of them.
In the diagram below, 5 elements are in A only, 3 in both, 4 in B only and 2 in neither: 5 + 3 + 4 + 2 = 14 in all. The whole of A is A only together with the overlap, so n(A) = 5 + 3 = 8, and n(B) = 4 + 3 = 7.
The four regions of two sets: 5 in A only, 3 in both, 4 in B only and 2 outside both circles.
Why each circle doubles the count
An element is either in A or not, and either in B or not. That is 2 × 2 = 4 combinations, and each combination is one region. A third circle C splits every one of those regions into a part in C and a part not in C, so three sets make 2 × 2 × 2 = 8 regions.
The region outside every circle is one of them. Counting only the regions inside the circles gives 3 for two sets and 7 for three.
The eight regions of three sets
With three sets A, B and C the eight regions are: in all three, the middle; in exactly two, which is A and B only, A and C only, or B and C only; in exactly one, which is A only, B only or C only; and in none.
In set notation the middle is , "A and B only" is , "A only" is , and "none" is .
Three circles inside a box. Count the pieces: one middle, three pieces in exactly two circles, three in exactly one, and the outside, 8 in all.
Filling a three-set diagram from the totals
A survey of 60 people asks which of tea, coffee and juice they drink. 30 drink tea, 25 coffee and 20 juice. 10 drink tea and coffee, 8 tea and juice, and 6 coffee and juice. 3 drink all three.
Start in the middle: 3. Each pair total includes those 3, so the regions for exactly two drinks are 10 − 3 = 7 for tea and coffee, 8 − 3 = 5 for tea and juice, and 6 − 3 = 3 for coffee and juice.
Now each circle. Take away everything already written inside it: tea only is 30 − 7 − 5 − 3 = 15, coffee only is 25 − 7 − 3 − 3 = 12, and juice only is 20 − 5 − 3 − 3 = 9.
The seven regions inside the circles add to 15 + 12 + 9 + 7 + 5 + 3 + 3 = 54, so 60 − 54 = 6 people drink none of the three.
The survey of 60, filled from the middle outward: 3 drink all three, 7, 5 and 3 drink exactly two, 15, 12 and 9 exactly one, and 6 none.
Why the middle comes first
"10 drink tea and coffee" counts everyone in both circles, and that includes the 3 who also drink juice. The tea-and-coffee overlap is made of two pieces: 7 who drink tea and coffee but not juice, and the 3 in the middle.
Writing 10 in the "tea and coffee only" region counts those 3 people twice, once there and once in the middle. Filling the middle first, then subtracting it from each pair, then subtracting everything from each circle, keeps every person in exactly one region.
Tea coffee shaded: the 7 who drink tea and coffee only, and the 3 in the middle, 10 in all.
A check by inclusion-exclusion
Add the three totals, take off the three pair totals, and add back the middle: 30 + 25 + 20 − 10 − 8 − 6 + 3 = 54, the number inside at least one circle, which agrees with the regions.
The middle is added back because each of those 3 people was counted three times in the circle totals and taken off three times with the pairs, leaving them not counted at all.
Four sets
Four sets need regions. Four circles cannot make them all: however they are placed, four circles cut a box into at most 14 regions, so a Venn diagram of four sets is drawn with ovals or other shapes instead.
The usual mistakes
Leaving out the region outside the circles. Two sets make 4 regions, not 3, and three sets make 8, not 7.
Writing a pair total in the "exactly two" region. Subtract the middle first.
Writing a set total in its "only" region. Tea only is what is left of the 30 after the overlaps are taken away.
Forgetting to check that every region adds back to the universal set.
Three languages
In the application below, the number of students studying all three languages is not given. It is found from the totals by inclusion-exclusion, and then the diagram is filled from the middle outward.
Worked example: A Survey of Three Languages, Where the Students Taking All Three Are Found from the Totals
Question A school surveyed 120 students about the languages they study. 50 study French, 40 study German and 45 study Spanish. 15 study both French and German, 18 study both French and Spanish, and 12 study both German and Spanish. 24 students study none of the three languages. (a) How many students study all three languages? (b) How many students study exactly one of the three languages?
1.Let x = n(F ∩ G ∩ S), the number who study all three languages. The students who study at least one language are n(F ∪ G ∪ S) = 120 − 24 = 96.
The students who study at least one language are n(F ∪ G ∪ S) = 120 − 24 = 96. 2.Write the union by inclusion-exclusion: 96 = 50 + 40 + 45 − 15 − 18 − 12 + x.
Inclusion-exclusion: add the three totals, take off the three pair totals, and add back the center x. 3.Simplify the right-hand side: 96 = 135 − 45 + x = 90 + x, so x = 6. (a) 6 students study all three languages.
(a) 96 = 90 + x, so x = 6 students study all three languages. 4.Put 6 in the center of the Venn diagram. Each pair total includes the center, so the regions for exactly two languages are 15 − 6 = 9 (French and German), 18 − 6 = 12 (French and Spanish) and 12 − 6 = 6 (German and Spanish).
Each pair total includes the center, so take 6 off each: 9, 12 and 6. 5.Take everything else in each circle away from its total. French only: 50 − 9 − 12 − 6 = 23. German only: 40 − 9 − 6 − 6 = 19. Spanish only: 45 − 12 − 6 − 6 = 21.
Take the other regions in each circle away from its total: 23, 19 and 21. 6.(b) Exactly one language: 23 + 19 + 21 = 63 students. Check: the seven regions add up to 63 + 9 + 12 + 6 + 6 = 96, the number who study at least one language.
(b) Exactly one language: 23 + 19 + 21 = 63 students.
Answer: (a) 6 students; (b) 63 students
Common mistakes
- Adding the three totals and taking off the three pair totals, 135 − 45 = 90, and stopping there. The students in all three sets were counted three times and then taken off three times, so they are not counted at all; the +x puts them back.
- Writing 15 in the French and German region of the diagram. The 15 students who study French and German include the 6 who also study Spanish, so the region for those two languages only holds 15 − 6 = 9.