Venn diagram basics
What is Venn diagram: Visual representation of sets and relationships.
Rectangle=universal set.
Circles=subsets.
Overlaps show intersection.
Fill the overlap (A∩B) first, then 'only A' and 'only B', then 'neither' — every region adds up to the total.
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| Notation | Region | In the diagram (A football, B basketball) |
|---|---|---|
| A ∩ B | Both: the overlap | 6 |
| A ∪ B | A or B or both: inside either circle | 13 + 6 + 4 = 23 |
| A′ | Not in A: B only and the outside | 4 + 7 = 11 |
| (A ∪ B)′ | Neither: outside both circles | 7 (or 30 − 23) |
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Set operations on Venn
Worked example
Universal set U={1-10}.
A={even}={2,4,6,8,10}.
B={>5}={6,7,8,9,10}.
Show on Venn.
Solution
- A∩B={6,8,10} (even AND >5) in overlap
- A only={2,4} (even but ≤5)
- B only={7,9} (>5 but odd)
- Outside both={1,3,5} (odd and ≤5)
Final answer
Overlap: 6, 8, 10. A only: 2, 4. B only: 7, 9. Outside: 1, 3, 5.
Operations: In the 1 to 10 example (A even, B greater than 5):
Union A ∪ B: inside A or B or both. Not the outside. 2, 4, 6, 7, 8, 9, 10.
Intersection A ∩ B: only the overlap. 6, 8, 10.
Complement A′: everything not in A, so B only and the outside. 1, 3, 5, 7, 9.
Outside both circles: (A ∪ B)′. 1, 3, 5.
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Three-set Venn diagrams
Three circles: More complex: 7 regions.
Central: A∩B∩C.
Pairwise overlaps: A∩B (not C), etc. Singles and outside.
Worked example
50 students were asked which clubs they go to: music (M), drama (D) and sport (S).
20 go to music, 18 to drama and 21 to sport. 8 go to music and drama, 5 to music and sport, 7 to drama and sport, and 3 go to all three.
Draw a Venn diagram to show this.
Solution
- Centre: all three = 3.
- Pairs only: 8 − 3 = 5, 5 − 3 = 2, 7 − 3 = 4.
- Music only = 20 − 3 − 5 − 2 = 10. Drama only = 18 − 3 − 5 − 4 = 6. Sport only = 21 − 3 − 2 − 4 = 12.
- Inside the circles: 3 + 5 + 2 + 4 + 10 + 6 + 12 = 42. Outside: 50 − 42 = 8.
Final answer
All three 3; M and D only 5; M and S only 2; D and S only 4; M only 10; D only 6; S only 12; none 8.
Probability from Venn diagrams
Worked example
Venn with |A|=30, |B|=25, |A∩B|=10, |outside|=20.
Total 65 people.
Find P(A), P(B), P(A∩B).
Solution
- P(A)=30/65=6/13
- P(B)=25/65=5/13
- P(A∩B)=10/65=2/13
- Verify: P(A∪B)=(30+25-10)/65=45/65=9/13
Final answer
Probabilities extracted from counts in regions.
IB-style question — Venn with an unknown [6 marks]
A gym has 50 members. 30 attend yoga (Y) and 26 attend pilates (M). 5 members attend neither class. Let x be the number of members who attend both classes.
(a) Find the value of x.
Step by step
- The members attending at least one class is the total minus those attending neither.
- Apply the union rule with the overlap as x.
- Solve for x.
Final answer
x = 11.
Part (b)
(b) Find the probability that a randomly chosen member attends yoga only.
Step by step
- 'Yoga only' is the yoga total minus the overlap, over the whole group.
Final answer
P(yoga only) = 19/50 = 0.38.
Part (c)
(c) Find the probability that a member attends yoga, given that they attend pilates.
Step by step
- Restrict to the 26 pilates members; 11 of them also attend yoga.
Final answer
P(Y | M) = 11/26 = 0.423.
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Filling a Venn diagram from the words
Overlap first, then subtract: "62% have a car" counts everyone in the car circle, the overlap too. Put the overlap in first, then take it away from each circle total.
Tap each number in order: overlap, each circle only, then neither. Percentages work the same way, with a total of 100.
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Percentages in the regions
In a town, 62% of households own a car, 35% own a bike and 20% own both.
(a) Find the percentage of households in each region of the Venn diagram.
Solution
- Both: 20%.
- Car only: 62 − 20 = 42%.
- Bike only: 35 − 20 = 15%.
- Neither: 100 − (42 + 20 + 15) = 23%.
Final answer
42%, 20%, 15%, 23%.
Part (b)
(b) Find the percentage of households that own a car or a bike or both.
Solution
- "Or both" = everything inside the circles.
Final answer
77%.
Exam tip: Adding 62 + 35 = 97 counts the overlap twice. Subtract it once: 62 + 35 − 20 = 77.
Three-set Venn diagrams from totals
Centre, pairs, singles, outside: Exam questions give totals that INCLUDE the inner regions. Work from the centre outwards: every region is a given total minus what is already inside it.
Tap each number in order: the centre, each pair only, each subject only, then none.
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Art, Music and Drama
100 students were asked which of Art (A), Music (M) and Drama (D) they take. 40 take Art, 35 take Music, 28 take Drama. 14 take Art and Music, 10 take Art and Drama, 9 take Music and Drama, and 6 take all three.
(a) Draw a Venn diagram to show this information.
Solution
- Centre: all three = 6.
- Pairs only: A and M only = 14 − 6 = 8; A and D only = 10 − 6 = 4; M and D only = 9 − 6 = 3.
- Singles: A only = 40 − 6 − 8 − 4 = 22; M only = 35 − 6 − 8 − 3 = 18; D only = 28 − 6 − 4 − 3 = 15.
Final answer
Regions: centre 6; pairs 8, 4, 3; singles 22, 18, 15; outside 24.
Part (b)
(b) Find the number of students who take none of the three.
Solution
- Inside the circles: 6 + 8 + 4 + 3 + 22 + 18 + 15 = 76. None = 100 − 76 = 24.
Final answer
24.
Part (c)
(c) Find the number of students who take exactly one of the three.
Solution
- Exactly one = the three single regions: 22 + 18 + 15 = 55.
Final answer
55.
Exactly two?
The three pair-only regions.
Example: 8 + 4 + 3 = 15.
At least two?
Pair-only regions plus the centre.
Example: 15 + 6 = 21.
Sometimes the question gives a region, not a total
If it says "8 take Art and Music but not Drama", that number goes straight into the region. Read "only" and "but not" carefully.
Exam tip: Draw the rectangle, three labelled circles and every number: each group of correct regions is a mark.
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Unknown regions: write one equation
All the regions add to the total: When regions are given in terms of x, add every region, set the sum equal to the total, and solve.
Regions in terms of x
In a Venn diagram of 60 people, the regions are: A only = 3x, A and B = x, B only = 12, neither = 8.
(a) Find x.
Solution
- Add every region and set it equal to the total.
- Collect the x terms and solve.
Final answer
x = 10.
Part (b)
(b) Find the number of people in A.
Solution
- A = A only + the overlap.
Final answer
40 people.
Exam tip: Check: put x back into every region; the regions must add to the total.
An unknown Venn region from independence
Use whole events: A region is not the same as an event. Add every region inside A to get P(A), and every region inside B to get P(B). Then use independence.
Find the missing region
A Venn diagram has P(A only) = x, P(A ∩ B) = 0.12 and P(B only) = 0.18. A and B are independent. Find x and the probability of neither event.
Step by step
- Find the probabilities of the whole events.
- Independence makes the overlap equal to the product.
- Divide by 0.30, then subtract 0.12.
- The four disjoint regions add to 1.
Final answer
x = 0.28; neither = 0.42.
After solving: 28% + 12% + 18% + 42% = 100%.
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