Venn Diagrams - Visualising Set Relationships
A Venn diagram is a visual tool that uses overlapping circles inside a rectangle to show the relationships between sets. The rectangle represents the universal set; each circle represents a set; and overlapping regions show what the sets share. Venn diagrams bring together everything you have learned about sets, union, intersection, and complement in a single picture.
Reading a Venn Diagram
| Region | Meaning | Set notation |
|---|---|---|
| Inside circle A only | Elements in A but not B | A ∩ B′ |
| Inside circle B only | Elements in B but not A | B ∩ A′ |
| Overlapping region | Elements in both A and B | A ∩ B |
| Inside either circle | Elements in A or B or both | A ∪ B |
| Outside both circles | Elements in neither A nor B | (A ∪ B)′ |
| Outside circle A | Elements not in A | A′ |
Three-Set Venn Diagrams
Three overlapping circles create eight regions. Working from the centre outwards:
- Centre: A ∩ B ∩ C (all three)
- Three pairwise overlaps (minus the centre): A∩B only, A∩C only, B∩C only
- Three single-set regions: A only, B only, C only
- Outside all: (A ∪ B ∪ C)′
Inclusion-Exclusion for Three Sets
|A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|
Worked Examples
A only: {1,2,3}. A ∩ B: {4,5}. B only: {6,7}. Outside: {8,9,10}.
The overlap region (shaded for A ∩ B) contains 4 and 5.
Art only = 25 − 10 = 15. Music only = 20 − 10 = 10. Both = 10.
Total accounted for = 15 + 10 + 10 = 35.
Neither = 40 − 35 = 5 students.
|M ∪ E ∪ S| = 30 + 25 + 20 − 10 − 8 − 7 + 4 = 54 students.
Students studying none = 60 − 54 = 6 students.
|A ∪ B| = 28 + 22 − 12 = 38.
|(A ∪ B)′| = 50 − 38 = 12. (A′ ∩ B′ = (A ∪ B)′ by De Morgan’s law.)
Common Mistakes
| Mistake | Correct approach |
|---|---|
| Placing the “both” value in circle A and again in circle B | The overlap region is counted only once – write “both” in the centre only |
| Forgetting the “outside both circles” region | Always account for elements in neither set; they live in the rectangle outside the circles |
| Confusing A only with A | |A only| = |A| − |A ∩ B|; the circle labelled A includes the overlap |
Key Takeaways
- The rectangle = universal set; circles = individual sets; overlap = intersection.
- Always fill in the Venn diagram from the centre (intersection) outwards.
- |A only| = |A| − |A ∩ B|. Neither = |ξ| − |A ∪ B|.
- Three-set problems use the extended inclusion-exclusion formula.
Practice Questions
- ξ = {1 to 12}, A = {even numbers}, B = {multiples of 3}. Draw a Venn diagram and find A ∩ B, A only, B only, and neither.
- 80 people were surveyed. 45 own a car, 30 own a bike, 15 own both. Find the number who own neither.
- Shade the region representing A′ ∩ B on a two-set Venn diagram. Describe in words what it contains.
- 100 students study at least one of three languages: French (60), Spanish (45), German (30). French and Spanish (20), French and German (15), Spanish and German (10), all three (5). How many study exactly one language?
- Draw a Venn diagram for three sets A, B, C and label all eight regions with their set notation.
You Have Completed the Set Theory Section!
Well done – you have worked through all 6 topics in the Set Theory section. Return to the Resources page to continue your mathematics journey.