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.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
| Notation | Meaning | Venn region |
|---|---|---|
| A∩B | Both A and B | Overlap |
| A∪B | A or B or both | Combined circles |
| Ac | Not in A | Outside circle |
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
Venn regions labeled with subsets.
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Set operations on Venn
Operations: Union (∪): everything in either set.
Intersection (∩): only overlap.
Complement (c): outside all given sets.
Worked example
From previous: n(A)=5, n(B)=5, n(A∩B)=3.
Find n(A∪B).
Solution
- n(A∪B)=n(A)+n(B)-n(A∩B)
- n(A∪B)=5+5-3=7
- Venn shows 7 elements in combined circles
Final answer
7 elements in A∪B.
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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
U={1-8}, A={2,4,6,8}, B={2,3,4,5}, C={1,2,3,4}.
Partition into regions.
Solution
- A∩B∩C={2,4}
- A∩B not C={none}
- A∩C not B={6,8}
- B∩C not A={3,5}
- A only={none}
- B only={none}
- C only={1}
Final answer
7 regions identified.
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.
(b) Find the probability that a randomly chosen member attends yoga only.
(c) Find the probability that a member attends yoga, given that they attend pilates.
Step by step
- (a) 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.
- (b) 'Yoga only' is the yoga total minus the overlap, over the whole group.
- (c) Restrict to the 26 pilates members; 11 of them also attend yoga.
Final answer
(a) x = 11. (b) P(yoga only) = 19/50 = 0.38. (c) P(Y | M) = 11/26 = 0.423.