Venn Diagrams & Set Notation – Grade 10 IGCSE Maths

Welcome to Venn Diagrams!

Venn diagrams organise information visually and are powerful tools for solving probability problems. In this lesson you'll master set notation, fill in two- and three-set Venn diagrams, and calculate conditional probabilities.

ξ = Universal set | A∩B = Intersection | A∪B = Union | A' = Complement

Cambridge IGCSE Exam Tips: Venn diagram questions are worth 5–10 marks. Always work from the CENTRE OUT when filling in regions. Show all region values clearly.

Two-Set Venn

Notation, filling from data

Three-Set Venn

7 regions, centre-outward method

Probability

P(A|B), independence, mutually exclusive

Examples

6 fully worked examples

Visualiser

Interactive Venn calculator

Practice

25 self-marking questions

Learn 1 — Two-Set Venn Diagrams and Set Notation

The Universal Set ξ

The universal set ξ contains everything in the context of the problem. It is represented by a rectangle in a Venn diagram. All sets A, B are drawn as circles inside this rectangle.

Core Set Notation:
• A∩B (intersection) — elements in both A and B (the overlap region)
• A∪B (union) — elements in either A or B or both
• A' (complement) — elements not in A (everything else in ξ)
• n(A) — number of elements in set A

De Morgan's Laws

(A∩B)' = A'∪B' (A∪B)' = A'∩B'

De Morgan's Law: the complement of an intersection is the union of complements. Think: "NOT (both)" = "NOT A or NOT B".

Filling in a Two-Set Venn Diagram — Work INSIDE OUT

Given n(ξ), n(A), n(B), n(A∩B):

Step 1: Write n(A∩B) in the overlap (centre).
Step 2: A only = n(A) − n(A∩B)
Step 3: B only = n(B) − n(A∩B)
Step 4: Outside = n(ξ) − n(A only) − n(A∩B) − n(B only)

n(A∪B) = n(A) + n(B) − n(A∩B)

Worked Example

50 students: 28 play football (F), 19 play tennis (T), 11 play both.
F∩T = 11 (centre)
F only = 28 − 11 = 17
T only = 19 − 11 = 8
n(F∪T) = 17 + 11 + 8 = 36
Neither = 50 − 36 = 14
Check: 17 + 11 + 8 + 14 = 50 ✓

Always check: all regions must sum to n(ξ). This is your built-in error checker!

Learn 2 — Three-Set Venn Diagrams

The 8 Regions

Three overlapping circles create 7 regions inside the rectangle, plus the outside region (neither A, B, nor C):

Region 1: A∩B∩C (centre — in all three)
Region 2: A∩B only (in A and B, not C)
Region 3: A∩C only (in A and C, not B)
Region 4: B∩C only (in B and C, not A)
Region 5: A only (in A, not B or C)
Region 6: B only (in B, not A or C)
Region 7: C only (in C, not A or B)
Region 8: Outside (none of A, B, C)

Fill from CENTRE OUTWARD — 7-Step Method

Place n(A∩B∩C) in the centre.
A∩B only = n(A∩B) − n(A∩B∩C)
A∩C only = n(A∩C) − n(A∩B∩C)
B∩C only = n(B∩C) − n(A∩B∩C)
A only = n(A) − (A∩B only) − (A∩C only) − n(A∩B∩C)
B only = n(B) − (A∩B only) − (B∩C only) − n(A∩B∩C)
C only = n(C) − (A∩C only) − (B∩C only) − n(A∩B∩C)
Outside = n(ξ) − sum of all 7 inside regions

Worked Example: 40 Students, Three Subjects

n(ξ)=40, n(M)=20, n(E)=18, n(S)=15
n(M∩E)=7, n(M∩S)=5, n(E∩S)=6, n(M∩E∩S)=2

Step 1: Centre = 2
Step 2: M∩E only = 7 − 2 = 5
Step 3: M∩S only = 5 − 2 = 3
Step 4: E∩S only = 6 − 2 = 4
Step 5: M only = 20 − 5 − 3 − 2 = 10
Step 6: E only = 18 − 5 − 4 − 2 = 7
Step 7: S only = 15 − 3 − 4 − 2 = 6
Sum inside: 10+5+3+2+7+4+6 = 37
Outside = 40 − 37 = 3 ✓

After filling in, always sum all 8 values. They must equal n(ξ). If they don't — recheck your subtraction steps.

Learn 3 — Probability from Venn Diagrams

Basic Probability from Venn Data

P(A) = n(A) / n(ξ) P(A∩B) = n(A∩B) / n(ξ) P(A') = 1 − P(A)

P(A∪B) = P(A) + P(B) − P(A∩B)

Conditional Probability

P(A|B) means "the probability of A, given that B has already occurred." The sample space is restricted to B only.

P(A|B) = n(A∩B) / n(B) = P(A∩B) / P(B)

Example: From the 50-student diagram (F only=17, F∩T=11, T only=8, neither=14).
P(F|T) = n(F∩T)/n(T) = 11/19
Interpretation: "Of those who play tennis, 11 out of 19 also play football."

Independence

A and B are independent if knowing B happened does not change the probability of A.

Independent ⟺ P(A∩B) = P(A) × P(B)

Equivalently: P(A|B) = P(A). Test: calculate both sides and check equality.

Mutually Exclusive

A and B are mutually exclusive if they cannot both occur — they share no elements.

Mutually exclusive ⟺ A∩B = ∅ ⟺ P(A∩B) = 0 ⟺ P(A∪B) = P(A) + P(B)

Key Distinction:
• Mutually exclusive: if one happens, the other CANNOT happen (they share no outcomes).
• Independent: one happening does NOT affect the probability of the other.
Two events that are mutually exclusive with non-zero probability are NEVER independent (because if A happens, P(B|A) = 0 ≠ P(B)).

Example 1 — Fill 2-Set Venn and Find Probabilities

In a class of 30: 18 like art (A), 12 like music (M), 7 like both. Find P(A only), P(A∪M), P(A'∩M).

A∩M = 7 (given). A only = 18−7 = 11. M only = 12−7 = 5. Neither = 30−11−7−5 = 7.

P(A only) = 11/30. P(A∪M) = (11+7+5)/30 = 23/30. P(A'∩M) = M only / 30 = 5/30 = 1/6.

Example 2 — Fill 3-Set Venn (40 Students)

n(ξ)=40, n(M)=20, n(E)=18, n(S)=15, n(M∩E)=7, n(M∩S)=5, n(E∩S)=6, n(M∩E∩S)=2.

Centre=2. M∩E only=5, M∩S only=3, E∩S only=4.

M only=10, E only=7, S only=6. Outside=40−37=3. All regions: 10,5,3,2,7,4,6,3 → sum=40 ✓

Example 3 — Conditional Probability P(A|B)

Using the 30-student art/music Venn. Find P(A|M) — probability likes art given likes music.

P(A|M) = n(A∩M)/n(M) = 7/12. Interpretation: of the 12 who like music, 7 also like art.

Example 4 — Test for Independence

In the 30-student example: n(ξ)=30, n(A)=18, n(M)=12, n(A∩M)=7. Are A and M independent?

P(A)×P(M) = (18/30)×(12/30) = 216/900 = 6/25 = 0.24

P(A∩M) = 7/30 ≈ 0.233. Since 7/30 ≠ 6/25, A and M are NOT independent.

Example 5 — Describe Shaded Region Using Set Notation

A shaded region covers A only (inside A, outside B) in a two-set Venn. What is the notation?

The region in A but NOT in B = A∩B'. Alternatively written as A\B (A minus B).

Other common regions: B only = B∩A'. Outside both = (A∪B)' = A'∩B'.

Example 6 — 3-Set Word Problem with Probability

From the 40-student example. A student is chosen at random. Find: (a) P(M∩E∩S), (b) P(studies exactly one subject), (c) P(M|E).

(a) P(M∩E∩S) = 2/40 = 1/20

(b) Exactly one = M only + E only + S only = 10+7+6 = 23. P = 23/40.

(c) P(M|E) = n(M∩E)/n(E) = 7/18.

Common Mistakes to Avoid

Mistake 1 — Three-set: not starting from the centre.
If you try to fill A∩B before accounting for A∩B∩C, you'll overcount the centre. Always place n(A∩B∩C) first, then subtract it from each pairwise intersection.

Mistake 2 — Wrong denominator for P(A|B).
P(A|B) = n(A∩B) / n(B), NOT n(A∩B) / n(ξ). Given B has occurred, the sample space is restricted to B.

Mistake 3 — Confusing mutually exclusive with independent.
Mutually exclusive: P(A∩B)=0 — they can't happen together.
Independent: P(A∩B) = P(A)×P(B) — one doesn't affect the other.
Non-zero mutually exclusive events are NEVER independent.

Mistake 4 — Complement formula wrong: P(A') = P(A) − 1.
WRONG. The correct formula is P(A') = 1 − P(A). Probability cannot be negative!

Mistake 5 — Double-counting in P(A∪B).
P(A∪B) = P(A) + P(B) − P(A∩B). If you forget to subtract P(A∩B), you count the overlap twice.

Key Formulas — Venn Diagrams & Set Notation

Formula	Description
n(A∪B) = n(A) + n(B) − n(A∩B)	Inclusion-exclusion for two sets
P(A') = 1 − P(A)	Complement rule
P(A∪B) = P(A) + P(B) − P(A∩B)	Addition rule
P(A\|B) = n(A∩B) / n(B)	Conditional probability
P(A\|B) = P(A∩B) / P(B)	Conditional probability (probability form)
P(A∩B) = P(A) × P(B)	Independence test
P(A∩B) = 0	Mutually exclusive
(A∩B)' = A'∪B'	De Morgan's Law 1
(A∪B)' = A'∩B'	De Morgan's Law 2

Two-Set Filling Method

A only = n(A)−n(A∩B) | B only = n(B)−n(A∩B) | Neither = n(ξ)−n(A∪B)

Three-Set Filling Method (Centre Outward)

1. Centre = n(A∩B∩C)
2. A∩B only = n(A∩B) − centre
3. A∩C only = n(A∩C) − centre
4. B∩C only = n(B∩C) − centre
5. A only = n(A) − (A∩B only) − (A∩C only) − centre
6. B only = n(B) − (A∩B only) − (B∩C only) − centre
7. C only = n(C) − (A∩C only) − (B∩C only) − centre
8. Outside = n(ξ) − (sum of all 7 inside regions)

Interactive Venn Diagram Calculator

Enter values for your two-set Venn diagram. The calculator will fill all regions, draw the diagram, and compute key probabilities.

n(ξ): n(A): n(B): n(A∩B):

Click "Draw Venn" to see results.

Exercise 1 — Two-Set Venn Fill & Compute

Q1. n(ξ)=40, n(A)=22, n(B)=17, n(A∩B)=9. Find n(A only).

Q2. Using Q1 data, find n(B only).

Q3. Using Q1 data, find n(neither A nor B).

Q4. Using Q1 data, find P(A∪B) as a fraction over 40.

/ 40

Q5. n(ξ)=60, n(X)=35, n(Y)=28, n(X∪Y)=50. Find n(X∩Y).

Exercise 2 — Three-Set Venn Fill

Use this data: n(ξ)=50, n(A)=25, n(B)=20, n(C)=18, n(A∩B)=8, n(A∩C)=6, n(B∩C)=7, n(A∩B∩C)=3.

Q1. Find n(A∩B only) — in A and B but not C.

Q2. Find n(A∩C only) — in A and C but not B.

Q3. Find n(B∩C only) — in B and C but not A.

Q4. Find n(A only).

Q5. Find n(outside) — in none of A, B, C.

Exercise 3 — Probabilities from Venn Diagrams

n(ξ)=80, n(A)=45, n(B)=38, n(A∩B)=20. Give answers as fractions with denominator 80 (numerators only).

Q1. Find P(A) — enter numerator (denominator is 80).

/ 80

Q2. Find n(A∪B).

Q3. Find P(A') — enter numerator (denominator 80).

/ 80

Q4. Find n(A∩B') — in A but not B.

Q5. Find P(A∪B) — enter numerator (denominator 80).

/ 80

Exercise 4 — Conditional Probability

n(ξ)=60, n(P)=30, n(Q)=24, n(P∩Q)=12. Give answers as decimals to 3 d.p.

Q1. Find P(P|Q). Give as a decimal.

Q2. Find P(Q|P). Give as a decimal.

Q3. Find P(P)×P(Q). Give as a decimal.

Q4. Find P(P∩Q). Give as a decimal.

Q5. Are P and Q independent? Enter 1 for Yes, 0 for No.

Exercise 5 — Independence and Mutual Exclusivity Tests

Q1. n(ξ)=100, n(A)=40, n(B)=25, n(A∩B)=10. Is P(A∩B) = P(A)×P(B)? Enter 1 for Yes, 0 for No.

Q2. P(A)=0.4, P(B)=0.3, and A,B are independent. Find P(A∩B) × 100 (enter as integer).

Q3. If A and B are mutually exclusive, P(A)=0.3, P(B)=0.5. Find P(A∪B) × 10 (enter as integer).

Q4. P(A∩B)=0.15, P(B)=0.5. Find P(A|B) × 100 (enter as integer).

Q5. Events A and B are mutually exclusive with P(A)>0 and P(B)>0. Are they independent? Enter 1 for Yes, 0 for No.

Practice — 25 Questions

Mixed questions on set notation, Venn filling, and probability. Decimals to 3 d.p. where needed.

1. n(ξ)=30, n(A)=16, n(B)=14, n(A∩B)=6. Find n(A only).

2. Same data as Q1. Find n(B only).

3. Same data as Q1. Find n(neither).

4. n(ξ)=50, n(A)=20, n(B)=18, n(A∩B)=8. Find n(A∪B).

5. n(ξ)=60, n(X)=40, n(Y)=30, n(X∩Y)=15. Find n(outside both).

6. n(ξ)=40, n(A)=24, n(B)=16, n(A∩B)=8. Find P(A). Enter as decimal to 3 d.p.

7. Same as Q6. Find P(A∩B). Decimal to 3 d.p.

8. Same as Q6. Find P(A∪B). Decimal to 3 d.p.

9. Same as Q6. Find P(A'). Decimal to 3 d.p.

10. Same as Q6. Find P((A∪B)'). Decimal to 3 d.p.

11. Same as Q6. Find P(A|B). Decimal to 3 d.p.

12. Same as Q6. Find P(B|A). Decimal to 3 d.p.

13. P(A)=0.6, P(B)=0.4, A and B are independent. Find P(A∩B). Decimal to 3 d.p.

14. P(A)=0.5, P(B)=0.3, A,B mutually exclusive. Find P(A∪B). Decimal to 3 d.p.

15. P(A∩B)=0.2, P(B)=0.5. Find P(A|B). Decimal to 3 d.p.

16. n(ξ)=60, n(A)=30, n(B)=25, n(C)=20, n(A∩B)=10, n(A∩C)=8, n(B∩C)=9, n(A∩B∩C)=4. Find n(A∩B only).

17. Same as Q16. Find n(A only).

18. Same as Q16. Find n(C only).

19. Same as Q16. Find n(outside).

20. Same as Q16. Find P(A∩B∩C). Decimal to 3 d.p.

21. n(ξ)=80, n(A)=50, n(B)=40, n(A∩B)=25. Find P(A|B). Decimal to 3 d.p.

22. n(ξ)=100, n(A)=60, n(B)=50, n(A∩B)=30. Is A independent of B? (1=yes, 0=no)

23. P(A)=0.7, P(A') = ? Decimal to 3 d.p.

24. n(ξ)=45, n(A)=25, n(B)=20, n(A∩B)=10. Find n(A'∩B').

25. P(A)=0.4, P(B)=0.3, P(A∩B)=0.12. Find P(A∪B). Decimal to 3 d.p.

Challenge — 12 Questions

Harder multi-step problems. Decimals to 3 d.p. unless stated.

1. n(ξ)=120, n(A)=70, n(B)=55, n(A∩B)=35. Find P(A'∩B'). Decimal to 3 d.p.

2. In Q1, are A and B independent? (1=yes, 0=no)

3. P(A)=0.55, P(B)=0.40, P(A|B)=0.55. Find P(A∩B). Decimal to 3 d.p.

4. In Q3, are A and B independent? (1=yes, 0=no)

5. n(ξ)=80, n(A)=48, n(B)=36, n(C)=24, n(A∩B)=14, n(A∩C)=10, n(B∩C)=8, n(A∩B∩C)=4. Find n(exactly one of A,B,C).

6. Same as Q5. Find P(B|A∩C). Decimal to 3 d.p.

7. P(A∪B)=0.75, P(A)=0.5, P(B)=0.4. Find P(A∩B). Decimal to 3 d.p.

8. 200 people: 120 speak English (E), 90 speak French (F), 40 speak both. Find P(E|F). Decimal to 3 d.p.

9. Same as Q8. Find P(E'∩F'). Decimal to 3 d.p.

10. P(A)=0.6, P(B)=0.5. If A and B are mutually exclusive, find P(A∪B). Decimal to 3 d.p.

11. From Q5 data. Find P(A only). Decimal to 3 d.p.

12. P(A∩B')=0.3, P(A)=0.5. Find P(A∩B). Decimal to 3 d.p.

Exam Style — 5 Questions

IGCSE-style questions. Show all working in an exam — here, just enter final answers.

Q1 [4 marks]. In a survey of 100 students: 62 study Biology (B), 48 study Chemistry (C), and 25 study both. A student is chosen at random. Find P(B∪C). Decimal to 3 d.p.

Q2 [3 marks]. Using Q1 data, find P(B|C). Decimal to 3 d.p.

Q3 [4 marks]. 60 students; 35 play Sport (S), 28 play an Instrument (I), 15 play both. How many play neither? Enter as integer.

Q4 [4 marks]. n(ξ)=80, n(P)=50, n(Q)=36, n(P∩Q)=18. Are P and Q independent? (1=yes, 0=no)

Q5 [5 marks]. In a three-set Venn: n(ξ)=50, n(A)=22, n(B)=18, n(C)=14, n(A∩B)=6, n(A∩C)=5, n(B∩C)=4, n(A∩B∩C)=2. Find P(exactly one set). Decimal to 3 d.p.