Three-Set Venn Diagrams

Fill a three-circle Venn diagram from the middle out, use an unknown x when one region is missing, then read off any count.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 30 real past-paper questions below

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What a Three-Set Venn Diagram Shows

Three circles overlap inside a box. The box is everyone in the question. The circles cut it into 8 regions.

Region	Holds
The centre	in all three sets
Three lenses	in exactly two sets
Three outer parts	in one set only
Outside the circles	in none of the sets

Rule

Every person goes in exactly one region. A circle's total is the sum of the 4 regions inside it.

Filling From the Middle Out

The 3-mark question: you get totals for each set, each pair, and all three. Start in the centre and work outwards.

50 students: 20 study French, 18 German, 16 Spanish. 8 study French and German, 7 German and Spanish, 5 French and Spanish, and 3 study all three.

Worked example

1
$all three = 3$ — Start in the centre. The 3 who study all three belong in one region only, so they go in first.
2
$F, G only = 8 - 3 = 5$ — The 8 who study French and German include the 3 in the centre, so take them away.
3
$G, S only = 7 - 3 = 4$ — The 7 who study German and Spanish also include the centre, so take 3 away again.
4
$F, S only = 5 - 3 = 2$ — Same reason: every pair total includes the centre.
5
$F only = 20 - 5 - 2 - 3 = 10$ — French's 20 includes its two lenses (5 and 2) and the centre, so take all three away.
6
$G only = 18 - 5 - 4 - 3 = 6$ — Take away German's two lenses (5 and 4) and the centre.
7
$S only = 16 - 2 - 4 - 3 = 7$ — Take away Spanish's two lenses (2 and 4) and the centre.
8
$10 + 6 + 7 + 5 + 4 + 2 + 3 = 37$ — Add the 7 regions inside the circles. 37 students study at least one language.
9
$none = 50 - 37 = 13$ — The rest of the 50 study none, so 13 go outside the circles.

Common mistake

Writing 8 in the French and German lens. The 8 includes the 3 who study all three, so the lens is 5.

Key idea

Marks come for each correct region, so fill in every region you can. If a region has nobody in it, write 0 rather than leaving it blank.

Reading a Three-Set Diagram

The 1 or 2-mark question: how many are in some part of the diagram. Exams often write it in symbols: $\cap$ means and, $\cup$ means or, and a dash ′ means not. Find the regions, then add them. These use the fifty students from before.

Steps

1
$n (F \cap G)$ : the lens and the centre: $5 + 3 = 8$
2
$n (F \cap G \cap S^{'})$ , French and German but not Spanish: the lens only: $5$
3
Exactly two languages: the three lenses: $5 + 4 + 2 = 11$
4
$n (F^{'} \cap G^{'})$ , neither French nor German: Spanish only and outside: $7 + 13 = 20$
5
$n (G^{'} \cap S)$ , Spanish but not German: Spanish only and the French and Spanish lens: $7 + 2 = 9$
6
$n (F \cup G)$ , French or German or both: every region in those two circles: $10 + 6 + 5 + 2 + 4 + 3 = 30$

Common mistake

Forgetting the centre. People who study all three also study French and German, so $F \cap G$ includes them.

Placing Numbers in Three Sets

The 3-mark question with numbers, not counts: you get $E$ and three rules. Write each set out, then test every number against all three rules before you place it.

$E = {1, 2, \dots, 12}$ , A = odd numbers, B = prime numbers, C = numbers less than 7.

Worked example

1
$A = {1, 3, 5, 7, 9, 11}$ — Write each set out in full first, so you can check every number against it.
2
$B = {2, 3, 5, 7, 11}$ — The primes up to 12. Remember 1 is not prime.
3
$C = {1, 2, 3, 4, 5, 6}$ — Less than 7 means 6 is the biggest.
4
$A \cap B \cap C = {3, 5}$ — In all three lists, so they go in the centre.
5
$A \cap B only = {7, 11}$ — Odd and prime, but not less than 7, so they sit in the A and B lens.
6
$A \cap C only = {1}$ — 1 is odd and less than 7, but it is not prime.
7
$B \cap C only = {2}$ — 2 is prime and less than 7, but it is even, so it is not in A.
8
$A only = {9}$ — 9 is odd, but it is not prime and not less than 7.
9
$B only = \emptyset$ — Every prime is already placed, so this region stays empty.
10
$C only = {4, 6}$ — Less than 7, but even and not prime.
11
$outside = {8, 10, 12}$ — These pass none of the rules, so they go outside the circles but inside the box.
12
$2 + 2 + 1 + 1 + 1 + 0 + 2 + 3 = 12$ — Check: all 12 numbers are placed, each one exactly once.

Common mistake

Putting 3 in the A and B lens. 3 is also less than 7, so it is in all three sets and goes in the centre. Test every number against all three rules.

When One Region Is x

The 3 + 3 mark question: one region is missing, so call it $x$ . Fill the rest, add every region, and make it equal the total.

65 people. 2 do all three sports. 6 swim and play tennis, 9 play tennis and hockey, 5 swim and play hockey. 20 play hockey and 9 do none. Swim only is 3 times tennis only. How many swim?

Worked example

1
$S, T only = 6 - 2 = 4$ — Middle out, as before. The 6 who swim and play tennis include the 2 in the centre, so take 2 away.
2
$T, H only = 9 - 2 = 7$ — Same reason for tennis and hockey.
3
$S, H only = 5 - 2 = 3$ — Same reason for swimming and hockey.
4
$H only = 20 - 7 - 3 - 2 = 8$ — Hockey's 20 includes its two lenses and the centre, so take them away.
5
$T only = x$ — Tennis has no total, so call this region x.
6
$S only = 3 x$ — Swim only is 3 times tennis only, so it is 3x.
7
$3 x + x + 4 + 7 + 3 + 2 + 8 + 9 = 65$ — All 8 regions, including the 9 who do none, add up to the 65 people.
8
$4 x + 33 = 65$ — Collect: $3 x + x = 4 x$ and $4 + 7 + 3 + 2 + 8 + 9 = 33$ .
9
$4 x = 32$ — Take 33 from both sides.
10
$x = 8$ — Divide both sides by 4.
11
$S only = 3 \times 8 = 24$ — Put x = 8 back into 3x.
12
$24 + 4 + 3 + 2 = 33$ — How many swim means the whole swim circle: its outer part, two lenses and the centre.

Common mistake

Giving 24 as the answer. The question asks how many swim, so add every region in the swim circle.

When x Is in the Centre

Sometimes the unknown is in the centre. Then each lens becomes its pair total minus x. Fill those in, add every region, and solve.

35 people. $n (A \cap B) = 10$ , $n (A \cap C) = 8$ , $n (B \cap C) = 7$ and x are in all three. The diagram shows A only 6, B only 5, C only 4, and 3 outside. Find x.

Worked example

1
$centre = x$ — The number in all three is the unknown, so x goes in the centre.
2
$A, B only = 10 - x$ — The 10 in A and B include the x in the centre, so take x away.
3
$A, C only = 8 - x$ — Same reason: the 8 includes the centre.
4
$B, C only = 7 - x$ — Same again for B and C.
5
$x + (10 - x) + (8 - x) + (7 - x) + 6 + 5 + 4 + 3 = 35$ — All 8 regions add up to the 35 people.
6
$x - x - x - x = - 2 x$ — Collect the x terms first. There is one +x and three −x, so they leave −2x.
7
$- 2 x + 43 = 35$ — The numbers: $10 + 8 + 7 + 6 + 5 + 4 + 3 = 43$ .
8
$- 2 x = - 8$ — Take 43 from both sides: $35 - 43 = - 8$ .
9
$x = 4$ — Divide both sides by −2. A negative divided by a negative is positive.
10
$10 - 4 = 6, 8 - 4 = 4, 7 - 4 = 3$ — Put x = 4 back into each lens. None is negative, so the answer makes sense.

Common mistake

Writing 10 in the A and B lens. The 10 includes the x people in the centre, so the lens is 10 − x.

Key idea

If a region comes out negative, check your working. A number of people cannot be negative.

Which Method? Read the Question

Look at what the question gives you and what it wants.

You see	You do
Totals for pairs and all three	centre first, then lenses, then outer parts
One region unknown	call it x, add all regions, equal the total
x in the centre	each lens is its pair total minus x
Numbers and three rules	write each set out, test every number
How many in A and B	the lens and the centre
Exactly two	the three lenses, no centre
A total for A ∪ B only	add only the regions inside A or B
Only two circles	Venn Diagrams
What does a symbol mean?	Set Notation

Key idea

Check at the end: all 8 regions must add up to the total in the box.

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