Edexcel International GCSE Maths Sets and Venn Diagrams Past Paper Questions and Solutions (4MA1 Higher)

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Edexcel International GCSE Maths A (4MA1) · Higher · Numbers and the Number System

Sets and Venn Diagrams: Edexcel 4MA1 Past Paper Questions and Worked Solutions

This page collects every sets and venn diagrams question from the Pearson Edexcel International GCSE Mathematics A (4MA1) Higher tier Paper 1H and Paper 1HR, January 2020 to June 2025: 11 questions worth 39 marks. Questions run from easier to harder. Try each one first, then click “Show solution” for a step-by-step method with the reason for every step. All answers are checked against the official Pearson mark schemes.

14 marksHigherJune 2020 · Paper 1H (4MA1/1H) · Q1

The numbers from 1 to 14 are shown in the Venn diagram. A only contains 8, 14 and 10. A ∩ B contains 2, 4, 6 and 12. B only contains 1 and 3. Outside both sets are 5, 7, 9, 11 and 13.

ℰAB8141026412139131175

(a) List the members of the set A ∩ B (1)

(b) List the members of the set B′ (1)

A number is picked at random from the numbers in the Venn diagram.

(c) Find the probability that this number is in set A but is not in set B. (2)

Show solution

Part (a)

  1. A ∩ B is the overlap of the two circles.

Answer: 2, 4, 6, 12

Part (b)

  1. B′ is everything outside circle B.

Answer: 5, 7, 8, 9, 10, 11, 13, 14

Part (c)

  1. In A but not in B: 8, 10 and 14, which is 3 numbers out of 14.

Answer: 3⁄14

23 marksHigherJune 2023 · Paper 1HR (4MA1/1HR) · Q2

Here is a Venn diagram. Set A contains 7, 11, 13 and 5 (5 is also in B). Set B contains 5, 15 and 10 (10 is also in C). Set C contains 10, 6, 8, 12 and 14. The number 9 is outside all three sets.

ℰABC71113515106812149

(a) Write down the numbers that are in the set (i) A (ii) B ∪ C (2)

Dominic writes down 9 ∉ C

(b) Explain why Dominic is correct. (1)

Show solution

Part (a)(i)

  1. Everything inside circle A, including the overlap with B.

Answer: 5, 7, 11, 13

Part (a)(ii)

  1. Everything inside circle B or circle C. Write each number once.

Answer: 5, 6, 8, 10, 12, 14, 15

Part (b)

  1. ∉ means “is not a member of”.

Answer: 9 is not inside circle C, so 9 is not a member of C

33 marksHigherJanuary 2020 · Paper 1HR (4MA1/1HR) · Q4

B = {b, l, u, e}, G = {g, r, e, y} and W = {w, h, i, t, e}

(a) List all the members of the set (i) B ∪ G (ii) W ∩ G′ (2)

Serena writes down the statement B ∩ G ∩ W = ∅

(b) Is Serena’s statement correct? You must give a reason for your answer. (1)

Show solution

Part (a)(i)

  1. Everything in B or G, each letter once.

Answer: b, l, u, e, g, r, y

Part (a)(ii)

  1. Letters in W that are not in G: remove e.

Answer: w, h, i, t

Part (b)

  1. The letter e is in all three sets, so the intersection is not empty.

Answer: No, because e is in B, G and W

42 marksHigherJanuary 2021 · Paper 1H (4MA1/1H) · Q4

ℰ = {20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, A = {odd numbers}, B = {multiples of 3}. List the members of the set (i) A ∩ B (ii) A ∪ B

Show solution
  1. A = {21, 23, 25, 27, 29} and B = {21, 24, 27}

Part (i)

  1. In both: the odd multiples of 3.

Answer: 21, 27

Part (ii)

  1. In either set, each number once.

Answer: 21, 23, 24, 25, 27, 29

54 marksHigherJune 2025 · Paper 1HR (4MA1/1HR) · Q5

ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}, C = {1, 3, 5, 7, 9, 11}

(a) List the members of the set (i) A ∪ B (ii) B′ (2)

(b) Write a symbol from the box (ℰ, ∩, ∪, ∅, ∈, ∉) on each dotted line to make each statement true: (i) A ∩ C = …… (ii) 13 …… ℰ (2)

Show solution

Part (a)(i)

  1. A ∪ B is everything in A or B (or both). Write each number once.

Answer: 2, 3, 4, 6, 8, 9, 10, 12

Part (a)(ii)

  1. B′ means everything in ℰ that is not in B.

Answer: 1, 2, 4, 5, 7, 8, 10, 11

Part (b)(i)

  1. A is the even numbers and C is the odd numbers, so they have nothing in common. The intersection is the empty set.

Answer: A ∩ C = ∅

Part (b)(ii)

  1. 13 is not in the universal set, so it is not a member of ℰ.

Answer: 13 ∉ ℰ

63 marksHigherJanuary 2022 · Paper 1H (4MA1/1H) · Q7

ℰ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, A ∩ B = {5, 10, 15}, B′ = {7, 8, 9, 11, 12, 13, 14} and A′ = {4, 6, 7, 8, 14}. Complete the Venn diagram for this information.

Show solution
  1. Overlap: A ∩ B = 5, 10, 15
  2. Outside both = numbers in A′ and B′: 7, 8, 14
  3. A only = numbers in B′ but not in A′: 9, 11, 12, 13
  4. B only = numbers in A′ but not in B′: 4, 6
  5. Check: every number from 4 to 15 is used exactly once ✓
ℰAB9, 1112, 135, 10,154, 67, 8, 14

Answer: A only: 9, 11, 12, 13; A ∩ B: 5, 10, 15; B only: 4, 6; outside: 7, 8, 14

75 marksHigherJune 2024 · Paper 1H (4MA1/1H) · Q7

ℰ = {23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, A = {even numbers}, B = {23, 29, 31}, C = {multiples of 3}

(a) List the members of the set

(i) B ∪ C (1)

(ii) A′ ∩ C (1)

(b) Is it true that B ∩ C = ∅? Tick Yes or No and give a reason for your answer. (1)

The set D has 4 members and is such that D ∩ (A ∪ C) = ∅

(c) List the members of set D. (2)

Show solution
  1. First list C, the multiples of 3 in ℰ: C = {24, 27, 30, 33}

Part (a)(i)

  1. ∪ (union) means everything that is in B or C (or both).

Answer: 23, 24, 27, 29, 30, 31, 33

Part (a)(ii)

  1. A′ means “not in A“, so the odd numbers. ∩ means in both sets.
  2. So we need the odd multiples of 3.

Answer: 27, 33

Part (b)

  1. B ∩ C = ∅ means B and C have no members in common.
  2. 23, 29 and 31 are all prime, so none of them is a multiple of 3.

Answer: Yes, because no member of B is a multiple of 3

Part (c)

  1. D ∩ (A ∪ C) = ∅ means no member of D is even and no member of D is a multiple of 3.
  2. The odd numbers in ℰ are 23, 25, 27, 29, 31, 33. Remove the multiples of 3 (27 and 33).

Answer: 23, 25, 29, 31

85 marksHigherNovember 2021 · Paper 1H (4MA1/1H) · Q16

There are 32 students in a class. Each took a test in Maths (M), English (E) and French (F). 25 passed Maths, 20 passed English, 14 passed French, 18 passed Maths and English, 11 passed Maths and French, 4 failed all three tests and x passed all three. The incomplete Venn diagram shows that 3 students passed Maths only and 2 students passed English and French but not Maths.

(a) Use all the given information about the results of students who passed the test in Maths to find the value of x. (2)

(b) Use your value of x to complete the Venn diagram to show the number of students in each subset. (2)

A student who passed the test in Maths is chosen at random.

(c) Find the probability that this student failed the test in French. (1)

Show solution

Part (a)

  1. Inside M there are four regions: M only (3), M and E only (18 − x), M and F only (11 − x), and all three (x).
  2. They add to 25: 3 + (18 − x) + (11 − x) + x = 25, so 32 − x = 25

Answer: x = 7

Part (b)

  1. M ∩ E only = 18 − 7 = 11 and M ∩ F only = 11 − 7 = 4
  2. E only = 20 − (11 + 7 + 2) = 0 and F only = 14 − (4 + 7 + 2) = 1
  3. Outside = 4. Check: 3 + 11 + 0 + 4 + 7 + 2 + 1 + 4 = 32 ✓
ℰMEF311047214

Answer: M∩E only 11, E only 0, M∩F only 4, all three 7, F only 1, outside 4

Part (c)

  1. 25 students passed Maths. Of these, the ones who failed French are M only (3) and M ∩ E only (11): 14 students.

Answer: 14⁄25

93 marksHigherJune 2023 · Paper 1H (4MA1/1H) · Q16

The Venn diagram shows a universal set ℰ and sets A, B and C. The numbers and the letter x represent numbers of elements: A only = 9, A ∩ B only = x, B only = 12, A ∩ C only = 5, A ∩ B ∩ C = 3, B ∩ C only = 7, C only = 10, outside all three = 7.

ℰABC9x12537107

Given that n(A ∪ B) = 42

(a) find the value of x (1)

(b) Find n(A′) (1)

(c) Find n(B′ ∩ C) (1)

Show solution

Part (a)

  1. A ∪ B is every region inside circle A or circle B: 9 + x + 12 + 5 + 3 + 7 = 36 + x
  2. 36 + x = 42

Answer: x = 6

Part (b)

  1. A′ is every region outside circle A: 12 + 7 + 10 + 7

Answer: 36

Part (c)

  1. B′ ∩ C means inside C but outside B: 5 + 10

Answer: 15

104 marksHigherJanuary 2022 · Paper 1HR (4MA1/1HR) · Q17

The Venn diagram shows a universal set ℰ and three sets A, B and C. The numbers represent the numbers of elements: A only 6, A ∩ B 3, B only 8, B ∩ C 2, C only 5, outside all sets 4. (A and C do not overlap.)

ℰACB638254

Find (i) n(A ∪ B) (ii) n(A ∩ C) (iii) n(B ∩ C′) (iv) n(A′ ∪ B′ ∪ C′) (4)

Show solution

Part (i)

  1. Everything in A or B: 6 + 3 + 8 + 2

Answer: 19

Part (ii)

  1. A and C do not overlap.

Answer: 0

Part (iii)

  1. In B but not in C: 3 + 8

Answer: 11

Part (iv)

  1. A′ ∪ B′ ∪ C′ is everything except what is in all three sets. Nothing is in all three, so it is the whole universal set: 6 + 3 + 8 + 2 + 5 + 4

Answer: 28

113 marksHigherJune 2021 · Paper 1H (4MA1/1H) · Q19

The Venn diagram shows a universal set ℰ and sets A, B and C. The numbers represent the numbers of elements: A only 12, A ∩ B only 5, B only 9, A ∩ C only 10, A ∩ B ∩ C 6, B ∩ C only 3, C only 4, outside all three 8.

ℰABC1259106348

Find (i) n(A ∪ B) (ii) n(A′ ∩ B′) (iii) n([A ∩ B] ∪ C) (3)

Show solution

Part (i)

  1. Every region inside A or B: 12 + 5 + 9 + 10 + 6 + 3

Answer: 45

Part (ii)

  1. Outside A and outside B: only C only (4) and outside everything (8).

Answer: 12

Part (iii)

  1. A ∩ B is 5 + 6. Union with all of C (10 + 6 + 3 + 4). Count 6 once: 5 + 6 + 10 + 3 + 4

Answer: 28

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