Every sets and venn diagrams question from 16 Cambridge IGCSE Mathematics (0580) past papers (May/June 2025 to May/June 2026), syllabus section 1.2. 16 questions covering Core and Extended tiers, ordered from Core to Extended. Each question is labelled with its question paper code (for example 0580/11/M/J/26) and every final answer matches the official Cambridge mark scheme. Try each question first, then click “Show solution” to check your working. This page is part of our IGCSE Maths 0580 Number past paper questions.
0580 Sets and Venn Diagrams past paper questions: Core (Papers 1 and 3)
May/June 2026 · Paper 1 · 0580/11/M/J/26 · Q15
(a) 60 people complete a survey about hobbies. 41 people like reading books (R). 31 people like cookery (C). 11 do not like reading books and do not like cookery.
(i) Complete the Venn diagram. (ii) One of these 60 people is picked at random. Find the probability that they like cookery but they do not like reading books.
(b) Shade A ∩ B on the Venn diagram.
Show solution
- 11 people are outside both circles, so 60 − 11 = 49 people are inside at least one circle.
- Add the two circles: 41 + 31 = 72. This counts the overlap twice, so the overlap = 72 − 49 = 23.
- R only = 41 − 23 = 18. C only = 31 − 23 = 8.
- R only 18, both 23, C only 8, outside 11
- Check: 18 + 23 + 8 + 11 = 60 ✓
- “Likes cookery but not reading” is the C-only region: 8 people.
- P = 8⁄60 = 2⁄15
- A ∩ B means in A and in B at the same time, which is the overlap.
- Shade the region inside both circles
May/June 2026 · Paper 1 · 0580/12/M/J/26 · Q25
ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9}, P = {prime numbers}, N = {odd numbers}.
(a) Complete the Venn diagram. (b) Write down n(P ∪ N).
Show solution
- List the sets: P = {2, 3, 5, 7} (primes up to 9) and N = {1, 3, 5, 7, 9} (odd numbers).
- 3, 5 and 7 are in both sets, so they go in the overlap.
- 2 is prime but even (P only). 1 and 9 are odd but not prime (N only). 4, 6 and 8 are in neither.
- P only: 2; P ∩ N: 3, 5, 7; N only: 1, 9; outside: 4, 6, 8
- P ∪ N is everything in P or N or both: {1, 2, 3, 5, 7, 9}.
- n(P ∪ N) = 6
May/June 2026 · Paper 3 · 0580/33/M/J/26 · Q22
(a) In a Venn diagram, both circles P and Q are fully shaded. Use set notation to describe the shaded region.
(b) ℰ = {A, B, E, H, M, O, Q, R, S, U}, X = {R, H, O, M, B, U, S}, Y = {S, Q, U, A, R, E}.
(i) Complete the Venn diagram. (ii) List the elements of X′. (iii) Find n(X ∩ Y).
Show solution
- Both circles are fully shaded, so the region is everything in P or Q or both. That is the union.
- P ∪ Q
- Find the letters in both X and Y first: R, S and U.
- X only (in X but not Y): B, H, M, O.
- Y only (in Y but not X): A, E, Q.
- Every letter in ℰ is in X or Y, so nothing goes outside the circles.
- X only: B, H, M, O; both: R, S, U; Y only: A, E, Q
- X′ means everything in ℰ that is not in X.
- A, E, Q
- n(X ∩ Y) means the number of elements in both X and Y: R, S, U.
- 3
Feb/Mar 2026 · Paper 1 · 0580/12/F/M/26 · Q19
ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A = {multiples of 3}, B = {multiples of 4}.
(a) Complete the Venn diagram. (b) Write down n(A ∩ B).
Show solution
- List each set: A = {3, 6, 9, 12} and B = {4, 8, 12}.
- 12 is in both sets, so it goes in the overlap.
- Every number from 1 to 12 that is in neither set goes outside the circles.
- A only: 3, 6, 9; A ∩ B: 12; B only: 4, 8; outside: 1, 2, 5, 7, 10, 11
- n(A ∩ B) is how many numbers are in both A and B. Only 12 is.
- n(A ∩ B) = 1
Oct/Nov 2025 · Paper 1 · 0580/11/O/N/25 · Q24
ℰ = {students in a year group}, H = {students who study History}, G = {students who study Geography}. 80 students are in the year group. 40 students study History. The Venn diagram shows 10 students in H only and 15 students outside both sets.
(a) Complete the Venn diagram. (b) Find n(G).
(c) A student is chosen at random. Work out the probability that the student does not study History and does not study Geography.
Show solution
- 40 students study History, and 10 of them are in H only. The rest are in both: H ∩ G = 40 − 10 = 30.
- G only = total − outside − History = 80 − 15 − 40 = 25.
- H ∩ G: 30; G only: 25
- n(G) means the number in G = both + G only = 30 + 25 = 55
- Students who study neither are outside both circles: 15 students.
- P = 15⁄80 (which simplifies to 3⁄16)
Oct/Nov 2025 · Paper 1 · 0580/12/O/N/25 · Q20
ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {1, 3, 6, 10}.
(a) Complete the Venn diagram. (b) List the elements of A ∩ B. (c) Find n(B′).
Show solution
- 1 and 3 are in both A and B, so they go in the overlap.
- A only: 5, 7, 9. B only: 6, 10. The numbers in neither set, 2, 4, 8, go outside the circles.
- A only: 5, 7, 9; A ∩ B: 1, 3; B only: 6, 10; outside: 2, 4, 8
- A ∩ B is the overlap: 1, 3
- B′ means “not in B”: 2, 4, 5, 7, 8, 9.
- n(B′) = 6
Oct/Nov 2025 · Paper 1 · 0580/13/O/N/25 · Q23
(a) A Venn diagram shows sets A and B with only the overlap shaded. Use set notation to describe the shaded region.
(b) ℰ = {people in a club}, T = {people who play tennis}, S = {people who go swimming}. There are 60 people in the club. 36 people go swimming. The Venn diagram shows 30 in S only and 10 outside both sets.
(i) Complete the Venn diagram. (ii) Find n(T′).
Show solution
- Only the overlap is shaded, which is the part in A and in B.
- A ∩ B
- 36 people go swimming and 30 of them are in S only, so the overlap T ∩ S = 36 − 30 = 6.
- T only = total − outside − all of S = 60 − 10 − 36 = 14.
- T only: 14; T ∩ S: 6
- T′ means “not in T”: S only + outside = 30 + 10 = 40
May/June 2025 · Paper 1 · 0580/12/M/J/25 · Q20
Use set notation to describe the shaded region.
Show solution
- Both circles are fully shaded, including the overlap.
- “In J or in K or both” is the union, written ∪.
- J ∪ K
May/June 2025 · Paper 1 · 0580/13/M/J/25 · Q12
A music group has 30 members. 7 members play a flute (F) and play a clarinet (C). 12 members play a flute. 1 member does not play a flute and does not play a clarinet.
(a) Use this information to complete the Venn diagram.
(b) Find n(C).
Show solution
- Start in the middle: 7 members play both, so put 7 in the overlap.
- 12 play a flute, so flute only = 12 − 7 = 5.
- 1 member plays neither, so put 1 outside the circles.
- Clarinet only = 30 − 5 − 7 − 1 = 17.
- F only 5, F ∩ C 7, C only 17, outside 1
- n(C) is everyone in the C circle: 7 + 17 = 24
0580 Sets and Venn Diagrams past paper questions: Extended (Papers 2 and 4)
May/June 2026 · Paper 2 · 0580/21/M/J/26 · Q9
In a Venn diagram with sets A and B, shade the region A′ ∪ B.
Show solution
- A′ means “not in A”, so shade everything outside circle A.
- ∪ B means also shade everything in B. This adds back the overlap of A and B.
- The only part left unshaded is the part of A that is outside B.
- Shade everything except the part of A that is outside B
May/June 2026 · Paper 2 · 0580/23/M/J/26 · Q25
x is an integer. ℰ = {x : 1 ≤ x ≤ 12}, A = {x : x is a square number}, B = {x : 4 ≤ x ≤ 8}, C = {x : x is a factor of 12}.
(a) Complete the Venn diagram. (b) Find n(A ∩ B ∩ C′).
Show solution
- List each set: A (square numbers) = {1, 4, 9}, B = {4, 5, 6, 7, 8}, C (factors of 12) = {1, 2, 3, 4, 6, 12}.
- 4 is in all three sets, so it goes in the middle.
- 1 is in A and C only. 6 is in B and C only. Nothing is in A and B only.
- Then A only: 9. B only: 5, 7, 8. C only: 2, 3, 12. Outside: 10, 11.
- A ∩ B ∩ C: 4; A ∩ C only: 1; A only: 9; B ∩ C only: 6; B only: 5, 7, 8; C only: 2, 3, 12; outside: 10, 11
- A ∩ B ∩ C′ means in A, in B and not in C.
- The only number in both A and B is 4, but 4 is in C. So there are no numbers.
- n(A ∩ B ∩ C′) = 0
Feb/Mar 2026 · Paper 2 · 0580/22/F/M/26 · Q13
ℰ = {x : x is an integer where 40 < x < 50}, P = {x : x is a square number}, Q = {x : x is an odd number}, R = {x : x is a multiple of 3}.
(a) Use set notation to complete the statement P ∩ R = …
(b) Find n((P ∩ Q ∩ R)′).
Show solution
- List the sets. ℰ = {41, 42, 43, 44, 45, 46, 47, 48, 49}.
- P (square numbers) = {49}. Q (odd numbers) = {41, 43, 45, 47, 49}. R (multiples of 3) = {42, 45, 48}.
- P ∩ R is the numbers in both P and R. 49 is not a multiple of 3, so there are none.
- P ∩ R = ∅ (the empty set)
- P ∩ Q ∩ R has no elements (it is already empty, because P ∩ R is empty).
- Its complement contains every element of ℰ, which has 9 numbers.
- n((P ∩ Q ∩ R)′) = 9
Oct/Nov 2025 · Paper 2 · 0580/21/O/N/25 · Q10
ℰ = {n : n is an integer and 1 ≤ n ≤ 8}, A = {factors of 12}, B = {odd numbers}. Find (a) A ∩ B (b) n(A′ ∪ B).
Show solution
- List the sets first. ℰ = {1, 2, 3, 4, 5, 6, 7, 8}.
- A = factors of 12 that are in ℰ = {1, 2, 3, 4, 6}. B = odd numbers = {1, 3, 5, 7}.
- A ∩ B is the numbers in both A and B.
- A ∩ B = {1, 3}
- A′ is everything not in A: {5, 7, 8}.
- A′ ∪ B joins A′ and B: {1, 3, 5, 7, 8}. Count the elements.
- n(A′ ∪ B) = 5
Oct/Nov 2025 · Paper 2 · 0580/22/O/N/25 · Q12
A fitness club has 100 members. 60 swim (S). 70 cycle (C). 25 do not swim or cycle.
(a) Complete the Venn diagram.
(b) One member is chosen at random. Find (i) P(C) (ii) P(S ∩ C) (iii) P(S ∪ C′).
Show solution
- 25 members do neither, so 100 − 25 = 75 members are in S or C or both.
- Add the two groups: 60 + 70 = 130. This counts the overlap twice, so the overlap = 130 − 75 = 55.
- S only = 60 − 55 = 5. C only = 70 − 55 = 15.
- S only: 5; S ∩ C: 55; C only: 15; outside: 25
- Check: 5 + 55 + 15 + 25 = 100 ✓
- 70 members cycle: P(C) = 70⁄100
- 55 members do both: P(S ∩ C) = 55⁄100
- S ∪ C′ means “swims, or does not cycle, or both”. The only members left out are those in “C only”.
- 100 − 15 = 85 members, so P(S ∪ C′) = 85⁄100
Oct/Nov 2025 · Paper 2 · 0580/23/O/N/25 · Q18
Shade the region in each Venn diagram. (a) (A ∩ B)′ (b) (C ∪ D) ∩ E′
Show solution
- A ∩ B is the overlap. The ′ means “not”, so (A ∩ B)′ is everything outside the overlap.
- Shade everything except the overlap of A and B
- C ∪ D is everything in C or D. ∩ E′ keeps only the parts that are also outside E.
- Shade the parts of C and D that are outside E
Oct/Nov 2025 · Paper 4 · 0580/41/O/N/25 · Q12
Some students are asked if they like football (F) or rugby (R). The Venn diagram shows 33 in F only, 7 in F ∩ R, 6 in R only and 4 outside both.
(a) Find the number of students who do not like rugby. (b) Use set notation to describe the region containing students who like rugby but not football.
Show solution
- People who do not like rugby are everyone outside circle R.
- That is “F only” plus “outside both”: 33 + 4 = 37
- “Likes rugby” means inside R. “Not football” means outside F, written F′.
- Both at the same time means the intersection: R ∩ F′
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