IGCSE Mathematics: Sets and Venn Diagrams Practice Questions
A set is a collection of elements. The intersection of two sets contains the elements in both, the union contains the elements in either, and the complement of a set contains everything in the universal set that is not in it.
Topic 1.2 of Cambridge IGCSE Mathematics 0580 is where marks are lost to notation rather than to reasoning. Filling a Venn diagram from the centre outwards solves almost every problem in this topic. The questions below apply that method and test the notation directly.
What you need to know for Sets and Venn Diagrams
- NotationThe intersection symbol means and, the union symbol means or, a dash or prime means complement, and n(A) means the number of elements in A.
- Universal setThe set containing every element under consideration, drawn as the rectangle enclosing the Venn diagram.
- IntersectionThe elements that belong to both sets, shown by the overlapping region.
- UnionAll elements that belong to at least one of the sets, shown by both circles together including the overlap.
- ComplementEverything inside the universal set but outside the named set. A dash after the letter denotes it.
- Filling a Venn diagramAlways start with the intersection, the region belonging to the most sets, then work outwards subtracting what you have already placed.
IGCSE Mathematics Sets and Venn Diagrams questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Set A is the set of factors of 12 and set B is the set of factors of 18. List the elements of the intersection of A and B, and state the number of elements in it.
Show the worked answer
- Factors of 12: 1, 2, 3, 4, 6, 12.
- Factors of 18: 1, 2, 3, 6, 9, 18.
- The intersection contains the elements appearing in both lists: 1, 2, 3 and 6.
- There are 4 elements, so n of the intersection equals 4. Note that the largest element, 6, is the highest common factor of 12 and 18.
In a class of 30 students, 18 study French and 15 study Spanish. 7 study both. Using a Venn diagram, find how many study neither language.
Show the worked answer
- Start with the intersection: 7 students study both languages, so 7 goes in the overlap.
- French only = 18 minus 7 = 11 students. Spanish only = 15 minus 7 = 8 students.
- Total studying at least one language = 11 + 7 + 8 = 26 students.
- Studying neither = 30 minus 26 = 4 students. These sit inside the rectangle but outside both circles.
In a survey of 50 people, 28 own a car, 22 own a bicycle and 10 own both. Find the number of people who own a car but not a bicycle, and the number who own exactly one of the two.
Show the worked answer
- Place 10 in the intersection, since 10 people own both.
- Car only = 28 minus 10 = 18 people.
- Bicycle only = 22 minus 10 = 12 people.
- Exactly one means the two outer regions added together but not the overlap: 18 + 12 = 30 people.
The universal set contains the integers from 1 to 10. Set P contains the even numbers and set Q contains the numbers greater than 6. List the elements of the complement of the union of P and Q.
Show the worked answer
- P, the even numbers from 1 to 10, is 2, 4, 6, 8, 10.
- Q, the numbers greater than 6, is 7, 8, 9, 10.
- The union of P and Q contains everything in either set: 2, 4, 6, 7, 8, 9, 10.
- The complement is everything in the universal set that is not in that union: 1, 3, 5.
Common mistakes in this topic
- Confusing the union and intersection symbols. Union means or and is the larger set.
- Filling a Venn diagram from the outside in, which double counts the overlap.
- Forgetting the region outside both circles when the totals are given.
- Reading exactly one as at least one.
- Repeating elements when listing a set.
Exam tips
- Always fill the intersection first, then subtract to find each remaining region.
- Check that all your regions add to the total in the universal set before finishing.
- Read the wording carefully: only, both, exactly one, at least one and neither all mean different regions.
- Write the notation out in words next to the symbols in the margin. It prevents symbol confusion under pressure.
- For three set problems, start with the central region belonging to all three and work outwards one layer at a time.
Practise 20 more questions like this, free
vStudyWise marks every answer instantly, tracks the topics you keep dropping marks on and turns them into a weekly study plan.
Sets and Venn Diagrams FAQs
What is the difference between union and intersection?
The intersection contains only the elements that appear in every set, shown by the overlapping region of a Venn diagram. The union contains every element appearing in at least one of the sets, which is both circles together including the overlap. Union means or, intersection means and.
How do I fill in a Venn diagram?
Always start with the region belonging to the most sets, which is the intersection, and write that number in first. Then work outwards, subtracting the numbers already placed from each set total. Finally subtract the total inside the circles from the universal set to find the outside region.
What does the complement of a set mean?
The complement of a set contains every element of the universal set that is not in that set. It is written with a dash or prime after the letter. On a Venn diagram it is everything inside the rectangle but outside the named circle.
What does exactly one mean in a set question?
Exactly one means an element belongs to one of the sets but not to the other, so it is the two outer regions of a two set Venn diagram added together, excluding the overlap. At least one, by contrast, means the union and does include the overlap.
Continue through the IGCSE Mathematics syllabus
Related IGCSE Mathematics topics
See all 41 IGCSE Mathematics practice topics ›
Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.