IGCSE Maths / Sets
Venn diagrams with three sets
Edexcel International GCSE Maths (4MA1), chapter 18: Sets, specification point 1.5b. Free lesson.
Read the revision notes · IGCSE Maths past papers
A three-set Venn diagram has eight regions, including the part outside all three sets. Fill it from the centre outwards so overlaps are counted once, then use the regions to read set notation and probabilities.
How to fill in three overlapping sets
Each circle represents a set, and the box contains everyone being counted. Start in the centre, where all three circles overlap. Each pair total includes the centre, so subtract it to find the pair-only region. Then subtract each circle’s inner regions from its total to find the part belonging only to that set. For football, tennis and swimming, put 4 in the centre. Pair-only regions: football and tennis, 10 − 4 = 6; football and swimming, 9 − 4 = 5; tennis and swimming, 7 − 4 = 3. Single-set regions: football, 27 − 6 − 5 − 4 = 12; tennis, 22 − 6 − 3 − 4 = 9; swimming, 20 − 5 − 3 − 4 = 8.…
How to read intersections and unions
An intersection means members in both named sets. It includes the centre when all three circles overlap. A union means members in at least one of the named sets, so the union of three sets covers every region inside the circles. A ∩ B means football and tennis: 6 + 4 = 10. A ∪ B ∪ C means students who play at least one sport. It covers all seven regions inside the circles, giving 47.
How complements change the region
A complement means outside a set. A dash outside brackets applies to the whole expression in the brackets, so it can describe a different region from complementing each set separately. A ∩ B ∩ C' means in A and B, but not C. It is the pair-only region where A and B overlap. (A ∪ B ∪ C)' means outside all three circles: members who do none of the activities. It is not the same as A' ∪ B' ∪ C'. B ∩ C includes both regions shared by B and C, including the centre.
How to find a probability
For a member chosen at random, the required region or regions give the numerator. The denominator is everyone in the box, including members outside every circle. In a club of 50 members, 3 do all three activities, so the probability of choosing one of them is 3/50. Eight members do none of the activities, but they still count in the total of 50.
Check the regions and avoid double counting
There are eight regions: seven inside the circles and one outside. Add all eight at the end. They must add back to the number in the box; otherwise, an overlap may have been counted more than once. For art, drama and music, start with 5 in the centre. The pair-only regions are 8 − 5 = 3, 9 − 5 = 4 and 7 − 5 = 2. The single-set regions are art: 20 − 3 − 4 − 5 = 8; drama: 15 − 3 − 2 − 5 = 5; music: 18 − 4 − 2 − 5 = 7. The inside regions total 34, so 40 − 34 = 6 students do none.