To count people in exactly one of two sets, add the two regions outside their overlap but inside either circle. To count neither, subtract the number in the union from the total in the universal set.
The 2027 SEC G3 Mathematics K310 syllabus covers set notation and Venn diagrams. This guide works through the four regions and shows why adding the two set totals counts the overlap twice.
- Put the intersection in the middle first.
- Subtract it from each set total to find each only region.
- Add the two only regions for exactly one.
- Subtract the union from the universal set for neither.
CHAPTER 1 OF 4 · Back to contents
Give every person one region
With two sets A and B, a person can be in A only, in B only, in both or in neither. These four regions are separate and together account for the whole group.
The number in A includes the intersection unless the question says A only. The number in B also includes the intersection. Write the shared count in the centre before working out the outside parts of each circle.
At least one includes the overlap. Exactly one excludes it. If those words are easy to mix up, write out the included regions before calculating.
CHAPTER 2 OF 4 · Back to contents
Worked example | Two activities in a group of 40
Of 40 students, 24 chose activity A and 18 chose activity B. Ten chose both. Find the numbers who chose exactly one activity and neither activity.
| Region | Working | Students |
|---|---|---|
| A only | 24 − 10 | 14 |
| Both | Given | 10 |
| B only | 18 − 10 | 8 |
| Neither | 40 − (14 + 10 + 8) | 8 |
Exactly one is 14 + 8 = 22. At least one is 14 + 10 + 8 = 32. Neither is 40 − 32 = 8.
Check the totals. The two A regions give 24 and the two B regions give 18. The four separate regions add to 40.
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CHAPTER 3 OF 4 · Back to contents
Practise when neither is given
In a group of 50, 28 belong to A, 21 belong to B and 9 belong to neither. The union therefore contains 50 − 9 = 41.
Adding the set totals gives 49. The difference between 49 and 41 is 8, so eight people belong to both. A only is 20 and B only is 13, giving 33 in exactly one.
Check by adding 20, 8, 13 and 9. If the sum differs from the stated total, revisit the overlap before changing an outside count.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What does A ∪ B include?
It includes A only, B only and the intersection.
Can I use a direct formula for exactly one?
Yes. Add the two set sizes, then subtract twice the intersection size because the overlap is counted in both.
What is the complement of A ∪ B?
It consists of the members of the stated universal set who are in neither A nor B.
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