At a glance
- Cambridge 0580
- Core C1.2 (2 sets); Extended E1.2 (2 or 3 sets)
- Edexcel 4MA1
- Foundation 1.5A to E; Higher 1.5A to D
- Key formula
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- Method
- Fill from the centre outwards
Notation you must know
| Symbol | Meaning | Who needs it |
|---|---|---|
| ξ | Universal set: everything under consideration | All tiers |
| A ∩ B | Intersection: in A and in B | All tiers |
| A ∪ B | Union: in A or B or both | All tiers |
| A′ | Complement: not in A | All tiers |
| n(A) | Number of elements in A | All tiers (Edexcel Higher 1.5C) |
| ∈, ∉ | Is, or is not, an element of | Cambridge Extended; Edexcel Foundation and Higher |
| ∅ | The empty set | Cambridge Extended; Edexcel Foundation and Higher |
| A ⊆ B or A ⊂ B | A is a subset of B | Cambridge Extended (⊆, ⊈); Edexcel Higher (⊂) |
From Cambridge 0580 C1.2 and E1.2, and Edexcel 4MA1 1.5 and Appendix 6 (notation).
The key ideas
Sets can be listed, such as A = {2, 4, 6}, described in words, or defined with set-builder notation, such as {x: x is an integer, -2 ≤ x < 3}, which means 'the set of x such that...'. That example is {-2, -1, 0, 1, 2}, so n(A) = 5 elements.
In a two-set Venn diagram there are four regions: A only, both, B only, and neither. A three-set diagram (Cambridge Extended) has eight regions. Every element of ξ goes in exactly one region, so the regions add to n(ξ).
When you are given totals, such as '25 like tea and 8 like both', the 25 includes the 8. So tea only is 25 - 8 = 17. Starting from the overlap and subtracting outwards avoids double counting. The formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B) captures the same idea.
Shading questions ask you to show a set such as A ∩ B′ (in A but not in B) or (A ∪ B)′ (outside both). Work out each part separately, then combine.
Worked example 1: listing sets
- ξ = {1, 2, 3, ..., 12}. A = multiples of 3 = {3, 6, 9, 12}. B = even numbers = {2, 4, 6, 8, 10, 12}.
- A ∩ B = {6, 12}.
- A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}, so n(A ∪ B) = 8. Check: 4 + 6 - 2 = 8.
- A′ = {1, 2, 4, 5, 7, 8, 10, 11}. (A ∪ B)′ = {1, 5, 7, 11}, and 12 - 8 = 4 elements. Correct.
Worked example 2: a two-set problem with an unknown
- In a class of 30, 18 play football, 12 play tennis and 5 play neither. How many play both?
- Let x play both. Football only = 18 - x, tennis only = 12 - x.
- All regions add to 30: (18 - x) + x + (12 - x) + 5 = 30, so 35 - x = 30 and x = 5.
- Check: football only 13, both 5, tennis only 7, neither 5. Total 13 + 5 + 7 + 5 = 30. Correct.
Worked example 3 (Cambridge Extended): three sets
- Of 50 students, 20 study French, 25 Spanish and 15 German. 8 study French and Spanish, 5 French and German, 6 Spanish and German, and 3 study all three. How many study none?
- Centre (all three): 3.
- Pairs only: French and Spanish only = 8 - 3 = 5; French and German only = 5 - 3 = 2; Spanish and German only = 6 - 3 = 3.
- Single languages: French only = 20 - 5 - 2 - 3 = 10; Spanish only = 25 - 5 - 3 - 3 = 14; German only = 15 - 2 - 3 - 3 = 7.
- Total studying at least one: 3 + 5 + 2 + 3 + 10 + 14 + 7 = 44, so 50 - 44 = 6 study none. Check by the formula: 20 + 25 + 15 - 8 - 5 - 6 + 3 = 44. Correct.
Common mistakes that cost marks
- Putting a total like 25 in the 'A only' region when it includes the overlap.
- Forgetting the 'neither' region, which sits inside ξ but outside every circle.
- Confusing ∪ and ∩. A memory aid: ∪ looks like a cup that holds everything from both sets.
- Listing an element twice in a union.
- Giving n(A) (a number) when the question asks for A (a set listed in braces), or the reverse.
- In three-set diagrams, subtracting the centre from the pair totals but forgetting to do it before finding the single regions.
Exam technique and how a tutor helps
Sets questions are usually short and early on the paper, and they connect to probability: once a Venn diagram is complete, P(A) = n(A) ÷ n(ξ), and 'given that' questions use only the relevant circle. Edexcel Foundation tests finding probabilities from a Venn diagram (6.3D), and Cambridge Extended tests conditional probability from Venn diagrams (E8.4).
When an unknown appears, write each region as an expression in x first, then form one equation from the total. Always add your regions at the end as a check.
Most errors in this topic come from the order of filling the diagram. In one-to-one lessons a tutor fills Venn diagrams with the student on the shared whiteboard, insisting on the centre-out order until it is a habit, and then mixes in notation and shading questions so that symbols like A ∩ B′ become as easy to read as words.
Self-check: can you do these?
- ξ = {1, 2, ..., 10}, P = primes, Q = odd numbers. List P ∩ Q. (Answer: {3, 5, 7})
- Using the same sets, find n(P ∪ Q). (Answer: 6)
- Using the same sets, list P′ ∩ Q. (Answer: {1, 9})
- n(A) = 14, n(B) = 9, n(A ∩ B) = 4. Find n(A ∪ B). (Answer: 19)
- List {x: x is an integer, -1 < x ≤ 3}. (Answer: {0, 1, 2, 3})
Common questions
What is the difference between ∪ and ∩?
∪ (union) includes everything in either set or both. ∩ (intersection) includes only what is in both. A ∪ B is always at least as big as A ∩ B.
Are three-set Venn diagrams on the syllabus?
On Cambridge 0580 Extended, yes: Venn diagrams are limited to two or three sets. Cambridge Core is limited to two sets. Edexcel 4MA1 does not set a limit in its specification, so practise both.
Is the empty set a subset of every set?
Yes. ∅ has no elements, so every element it has (none) is in any set A. Subsets are Cambridge Extended and Edexcel Higher content.
How are Venn diagrams used in probability?
Each region's count divided by n(ξ) gives a probability. For 'given that' questions, divide by the number in the condition's circle instead of the total.
What does a LiveTutor lesson cost?
$15 for a 60-minute, one-to-one online lesson. You can book 1 to 5 lessons a week on a monthly plan, and the first lesson is a free trial with a tutor who teaches your child's board.
Sources
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