Sets and Venn diagrams
Translate statements into regions, so overlapping groups are counted once and nobody disappears.
Section Paper 2 · Extended · Non-calculator
Core: two sets, counts, union, intersection and complement. Extended also includes three sets, membership, empty-set and subset notation. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Add and subtract whole numbers.
- Distinguish 'both' from 'only one'.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A club has 10 dancers, including 3 who also sing. How many are dancers who do not sing?
What you will learn
- Read membership, union, intersection and complement notation.
- Complete two-set and three-set diagrams.
- Check a diagram against its group totals.
Define the world before the groups
A set is a collection of distinct elements. Curly brackets list its members: A = {2, 4, 6}. Order does not matter, and repeating a member does not make it a new element. The universal set U contains everything under consideration. If U = {1, 2, 3, 4, 5, 6}, the complement A′ is {1, 3, 5}. A complement always depends on the chosen universal set.
The notation n(A) means the number of elements in A; here n(A) = 3. A rule can define a set instead of a list: {x: x is an integer and 1 ≤ x ≤ 4} means {1, 2, 3, 4}. The colon means 'such that'. The word integer matters: without it, an interval could contain infinitely many real numbers.
Two groups can overlap
A ∩ B, the intersection, contains members in both sets. A ∪ B, the union, contains members in A or B or both. Mathematical 'or' in a union includes the overlap. If A = {1, 2, 3} and B = {3, 4}, then A ∩ B = {3} and A ∪ B = {1, 2, 3, 4}.
In a Venn diagram, the rectangle represents U and each circle a set. Fill the overlap first. If 12 students study art, 9 study music, and 4 do both, art-only is 8 and music-only is 5. The union is 8 + 4 + 5 = 17, not 21. For a class of 20, the 3 outside the circles do neither.
Extended: membership and subsets
Write 3 ∈ A to say 3 is an element of A, and 5 ∉ A to say it is not. The empty set ∅ has no elements, so n(∅) = 0. The set {0} is not empty: it contains one element, zero.
A ⊆ B means every member of A is also in B. Equality is allowed in this subset notation. A ⊈ B means at least one member of A is missing from B. Membership compares an element with a set; subset compares two sets. For example, 2 ∈ {1, 2, 3}, while {2} ⊆ {1, 2, 3}. A set of ordered pairs, such as {(x, y): y = 2x}, is a set whose members are coordinate pairs.
Extended: build three-set diagrams from the centre
For three groups, the region inside all three circles must be filled first. A reported total for 'A and B' normally includes people who also belong to C. Subtract the triple intersection to find the region in A and B only. Repeat for the other pairwise overlaps; then complete each single-only region.
Example: in a group of 40, n(A) = 18, n(B) = 16, n(C) = 14. The inclusive pairwise totals are 7, 6 and 5 for A∩B, A∩C and B∩C; all three contain 2. The pair-only counts are therefore 5, 4 and 3. The single-only counts are A: 18−5−4−2 = 7; B: 16−5−3−2 = 6; C: 14−4−3−2 = 5. The seven inside regions sum to 32, leaving 8 outside. Check every circle total before accepting the result.
Pause and explain
If someone belongs to both A and B, are they in A ∪ B?
Worked example
Of 30 students, 17 play football, 14 swim, and 6 do both. How many do exactly one activity and how many do neither?
Show the worked solution
- Put 6 in the overlap. Football-only is 17 − 6 = 11; swimming-only is 14 − 6 = 8.
- Exactly one means the two single-only regions: 11 + 8 = 19.
- At least one includes the overlap: 11 + 6 + 8 = 25. Neither = 30 − 25 = 5.
Answer 19 do exactly one; 5 do neither.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
In a class of 24, 13 like apples, 10 like pears and 5 like both. Find apples-only, pears-only and neither.
Give me a hint
Fill the overlap with 5 before finding the single-only regions.
Compare my reasoning
- Apples-only: 13 − 5 = 8. Pears-only: 10 − 5 = 5.
- At least one: 8 + 5 + 5 = 18.
- Neither: 24 − 18 = 6. All four regions total 24.
8 apples-only; 5 pears-only; 6 neither.
Look for these in your work
- Subtracted the overlap from both group totals.
- Checked all regions add to the universal total.
Let U = {1,2,3,4,5,6,7,8}, A = {2,4,6,8} and B = {2,3,5,7}. Find A ∩ B, A ∪ B and n(A′).
Give me a hint
List shared elements, then list all elements used by either set without repeats.
Compare my reasoning
- Only 2 appears in both, so A ∩ B = {2}.
- A ∪ B = {2,3,4,5,6,7,8}.
- A′ = {1,3,5,7}, so n(A′) = 4.
{2}; {2,3,4,5,6,7,8}; 4.
Look for these in your work
- Did not duplicate 2 in the union.
- Gave a number, not a set, for n(A′).
Extended: in a group of 30, n(A)=15, n(B)=12 and n(C)=10. Inclusive pair totals are n(A∩B)=5, n(A∩C)=4 and n(B∩C)=3. Two belong to all three. Find how many belong to exactly two sets and how many belong to none.
Give me a hint
Put 2 in the centre. Subtract it from each pair total before calculating the single-only regions.
Compare my reasoning
- Pair-only regions: 5−2=3, 4−2=2 and 3−2=1. Exactly two sets: 3+2+1=6.
- Single-only regions: A has 15−3−2−2=8; B has 12−3−1−2=6; C has 10−2−1−2=5.
- The union contains 8+6+5+3+2+1+2=27 people. None: 30−27=3.
6 belong to exactly two; 3 belong to none.
Look for these in your work
- Excluded the triple intersection from exactly-two counts.
- Checked each circle total and the universal total.
A survey claims that of 30 people, 22 own a bicycle, 18 own a scooter and 5 own both. Explain why the figures cannot all be true. What is the smallest possible overlap if the first three figures are correct?
Give me a hint
Calculate the claimed number who own at least one. It cannot exceed 30.
Compare my reasoning
- The claimed union is 22 + 18 − 5 = 35, exceeding the 30 people surveyed.
- To fit within 30, the overlap must be at least 22 + 18 − 30 = 10.
- An overlap of 10 gives bicycle-only 12, scooter-only 8, neither 0, so the bound is possible.
The claim counts 35 distinct people; at least 10 must own both.
Look for these in your work
- Used the total as a constraint.
- Demonstrated a possible arrangement at the minimum.
Common mistakes
- The overlap is already included in each circle's total.
- The complement is everything outside that set within U, not everything outside both circles.
- In three-set questions, read whether a pairwise total says 'only'.
What can you explain now?
Let U = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {4, 5, 6}. Find A ∩ B and A′.
Compare with the explanation
A ∩ B = {4, 6}; A′ = {1, 3, 5}.
Intersection keeps the shared members. Complement removes every member of A from U.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Redraw the three-set example from its totals tomorrow. Explain why subtracting 2 from each inclusive pair total is necessary.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.