Sets and Venn diagrams
Translate statements into regions, so overlapping groups are counted once and nobody disappears.
Core: two sets, counts, union, intersection and complement. Extended also includes three sets, membership, empty-set and subset notation.
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.
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.
Your review choice appears on Today. Sign in to sync it across devices.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources