What is a Venn diagram?
A Venn diagram shows how different sets of data relate to each other, using overlapping circles inside a rectangle. The rectangle represents the universal set (everything being considered), and each circle represents a set of items with a shared property.
Key notation:
- ξ (xi) — the universal set, everything inside the rectangle
- A ∪ B ("A union B") — everything in A, in B, or in both
- A ∩ B ("A intersection B") — only the items in both A and B (the overlap)
- A′ ("A complement" or "not A") — everything outside set A
- The numbers in each region show how many items belong there
Constructing a Venn diagram
Worked Example 1
In a survey of 30 students: 18 study French (F), 14 study Spanish (S), and 7 study both. Construct a Venn diagram to show this information.
1
Start with the overlap: 7 students study both, so the intersection F ∩ S = 7
2
French only = 18 − 7 = 11. Spanish only = 14 − 7 = 7
3
Students in neither = 30 − (11 + 7 + 7) = 30 − 25 = 5
AnswerFrench only 11, both 7, Spanish only 7, neither 5
French only (11), Spanish only (7), both (7), and the 5 outside both circles but inside ξ study neither language.
Reading probabilities from a Venn diagram
Worked Example 2
Using the Venn diagram above, find P(a randomly chosen student studies French only).
1
"French only" is the part of circle F that doesn't overlap with S: 11 students
2
Divide by the total: P(French only) = 11 ÷ 30
Answer11/30
Union and intersection
Worked Example 3
Using the same Venn diagram, find P(F ∩ S) and P(F ∪ S).
1
F ∩ S is just the overlap: 7 students, so P(F ∩ S) = 7/30
2
F ∪ S is everyone in F, in S, or in both: 11 + 7 + 7 = 25 students
3
P(F ∪ S) = 25 ÷ 30 = 5/6
AnswerP(F ∩ S) = 7/30, P(F ∪ S) = 5/6
Finding a missing region
Worked Example 4 — Working backwards
In a survey of 40 people: 22 have a dog (D), 15 have a cat (C), and 6 have both. Find how many people have neither pet.
1
Dog only = 22 − 6 = 16. Cat only = 15 − 6 = 9
2
Total inside the two circles = 16 + 9 + 6 = 31
3
Neither = 40 − 31 = 9 people
Answer9 people
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A Venn diagram has ξ = 25. Set A has 12 members, set B has 10, and 4 are in both. Find n(A only).Foundation
Q2Using the data from Q1, find n(B only).Foundation
Q3Using the data from Q1, find how many items are in neither A nor B.Foundation
Q4True or false: A ∩ B represents everything in A, in B, or in both. Explain your answer.Foundation
Q5In a Venn diagram, n(ξ) = 50 and n(A) = 30. Find n(A′).Foundation
Higher (Grade 5–7)
Q6In a survey of 60 people: 35 like tea, 28 like coffee, and 15 like both. Find n(tea only).Higher
Q7Using the data from Q6, find n(coffee only).Higher
Q8Using the data from Q6, find how many people like neither tea nor coffee.Higher
Q9Using the data from Q6, find P(a randomly chosen person likes both tea and coffee).Higher
Q10Using the data from Q6, find P(tea ∪ coffee).Higher
Higher — Hard (Grade 8–9)
Q11In a survey of 45 students: n(Maths) = 27, n(Science) = 24, and 6 study neither subject. Find n(Maths ∩ Science).Grade 8–9
Q12In a Venn diagram, n(ξ) = 80, n(A) = 45, n(B) = 38, and 12 items are in neither set. Find n(A ∩ B).Grade 8–9
Q13A Venn diagram shows sets P and Q. n(P only) = 14, n(Q only) = 9, n(neither) = 5, and n(ξ) = 40. Find n(P ∩ Q).Grade 8–9
Q14Explain, using set notation, the difference between P(A ∩ B) and P(A ∪ B).Grade 8–9
Q15In a class of 32 students, every student studies at least one of French or Spanish. 20 study French and 18 study Spanish. Find the number of students who study both.Grade 8–9
Answers
Foundation (Q1–Q5)
Q18(12 − 4)
Q26(10 − 4)
Q37(25 − (8 + 6 + 4))
Q4False(A ∩ B is only the overlap — both at once. A ∪ B is everything in A, B, or both)
Q520(50 − 30)
Higher (Q6–Q10)
Q620(35 − 15)
Q713(28 − 15)
Q812(60 − (20 + 13 + 15))
Q91/4(15 ÷ 60)
Q104/5((20 + 13 + 15) ÷ 60 = 48 ÷ 60)
Higher — Hard (Q11–Q15)
Q116(n(M∪S) = 45 − 6 = 39; n(M∩S) = 27 + 24 − 39)
Q1215(n(A∪B) = 80 − 12 = 68; n(A∩B) = 45 + 38 − 68)
Q1312(14 + 9 + y + 5 = 40 → y = 12)
Q14P(A ∩ B) is the probability an item is in both A and B — only the overlap region. P(A ∪ B) is the probability an item is in A, in B, or in both — every region except "neither"
Q156(n(F∪S) = 32, since everyone studies at least one; n(F∩S) = 20 + 18 − 32)
Common mistakes
Common Mistake 1
Starting with the individual set totals instead of the overlap
Always fill in the intersection (both) first. Only once that number is placed can you correctly work out "A only" and "B only" by subtracting the overlap from each set's total.
Common Mistake 2
Confusing ∪ (union) and ∩ (intersection)
∩ looks like the top of a circle and means the overlap ("and"). ∪ looks like a cup and means everything in either set ("or"). Mixing these up is one of the most common errors on this topic.
Common Mistake 3
Forgetting to include the "neither" region
The total inside the rectangle (ξ) includes people or items outside both circles. Forgetting this region when finding totals or probabilities is a very common source of error.
Common Mistake 4
Double-counting the overlap when adding set totals
n(A) + n(B) counts the overlap twice. To find n(A ∪ B) correctly, use n(A) + n(B) − n(A ∩ B), subtracting the intersection once to avoid double-counting.
Exam tips
💡 Exam Tip 1
Fill in the diagram before answering the question
Even if a question only asks for one value, sketch and complete the whole Venn diagram first — it's much easier to read off multiple answers from a complete diagram than to calculate each one separately from scratch.
💡 Exam Tip 2
Use the "n(ξ) check" to catch errors
Once your diagram is complete, add up every region — both only, overlap, and neither. The total must equal n(ξ). If it doesn't, you've made an error somewhere.
💡 Exam Tip 3
Translate word problems into set notation early
Before calculating, write down exactly what's being asked in symbols — e.g. "find P(A ∩ B)" — so you know precisely which region(s) of the diagram to use.
💡 Exam Tip 4
Practise reading A′ (complement) carefully
A′ means everything outside A — including any part of B that doesn't overlap with A, and the region outside both circles. Don't mistake it for "B only".
Want to improve your grade faster?
If Venn diagrams are still causing problems, Alamin's diagnostic approach identifies exactly which skills are missing and builds a targeted plan to address them — with AI-powered practice between sessions.
Book an Assessment Session (£60)No upfront payment required — payment is taken after confirmation.