Set Notation and Venn Diagrams –

Set notation and Venn diagrams are Higher-tier topics that appear on every GCSE exam board — AQA, Edexcel and OCR. You need to understand the formal symbols for union, intersection and complement, read and shade Venn diagrams, and use them to solve probability problems. These skills are worth several marks and are often combined with other probability topics. This guide explains every symbol clearly, works through graded examples and highlights the errors that trip students up most. For the full specification overview, see our complete GCSE Maths topics list.

What Is Set Notation?

A set is a collection of distinct objects called elements or members. In GCSE Maths, set notation provides a formal way to describe groups and their relationships.

Key Symbols

• A ∪ B (A union B) — everything in A or B or both.
• A ∩ B (A intersection B) — everything in both A and B.
• A' (A complement) — everything not in A (but in the universal set).
• ξ (xi) — the universal set containing all elements under consideration.
• n(A) — the number of elements in set A.
• ∈ — "is a member of" (e.g. 3 ∈ A means 3 is in set A).
• ∅ — the empty set (a set with no elements).
• A ⊂ B — A is a subset of B (every element of A is also in B).

Key Formulas

Step-by-Step Method

1. Draw the universal set as a rectangle and label it ξ.
2. Draw circles inside for each set (usually two or three), overlapping where they share elements.
3. Fill in the intersection first — put the number of elements common to both sets in the overlap.
4. Subtract the intersection from each set total to fill in the remaining parts of each circle.
5. Find the outside value — subtract all circle values from n(ξ) to find elements in neither set.
6. Use the diagram to answer questions about unions, intersections, complements and probabilities.

Worked Example 1 — Foundation Level (Higher Paper)

Question: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. B = {3, 6, 9}. (a) Draw a Venn diagram. (b) List A ∩ B. (c) List A'.

Working:

A ∩ B = elements in both A and B = {6}.

A only = {2, 4, 8, 10}. B only = {3, 9}. Neither = {1, 5, 7}.

(b) A ∩ B = {6}.

(c) A' = everything not in A = {1, 3, 5, 7, 9}.

Answer: (a) Venn diagram with 6 in the overlap, {2,4,8,10} in A only, {3,9} in B only, {1,5,7} outside. (b) {6}. (c) {1, 3, 5, 7, 9}.

Worked Example 2 — Higher Level

Question: In a class of 30 students, 18 study French (F), 12 study Spanish (S) and 5 study both. (a) Draw a Venn diagram. (b) Find n(F ∪ S). (c) Find the probability that a randomly chosen student studies neither subject.

Working:

n(F ∩ S) = 5. F only = 18 − 5 = 13. S only = 12 − 5 = 7. Neither = 30 − 13 − 5 − 7 = 5.

(b) n(F ∪ S) = 13 + 5 + 7 = 25. Alternatively: 18 + 12 − 5 = 25.

(c) Students studying neither = 5. P(neither) = 5/30 = 1/6.

Answer: (a) Venn diagram with regions 13, 5, 7, 5. (b) 25. (c) 1/6.

Worked Example 3 — Exam Style

Question: 50 people were surveyed about hobbies. 28 like reading (R), 22 like cooking (C), and 8 like both. A person is chosen at random. Find (a) P(R ∪ C), (b) P(R' ∩ C), (c) P(R' ∩ C').

Working:

R ∩ C = 8. R only = 28 − 8 = 20. C only = 22 − 8 = 14. Neither = 50 − 20 − 8 − 14 = 8.

(a) n(R ∪ C) = 20 + 8 + 14 = 42. P(R ∪ C) = 42/50 = 21/25.

(b) R' ∩ C means "not R and C" = C only = 14. P(R' ∩ C) = 14/50 = 7/25.

(c) R' ∩ C' means neither R nor C = 8. P(R' ∩ C') = 8/50 = 4/25.

Answer: (a) 21/25 (b) 7/25 (c) 4/25.

Common Mistakes

• Confusing ∪ and ∩. Union (∪) means "or" and includes everything in either set. Intersection (∩) means "and" and only includes what is in both.
• Counting the intersection twice. When calculating n(A ∪ B), remember to subtract n(A ∩ B) once: n(A) + n(B) − n(A ∩ B).
• Forgetting the "neither" region. Always subtract the total in all circles from n(ξ) to find how many elements are outside all sets.
• Misreading complement notation. A' means everything in ξ that is NOT in A, not just B.

Exam Tips

• Always start by filling in the intersection region first — everything else follows from that.
• When a question uses set notation like (A ∪ B)', shade the diagram to visualise what is being asked.
• Check that all regions of your Venn diagram add up to n(ξ).
• For basic Venn diagram questions without formal notation, see Venn diagrams. For conditional probability extensions, see conditional probability.

Practice Questions

Q1 (Foundation): ξ = {1, 2, ..., 12}. A = {multiples of 3} and B = {even numbers}. List (a) A ∩ B, (b) A ∪ B.

Q2 (Foundation): Using Q1, find n(A') and list the elements.

Q3 (Higher): 60 students: 35 like maths (M), 30 like science (S), 15 like both. Find P(M ∪ S)' — the probability a student likes neither.

Practise set notation and Venn diagrams for free on GCSEMathsAI.

Related Topics

• Venn Diagrams
• Conditional Probability
• Probability Basics and Relative Frequency

Summary

Set notation provides a formal language for describing groups and their relationships. The union A ∪ B includes everything in either set; the intersection A ∩ B includes only what is in both; the complement A' includes everything not in A. Use the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) to avoid double-counting. On a Venn diagram, always fill in the intersection first, then the remaining parts of each circle, then the outside region. Check that all regions sum to n(ξ). These skills underpin Higher-tier probability questions and often appear alongside conditional probability.