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Sets

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Cambridge O Level Mathematics (Syllabus D) 4024

Sets

The big idea: A set is a collection of elements, and a Venn diagram shows how sets overlap. Read the notation carefully: almost every mark here is lost by confusing ∪ with ∩, or by forgetting the elements that are in none of the sets.

Summary — What This Chapter Covers

Describe sets in a list, in words, or with set-builder notation such as {x: −2 ⩽ x < 3}.
Use n(A), ∈, ∉, A′, ∅, ℰ, ⊆, ⊈, A ∪ B and A ∩ B.
Represent two or three sets on a Venn diagram, and read sets and regions from one.
Solve counting problems with Venn diagrams, including finding an unknown number of elements.

1. Describing Sets

Syllabus 1.2 · Paper 1 & 2

Lists, words and set-builder notation

A set can be described by listing its elements in curly brackets, D = {a, b, c, …}, or by a rule. In set-builder notation the colon reads “such that”:

Notation	Meaning
A = {x: x is a natural number}	A is the set of natural numbers 1, 2, 3, …
C = {x: a ⩽ x ⩽ b}	all numbers from a to b inclusive
B = {(x, y): y = mx + c}	all the points (x, y) on the line y = mx + c

Worked Example

A = {x: x is an integer, −2 ⩽ x < 3}. List the elements of A and write down n(A).

2. Set Notation

Syllabus 1.2 · Paper 1 & 2

The symbols the exam uses

Symbol	Meaning
n(A)	number of elements in set A
∈ / ∉	“is an element of” / “is not an element of”
ℰ	the universal set: everything under consideration
A′	complement of A: elements of ℰ that are not in A
∅	the empty set (no elements)
A ⊆ B / A ⊈ B	A is a subset of B (every element of A is in B) / A is not a subset of B
A ∪ B	union: elements in A or B or both
A ∩ B	intersection: elements in both A and B

Worked Example

ℰ = {1, 2, 3, …, 12}, A = {multiples of 3}, B = {factors of 12}.
List (a) A ∩ B (b) A ∪ B (c) A′ (d) (A ∪ B)′. (e) Write down n(A ∪ B).

Practice Question 1

Using the sets above, is each statement true or false?
(a) 9 ∈ B (b) {3, 6} ⊆ A (c) A ∩ B′ = {9} (d) A′ ∩ B = ∅

3. Venn Diagrams and Regions

Syllabus 1.2 · Paper 1 & 2

Two or three overlapping circles inside a rectangle

The rectangle is ℰ. Each set is a circle; overlaps are intersections. To fill in a Venn diagram, always start with the innermost region (the intersection) and work outwards, and put the elements in none of the sets outside the circles but inside the rectangle.

Region (two sets)	Notation
in A only	A ∩ B′
in both	A ∩ B
in B only	A′ ∩ B
in neither	(A ∪ B)′ = A′ ∩ B′
in exactly one set	(A ∩ B′) ∪ (A′ ∩ B)

Shading questions

To shade a region such as A′ ∪ B, shade A′ (everything outside A) and B separately, then keep every region shaded at least once. For ∩ keep only the regions shaded twice.

4. Counting with Venn Diagrams

Syllabus 1.2 · Paper 1 & 2

Numbers in regions, not lists

numbers of elements

Worked Example

Practice Question 2

Memorise

∪ and ∩

Complement

Subset

Fill order

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Past papers that test Sets

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4024 Jun 2026 · Paper 2 · Variant 2 Cambridge · mark scheme Open → 4024 Nov 2025 · Paper 2 · Variant 2 Cambridge · mark scheme Open → 4024 Jun 2025 · Paper 2 · Variant 2 Cambridge · mark scheme Open → 4024 Nov 2024 · Paper 1 · Variant 2 Cambridge · mark scheme Open → 4024 Jun 2024 · Paper 2 · Variant 2 Cambridge · mark scheme Open → 4024 Nov 2023 · Paper 1 · Variant 2 Cambridge · mark scheme Open →

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Sets

Sets

Lists, words and set-builder notation

The symbols the exam uses

Two or three overlapping circles inside a rectangle

Numbers in regions, not lists

∪ and ∩

Complement

Subset

Fill order

Past papers that test Sets

More Mathematics D topics