Set Notation & Venn Diagrams (Cambridge IGCSE Maths): Revision notes

Section 1

What is a set and how is it written?

A set is a well-defined collection of objects, called elements. Sets are usually named with capital letters and written with curly brackets, e.g. $A = \{2, 4, 6, 8\}$ .

Sets can also be defined using a rule, e.g. $A = \{x : x \text{ is a natural number}\}$ means "A is the set of all x such that x is a natural number". $C = \{x : a \leq x \leq b\}$ describes all values of x between a and b inclusive.

Key termssetelement

Section 2

What does each piece of set notation mean?

Symbol	Meaning
$n(A)$	the number of elements in set A
$A'$	the complement of A — everything NOT in A
$\mathcal{E}$	the universal set — everything under consideration
$A \cup B$	union — elements in A OR B (or both)
$A \cap B$	intersection — elements in BOTH A and B
$\emptyset$	the empty set — a set with no elements
$A \subseteq B$	A is a subset of B — every element of A is also in B
$x \in A$	x is an element of A
$x \notin A$	x is not an element of A

Key termscomplementuniversal setunionintersectionsubsetempty set

Section 3

How do I draw and read a Venn diagram?

A Venn diagram shows sets as overlapping circles inside a rectangle representing the universal set $\mathcal{E}$ .

Elements common to two sets go in the overlapping region (intersection)
Elements in only one set go in the non-overlapping part of that circle
Elements in neither set go outside the circles but inside the rectangle

Extended tier questions may use three overlapping circles; Core tier uses at most two.

Key termsVenn diagram

Exam tip

Fill in the innermost (most overlapping) regions of a Venn diagram first, then work outward — this avoids double-counting elements.

Section 4

How do I use Venn diagrams to solve problems?

Venn diagrams let you calculate $n(A \cup B)$ , $n(A \cap B)$ , and complements directly by counting regions.

Example: In a class of 30 students, 18 study French (F), 15 study Spanish (S), and 8 study both.

$n(F \cap S) = 8$
Only French = $18 - 8 = 10$
Only Spanish = $15 - 8 = 7$
$n(F \cup S) = 10 + 8 + 7 = 25$
Neither = $30 - 25 = 5$

Common mistake

Do not add n(F) and n(S) directly to find the union when there is overlap — this double-counts the intersection. Use n(A∪B) = n(A) + n(B) − n(A∩B).

Must Know

$n(A)$ = number of elements in A; $A'$ = complement; $\mathcal{E}$ = universal set
$A \cup B$ = union (in A or B); $A \cap B$ = intersection (in both)
$\emptyset$ = empty set; $\subseteq$ = subset; $\in$ / $\notin$ = element of / not an element of
Fill Venn diagram regions from the intersection outward
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Core tier uses two-set Venn diagrams; Extended can use three sets

That's the notes covered.

Carry on to the next subtopic.