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

Section 1

What is a set and how do we write one?

A set is a well-defined collection of objects, called elements or members.

Sets are usually named with capital letters: $A$ , $B$ , $\mathscr{E}$
Elements are listed inside curly brackets: $A = \{1, 2, 3, 4\}$
$x \in A$ means " $x$ is an element of $A$ "
$x \notin A$ means " $x$ is not an element of $A$ "
$n(A)$ means "the number of elements in set $A$ "

Example: If $A = \{2, 4, 6, 8\}$ , then $n(A) = 4$ , and $4 \in A$ but $5 \notin A$ .

Key termsSetElementn(A)

Example

If $B = \{\text{prime numbers less than } 10\} = \{2, 3, 5, 7\}$ , then $n(B) = 4$ .

Section 2

What are the universal set, empty set, and subsets?

The universal set, $\mathscr{E}$ (or sometimes $U$ ), contains all elements being considered in a given question.
The empty set, written $\varnothing$ or $\{\}$ , contains no elements: $n(\varnothing) = 0$ .
$A$ is a subset of $B$ , written $A \subseteq B$ , if every element of $A$ is also in $B$ .
$A$ is a proper subset of $B$ , written $A \subset B$ , if $A \subseteq B$ but $A \neq B$ .

Table of subset symbols:

Symbol	Meaning
$A \subseteq B$	$A$ is a subset of $B$ (could be equal to $B$ )
$A \subset B$	$A$ is a proper subset of $B$ (not equal to $B$ )
$A \not\subset B$	$A$ is not a subset of $B$

Key termsUniversal setEmpty setSubsetProper subset

Common mistake

Do not confuse $\in$ (an element belongs to a set) with $\subseteq$ (one set belongs inside another set) — $3 \in A$ but $\{3\} \subseteq A$ .

Section 3

What do union, intersection, and complement mean?

These three operations combine or restrict sets.

Union $A \cup B$ : everything in $A$ OR $B$ (or both) — combine the sets.
Intersection $A \cap B$ : only elements in BOTH $A$ AND $B$ — the overlap.
Complement $A'$ : everything in $\mathscr{E}$ that is NOT in $A$ .

Key counting rule: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

This avoids double-counting elements that are in both sets.

Example: If $\mathscr{E} = \{1,2,3,...,10\}$ , $A = \{2,4,6,8,10\}$ , $B = \{3,6,9\}$ , then:

$A \cup B = \{2,3,4,6,8,9,10\}$
$A \cap B = \{6\}$
$A' = \{1,3,5,7,9\}$

Key termsUnionIntersectionComplement

Exam tip

Remember: $\cup$ (union) looks like a cup that holds everything — think 'combine'. $\cap$ (intersection) looks like an upside-down cup — think 'overlap only'.

Common mistake

A common error is forgetting to subtract $n(A \cap B)$ once when finding $n(A \cup B)$ , which double-counts the overlap.

Section 4

How do Venn diagrams represent sets?

A Venn diagram shows sets as overlapping circles inside a rectangle representing $\mathscr{E}$ .

The rectangle = universal set $\mathscr{E}$
Each circle = one set (e.g. $A$ , $B$ )
The overlapping region = $A \cap B$
Outside all circles (but inside the rectangle) = elements in neither set, i.e. $(A \cup B)'$

Reading numbers from a Venn diagram (two sets): If a diagram shows 5 in the " $A$ only" region, 3 in the overlap, and 4 in the " $B$ only" region, with 2 outside both circles, then:

$n(A) = 5 + 3 = 8$
$n(B) = 3 + 4 = 7$
$n(A \cap B) = 3$
$n(A \cup B) = 5 + 3 + 4 = 12$
$n(\mathscr{E}) = 5 + 3 + 4 + 2 = 14$

Key termsVenn diagram

Think of it like this

Think of two overlapping circles as two friend groups at a party — the overlapping middle is people who know both groups; outside both circles are guests who know neither.

Section 5

How do I solve worded problems using sets?

Exam questions often describe real situations (e.g. students studying subjects) rather than drawing the diagram for you.

Method:

Define what each set represents (e.g. $F$ = students who study French).
Fill in the intersection region first (the number who satisfy both conditions).
Work outwards to fill in "only $A$ " and "only $B$ " regions by subtraction.
Use $n(\mathscr{E})$ minus everything placed so far to find the region outside both circles.
Check all regions sum to $n(\mathscr{E})$ .

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

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

Exam tip

Always fill in the intersection value first when working from a worded problem — everything else is found by subtraction from there.

Must Know

$\in$ means "is an element of"; $A \subseteq B$ means " $A$ is a subset of $B$ "
$A \cup B$ = everything in $A$ or $B$ (combine); $A \cap B$ = only the overlap
$A'$ = the complement of $A$ = everything in $\mathscr{E}$ not in $A$
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(\varnothing) = 0$ ; the empty set has no elements
In worded problems, always find the intersection first, then work outwards by subtraction

That's the notes covered.

Carry on to the next subtopic.