Sets and Venn Diagrams

A set is a collection of things, called elements. Venn diagrams draw sets as overlapping circles inside a rectangle (the universal set). Learn the symbols, fill diagrams from the middle outwards, and counting questions become simple bookkeeping.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use the notation $\xi$ , $\in$ , $\notin$ , $n(A)$ , $A'$ , $\cap$ , $\cup$ , $\subset$ and $\varnothing$ .
List the elements of sets such as $A\cap B'$ .
Complete Venn diagrams with two or three sets.
Solve counting problems, including with an unknown $x$ .

Pause and think

In a class, $18$ students play football and $15$ play cricket. The class has only $30$ students. How can that be?

Section 2 of 14

Core idea

Start hereCounting people twice

A teacher asks: "Who studies French?" 25 hands. "Spanish?" 18 hands. The class has 40, and 7 study neither. $25+18=43$ is more than the 33 who study something, because some students raised their hand twice. Venn diagrams make the double-counting visible.

The big idea

$A\cap B$ : in both; $A\cup B$ : in either; $A'$ : not in $A$ . $n(A\cup B)=n(A)+n(B)-n(A\cap B)$ .

1. Notation

$\mathscr E$ : universal set.
$x\in A$ : $x$ is in $A$ ; $n(A)$ : number of elements.
$A\subset B$ : every element of $A$ is in $B$ .
$\varnothing$ : empty set.

E = {1, …, 10}, A = even numbers, B = multiples of 3

A\cap B=\{6\}

(in both).

A\cup B=\{2,3,4,6,8,9,10\}

(in either).

A'=\{1,3,5,7,9\}

(not in A).

The addition rule

$n(A)+n(B)$
Counts the overlap $A\cap B$ twice.
$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
Subtract it once.

2. Counting with Venn diagrams

Worked exampleFrench and Spanish

40 students; 25 French, 18 Spanish, 7 neither.

Goal: Both; French only; Spanish only.
Strategy: Find the overlap first.

$n(F\cup S)=40-7=33$
At least one.
$n(F\cap S)=25+18-33=10$
Addition rule.
$25-10=15,\quad18-10=8$
Only regions.

AnswerBoth 10; French only 15; Spanish only 8.

✓ Check

15+10+8+7=40

✓.

Worked exampleUsing x

$n(\mathscr E)=30$ , $n(A)=17$ , $n(B)=14$ , $n((A\cup B)')=4$ . Find $n(A\cap B)$ .

Goal: $x$ .
Strategy: Label the overlap $x$ ; regions sum to 30.

$(17-x)+x+(14-x)+4=30$
Four regions.
$35-x=30\Rightarrow x=5$
Solve.

Answer

5

✓ Check

12+5+9+4=30

✓.

Common misconceptionWriting $n(A)$ inside A only 17 is the whole circle A, including the overlap.

Pause and think

List the subsets of $\{a,b\}$ .

Summary

Fill the overlap first.
Use $x$ for an unknown region.

Intersection: $A\cap B$ .
Union: $A\cup B$ .
Complement: $A'$ .

HistoryCantor's infinite sets. Georg Cantor founded set theory in the 1870s and showed something astonishing: some infinite sets are bigger than others. There are "more" real numbers than whole numbers, even though both sets go on forever.

Fun factThe barber paradox. A barber shaves everyone in the village who doesn't shave themselves. Who shaves the barber? Bertrand Russell's version of this paradox (1901) showed that careless set definitions lead to contradictions, and forced mathematicians to rebuild set theory carefully.

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