IGCSEMathsEdexcelMaths AHigherRevision NotesNumbers & the Number SystemSet Notation & Venn DiagramsVenn Diagrams with Three Sets

Venn Diagrams with Three Sets (Edexcel IGCSE Maths A): Revision Note

Exam code: 4MA1

Written by: Dan Finlay

Reviewed by: Roger B

Updated on 27 March 2026

Venn diagrams with three sets

What does a Venn diagram with three sets look like?

There is a rectangle representing the universal set
There are three circles
- One for each of the sets
  - E.g. $A$ , $B$ and $C$
The three circles intersect and split the rectangle into eight regions
- A region where all three circles intersect
  - $A \cap B \cap C$
- Three regions where exactly two circles intersect
  - $A \cap B \cap C'$ , $A \cap B' \cap C$ and $A' \cap B \cap C$
- Three regions where each circle does not intersect with any other circle
  - $A \cap B' \cap C'$ , $A' \cap B \cap C'$ and $A' \cap B' \cap C$
- A region outside the three circles
  - $A' \cap B' \cap C'$
  - This can also be written as $(A \cup B \cup C)'$

Venn diagram with three overlapping circles labelled A, B, C. Regions are marked with formulas like AnBnC and A'nB'nC for set operations. — **The intersections of three sets**

Venn diagram with three overlapping circles labelled A, B, C, showing numbers 1 to 11 in various sections, including intersections and outer areas. — **Example of a Venn diagram with three sets**

It is possible that two of the circles do not intersect
- E.g. if $A \cap C = \emptyset$

Venn diagram with three circles labelled A, B, and C. Numbers inside circles are 4, 11, 7, 5, 8. Outside circles are 9, ε. — **Example of a Venn diagram where two sets do not have an intersection**

How do I find the number of elements in a subset?

Identify the intersections which make up the subset
- E.g. the subset $A \cap B$ is made up of $A \cap B \cap C$ and $A \cap B \cap C'$
Add together the number of elements in the intersections

Two Venn diagrams with sets A, B, C. Top highlights A∩B: 11+3=14. Bottom highlights A: 4+1+11+3=19, with shaded areas shown. — **Example of finding number of elements in subsets**

How do I fill in a Venn diagram with three sets?

Start with the intersection of all three circles $A \cap B \cap C$
- Fill in the number or label it $x$ if it is unknown
Fill in regions where exactly two circles intersect
- You might be given the total number of elements in the intersection between those two sets
  - E.g. There are 20 elements that are in both set $A$ and set $C$
- Subtract the number in the intersection of all three circles to find the number of elements that are just in those two sets
  - E.g. $20 - x$ elements are in set $A$ and set $C$ but not set $B$

Venn diagram with three circles A, B, C. Overlapping areas marked with numbers and expressions. Notes on the left explaining intersections.

Fill in the parts of the circles which do not intersect other circles
- You might be given the total number of elements in a set
  - E.g. There are 60 elements in set $A$
- Subtract the numbers in the intersections which involve set $A$
  - E.g. subtract the number of elements in $A \cap B \cap C$ , $A \cap B \cap C'$ and $A \cap B' \cap C$ from the number of elements in $A$

Venn diagram with three sets A, B, C. Set A is shaded with numbers: 27, 13, x, and 20-x. Equation shown: 60 - x - (20 - x) - 13 = 27.

Fill in the number outside all the circles
- This is the total number of elements minus the number of elements in all the intersections

Worked Example

Some students were asked whether they like studying statistics $(S)$ , algebra $(A)$ and geometry $(G)$ .

5 said they like studying all three of statistics, algebra and geometry
11 said they like studying statistics and algebra
16 said they like studying algebra and geometry
8 said they like studying statistics and geometry
25 said they like studying geometry
4 said they do not like studying any of the three topics
the number who said they like studying statistics only is the same as the number who said they like studying algebra only

Let $x$ be the number of students who said they like studying statistics.

(a) Show all this information on the Venn diagram, giving the number of students in each appropriate subset, in terms of $x$ where necessary. Simplify all expressions.

Venn diagram with three overlapping circles labelled S, A, G, enclosed in a rectangle labelled ε, representing set relationships.

Answer:

Put 4 outside the circles

Put 5 in the intersection of all three circles

Find the number of students who like statistics and algebra only

11 - 5 = 6

Find the number of students who like algebra and geometry only

16 - 5 = 11

Find the number of students who like statistics and geometry only

8 - 5 = 3

Find the number of students who like geometry only

25 - 5 - 11 - 3 = 6

Find the number of students who like studying statistics only

Give your answer in terms of $x$

$x - 5 - 6 - 3 = x - 14$

The number of students who like algebra only is the same as the number of students who like statistics only

Venn diagram with three overlapping circles labelled S, A, and G, containing numbers and expressions. The area outside the circles has the number 4.

(b) Given that 30 students said they like studying algebra. Find the number of students who said they like studying statistics.

Answer:

Add together the numbers in the circle for algebra

$x - 14 + 6 + 5 + 11 = x + 8$

Set this equal to 30 and solve for $x$

$\begin{array}{rcl} x + 8 & = & 30 \\ x & = & 22 \end{array}$

22 students said they like studying statistics

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