What are Venn diagrams? - KS3 Maths

Key points

An image of a Venn diagram with three intersecting circles. — Image caption,
A Venn diagram can show data sets such as preferences in market research.

are a visual way of representing . They consist of one or more circles inside a rectangle. The circles usually overlap.
All the data being considered is contained inside the rectangle. This is called the and is represented by the Greek letter ξ (Xi). Each circle represents a set.
Set is used to describe different regions of a Venn diagram.

Understanding and interpreting a Venn diagram

Set notation is used to describe different on a Venn diagram:
- ξ represents the universal set.
- A represents set A and B represents set B, usually a capital letter is used.
- A∩B represents the of set A and set B.
- A∪B represents the of set A and set B.
- A’ represents the of set A.
The elements for each region of a Venn diagram can be written with the set notation equal to the list of inside curly brackets { }.
When sets intersect (overlap), the elements in the intersection are members of both sets. When the sets have no elements in common, the overlap is empty or the circles do not overlap.

Examples

Skip image gallery

Image caption,
Describe each region of the Venn diagram using set notation.
Image caption,
The whole rectangle contains all the elements being considered. This is called the universal set, represented by the Greek letter ξ (Xi).
Image caption,
A is the region representing set A, this is contained inside the circle labelled A. B is the region representing set B, this is contained inside the circle labelled B.
Image caption,
A’ is the complement of A. This is the region that is not set A. It is the region outside the circle for set A. B’ is the complement of B. This is the region that is not set B. It is the region outside the circle for set B.
Image caption,
The overlap between sets is called the intersection. The notation for the intersection of set A and set B is A∩B. The intersection contains elements that are in both set A and set B.
Image caption,
The combination of sets is the union. The notation for the union of set A and set B is A∪B. The union of set A and set B contains the elements that are in set A, the elements that are in set B and the elements that are in both set A and set B.
Image caption,
Describe and list the elements in set A, set B and the intersection of set A and set B. The universal set is the set of integers from 1 to 10 inclusive, ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Image caption,
Set A can be described as the set of odd numbers from 1 to 10 inclusive. In set notation this is A = {odd numbers from 1 to 10 inclusive}. The elements in set A are 1, 3, 5, 7 and 9. In set notation this is A = {1, 3, 5, 7, 9}.
Image caption,
Set B can be described as the set of square numbers from 1 to 10 inclusive. In set notation this is B = {square numbers from 1 to 10 inclusive}. The elements in set B are 1, 4, and 9. In set notation this is B = {1, 4, 9}.
Image caption,
The intersection of set A and set B, A∩B, contains the numbers from 1 to 10 inclusive that are odd and square. A∩B = {1,9}. The only odd square numbers from 1 to 10 inclusive are 1 and 9

Question

30 people answered a survey about whether they have a dog or a cat. The Venn diagram shows the information from the survey.

Answer the following questions:

How many people have got a dog?
How many people have only got a cat?
How many people have a cat and a dog?
How many people do not own a cat?

An image of a Venn diagram with two intersecting circles. The circle on the left is labelled, dog. The circle on the right is labelled, cat. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. The Venn diagram has been populated with numbers. The number, eight, is in the intersection. The number, five, is in the other part of the circle labelled dog. The number, seven, is in the other part of the circle labelled cat. The number, ten, is outside the circles, in the rectangular box.

Drawing Venn diagrams

To draw with two overlapping A and B:

Write each set as a list of .
Write the elements that are in both sets in the , A∩B.
Write the rest of the elements that are just in set A.
Write the rest of the elements that are just in set B.
Write any remaining elements from the in the space inside the rectangle and outside the circles.

To avoid errors:

Make sure that the number of elements in the Venn diagram is equal to the number of elements in the universal set.
Check that a set does not include an element that is not in the universal set.

Examples

Skip image gallery

Image caption,
Draw a Venn diagram for ξ = {positive whole numbers less than 10}, A = {prime numbers} and B = {even numbers}. Write each set as a list of elements. ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 3, 5, 7} and B = {2, 4, 6, 8}.
Image caption,
Draw a rectangle for the universal set and two overlapping circles, one for set A and one for set B. Write the elements that are in both sets in the intersection, A∩B. One number is prime and even. A∩B = 2
Image caption,
Write the rest of the elements that are just in set A. The number 2 has already been entered, 3 5 and 7 are the rest of the prime numbers that are in set A.
Image caption,
Write the rest of the elements that are just in set B. The number 2 has already been entered, 4, 6 and 8 are the rest of the even numbers in set B but not in set A.
Image caption,
Write any remaining elements from the universal set in the space inside the rectangle and outside the circles. The only numbers that have not yet been entered are 1 and 9, they are the odd numbers that are not prime. 1 and 9 are in the universal set but not in either set A or set B.
Image caption,
The Venn diagram shows the prime numbers and even numbers within the universal set of positive whole numbers less than 10
Image caption,
Find the mistake that has been made in this Venn diagram.
Image caption,
Sport is played by 20 people. 4 + 6 + 8 = 18, the Venn diagram only shows the data of 18 people. Assuming that the given data is correct, there should be 2 people placed inside the rectangle and outside the circles who do not play either hockey or football.
Image caption,
Find the mistake that has been made in this Venn diagram.
Image caption,
Set A will only contain the multiples of 3 given in the universal set. This limits the multiples of 3 to 3, 6, 12 and 24 only. 9 is incorrect because it is not a factor of 24

Question

Draw a Venn diagram to show the following data:

ξ = {whole numbers from 1 to 12 inclusive}
A = {prime numbers}
B = {factors of 12}

Practise understanding and interpreting Venn diagrams

Practise understanding and interpreting Venn diagrams with this quiz. You may need a pen and paper to help you with your answers.

Quiz

Real-life maths

A chocolate snack manufacturer may research the features that consumers favour when creating a new product, such as:

a biscuit base
inclusion of nougat, fruit or caramel
the snack size
whether it is low calorie or contains artificial colours

The manufacturer will look for the combination of features that will give a product the highest chance of success when competing with other similar products.

An image of a Venn diagram with three intersecting circles. One circle is labelled, biscuit base. One circle is labelled, caramel. The final circle is labelled, snack size. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ.

In this Venn diagram of market research, the responses collected suggest that inclusion of caramel and the size of the snack are the most favoured features of the new product. As a result, the manufacturer can focus on developing this for their target consumers.