Venn Diagrams are graphical figures which are used for the representation of relationships between 2 or more sets of data. They are widely used in the fields of mathematics, statistics and data analysis to visually represent similarities and distinctions between items in a set of data.

In this article, we will learn about the Venn diagram, the components of Venn diagrams, how to create Venn diagrams and some examples.

What are Venn Diagrams?

Venn diagrams are the diagrams which are used for illustrating and analysing Set data with the help of overlapping circles. These are used in finding relationships between logically similar data by performing various set operations. They aim to show how two Sets combine, intersect and complement each other.

Venn Diagram Symbols

A Venn diagram consists of multiple overlapping circles, each representing a set. these overlapping circles depict all possible relationships between the sets. The 3 main components in a Venn diagram are:

Circles/Oval

Each circle/oval represents a single set of data.

representation of datasets in venn diagram

Overlapping Regions

The region shared by 2 or more sets is the overlapping region, it represents the common data between multiple sets.

Overlapping region in Venn diagram

Non-overlapping Regions

These parts of the circle represent the data which are unique only to the set and is not common among any other set of data.

Non-overlapping region in venn diagram

How to Create a Venn Diagram?

To create a Venn diagram, we follow the following steps:

Step-1: Identify all sets

The first step to create a Venn diagram is to figure out all the data sets which are being worked with.

Step-2: Identify similarities and dis-similarities between the sets

In this step, we will find all the elements which are common in different sets and the elements which are only unique to one set with the help of given information. Find Union, Intersection and differences of all the given sets.

Step-3: Draw Venn diagram

In this step we draw the Venn diagram through the extracted information about the sets. Represent each set with circles, overlap the circles of the common sets, and set the unique elements apart.

Step-4: Labelling and shading the union and intersections

After making the Venn diagram label the whole diagram, label individual sets and then shade the overlapping regions with different colors to differentiate the common and the uncommon regions.

e.g. Let suppose we are making a two set Venn diagram with sets A and B with A and B having some common data with each other, While making the Venn diagram the common data is represented through a overlapping region between both the circles indicating that both the circles share same data.

Two set Venn diagram

Three Set Venn Diagram

A Three set Venn diagram is made similar to two set Venn diagram but here we compare 3 sets of data instead of two. Determining relationships between more than 2 sets of data can be bit complex. Here’s an Example for how to make a Three set Venn diagram –

Step 1: Let’s suppose we create a Venn diagram to identify how many students like French, English or Spanish. We will make 3 sets i.e. English, Spanish, French.

Step 2: Identify the students who like all the three languages, students who only like English, students who only like Spanish and students who only like French, students who like English and Spanish but not French and more.

set identification in venn diagram

Step 3: Making the Venn diagram by the gathered information, putting common elements in overlap with each other.

Three set Venn diagram example

Step 4: Labeling and shading the diagram.

Following the same steps, we can make four and five set Venn diagrams too.

Venn Diagram for Set Operations

Venn diagram consists of various sets operations which help us in evaluating different relationships like common regions, uncommon regions, complements etc. Some of important set operations in Venn diagram are:

Complement of a Set

Complement of a set refers to opposite of the set, which means everything except the complement will be included. Complement of a set is represented by , ` symbol.

e.g. If we are finding complement of a set named A, then its complement will be A` = U – A , where U is the Universal set which includes everything except set A. a Universal set is a set that contains the whole population i.e. every set in consideration.

Complement of a set

Union of Two Sets

Union in a Venn diagram represents all the elements which occur in given sets whether they are common or uncommon among one another. Union is represented by, ∪ symbol.

e.g. Let say we have 2 set of elements A and B, Union of A and B will be all elements in A and all Element in B.

Union in Venn diagram

Intersection of Two Sets

Intersection of data sets in the Venn diagram are the elements which are common between all the given sets. In the Venn diagram they represent the overlapping region. Intersection is represented by, ∩ symbol.

e.g. In set A and B the elements which occur in both set A and set B represents the intersection. these elements overlap in the Venn diagram

Intersection in Venn diagram

Difference Between Two Sets

Set Difference in dataset are the elements which occur distinctly only in one set and do not overlap with any other element. The difference is represented by , – symbol

e.g. Let suppose we have to find B-A (i.e. B difference A) for two sets A and B, we will include the elements which are uniquely present in set B, i.e. excluding common of set A and B.

vice-versa is true for B-A, only unique elements of set B are included in B difference A.

Difference in Venn diagram (B-A)

Complement of Union of Two Sets

Complement of union of sets which can be represented as (A ∪ B)’ depicts everything in the population except the union of set A and B. By applying De Morgan’s law from set theory (A ∪ B)’ can be written as,

(A ∪ B)’ = A’ ∩ B’, i.e. the intersection of individual complements of both the sets.

It can also be written as (A ∪ B)’ = U – (A ∪ B), Universal set – Union of set A and B.

e.g. Universal Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Set A: {2, 4, 6, 8} , Set B: {1, 3, 5, 7, 9}

Here, (A ∪ B)’ = U – (A ∪ B) = U – {1, 2, 3, 4, 5, 6, 7, 8, 9} = {10}

Complement of union of two sets Venn Diagram

Complement of Intersection of Two Sets

Similarly, as in the case of complement of union of 2 sets, complement of intersection of two sets depicts everything in the population except the Intersection of set A and B. By applying De Morgan’s Law,

(A ∩ B)’ = A’ ∪ B’, i.e. union of complements of the individual sets.

It can also be written as (A ∩ B)’ = U – (A ∩ B), Universal set – Intersection of set A and B.

e.g. Universal Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Set A: {2, 4, 6, 9}, Set B: {3, 6, 9}

Here, (A ∩ B)’ = U – (A ∩ B) = U – {6, 9} = {1, 2, 3, 4, 5, 7, 8, 10}

Complement of Intersection of Two Sets

Symmetric Difference Between Two Sets

Symmetric differences between two sets shows the union of uncommon elements of two sets. It is represented through ? symbol. It can also be written as, A ? B = (A-B) ∪ (B-A) i.e. unique elements of a and unique elements of B.

e.g. Set A: {1, 2, 3, 4} , Set B: {3, 4, 5, 6}

A ? B = (A-B) ∪ (B-A) = {1,2} ∪ {5,6} = {1,2,5,6}

Symmetric difference between two sets Venn Diagram

Solved Examples on Venn Diagrams

Example 1: In a class 10 students are having a discussion about which sports they like. out of 10, 7 like only cricket and 2 like only basketball. One of the students likes both cricket and basketball. Represent the data through Venn diagram.

Solution:

Here we are given data which involves only 2 sports, so we will have to make a two set Venn diagram.

Set A = Students who like cricket = 7

Set B = Students who like basketball = 2

After identifying all sets, we will find all relationships –

Student who likes both = 1, this will be represented by overlapping region between A and B representing the common between both sports.

Now, with the available data we will make our Venn diagram:

Example 1 -sport preferences

Example 2: In a survey it is found that out of 50 people, 10 people like bananas, 20 like apple and 30 like oranges. if 5 people like both apple and banana, 10 like both apples and orange and 2 like all of them, the how many students like only 1 specific fruit?

Solution:

In this problem we have to find how many people like only one kind of fruit. we can easily illustrate it using Venn diagram:

Here we are given 3 fruits and 50 people, so as the number of datasets which we are working with is 3 (apple, banana, orange)

People who like apple = 20

People who like banana= 10

People who like orange= 30

People who like both apple and banana = 5 , who like both apple and orange = 10

Those who like all of them = 2

Making Venn diagram with the given data,

Example 2 -fruit preferences

Hence, the people who like only 1 specific fruit are – (18 + 3 + 3) = 24 people.

Example 3: A study about favourite subjects of students shows that out of 80 students, 40 students choose math 50 choose Science, 30 choose English and 30 choose Hindi. The study also shows that the students also have more than 1 favourite subjects, 15 like both Math and Science, 10 students like both Science and English, 8 like English and Hindi, 5 like Hindi and Math, 4 likes Math, Science and English ,2 like Science, English and Hindi, 4 like English, Hindi and Math and 1 like all the subjects. Express the data using Venn diagram.

Solution:

Given, Total number of students – 80

Maths = 40, Science = 50, English = 30, Hindi = 30

Maths and Science = 15, Science and Hindi= 10, English and Hindi = 8, English and Maths = 5

Math, Science and English = 10, Science, English and Hindi = 2, English, Hindi and Math = 4, Math, Science and Hindi = 3

Likes all the subjects = 1

Now making Venn diagram using the available information:

Example 3 – favourite subject

Practice Questions on Venn Diagrams

Q1. At a wedding, the guests may have ice cream or custard with their dessert, 60 people had Ice-cream, 34 people only had custard, 9 people had both custard and ice-cream. If the total guests in the wedding were 110, How many people didn’t have ice-cream nor custard? Solve using Venn diagram

Q2. 300 people were asked if they own a dog or a cat. 160 people said they own a dog. 3/8 people who have a dog also have a cat. 40 people said they owned neither a cat nor a dog. Illustrate the data using Venn diagram.

Q3. At a function there are 210 people at a concert. 99 of the people are women, 98 of the people are wearing glasses, 142 of the people have straight hair, There are 16 women with straight hair wearing glasses, There are 44 women wearing glasses, There are 53 women with straight hair, All of the people with straight hair and glasses are women. Draw a Venn diagram to represent the above data.

Q4. Question: In a company, 100 employees were surveyed: 40 employees have a bachelor’s degree, 30 employees have a master’s degree, 20 employees have both a bachelor’s and a master’s degree, 10 employees have neither degree, How many employees have at least one degree?

Q5. In a group of 80 people: 30 people like action movies, 25 people like comedy movies, 20 people like drama movies, 10 people like all three genres, 15 people like action and comedy movies but not drama, 10 people like comedy and drama movies but not action. How many people like exactly one genre?

FAQs on Venn Diagrams

What is a Venn Diagram in Math?

Venn diagrams are pictorial form to represent sets and their relations.

What are Different Types of Venn Diagram?

Venn diagrams are classified on the basis of the number of sets:

• Two set Venn diagram
• Three set Venn diagram
• Four set Venn diagram
• Five set Venn diagram

What is Formula for Venn Diagram?

Formula for Venn diagram is:

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Can a Venn Diagram have Two Non-intersecting Circles?

Yes, sets with no common elements are represented as two non-intersecting circles in Venn diagram.

Can a Venn Diagram have Three Circles?

Yes, Venn diagram with three sets have 3 circles.