Venn diagrams are visual tools used to illustrate the logical relationships between different sets of items. They use overlapping circles to show how sets intersect, share elements, or remain distinct. By categorizing items in this way, Venn diagrams make it easier to understand similarities and differences.
Venn diagrams are also known as set diagrams or logical diagrams.
In mathematics, Venn diagrams are often drawn within a rectangle, which represents the universal set—the set containing all possible elements under consideration. This visual structure helps simplify and clarify complex relationships between sets.
Terms Related to Venn Diagram
The concept of the Venn diagram is very useful for solving a variety of problems in Mathematics and other areas. To understand more about it, let's learn some important terms related to it.
1) Set
A set is a collection of well-defined, distinct objects, considered as a single entity.
Example: A set of natural numbers: {1, 2, 3, 4, 5, 6, ...}
2) Universal Set
Universal Set is a large set that contains all the sets that we are considering in a particular situation.
For example, suppose we are considering the set of Honda cars in a society say set A, and let set B is the group of red car in the same society then the set of all the cars in that society is the universal set as it contains the values of both the sets, set A and set B in consideration.
3) Subset
Subset is actually a set of values that is contained inside another set i.e. we can say that set B is the subset of set A if all the values of set B are contained in set A.
We can say that N is a subset of W all the values of set N are contained in set W i.e. N ⊆ W
4) Disjoint Sets
Disjoint sets are sets that do not have any elements in common. If none of the elements of set A are present in set B, then sets A and B are disjoint. In other words, the intersection of two disjoint sets is an empty set.
For example, let A be the set of even numbers less than 10 and B be the set of odd numbers less than 10. Then, A = {2, 4, 6, 8} B = {1, 3, 5, 7, 9} We can say that A and B are disjoint sets because they do not share any elements, i.e., A ∩ B = ∅.
Venn Diagram Symbols
Some of the most important types of symbols used in drawing Venn diagrams are,
Venn Diagram Symbols
Name of Symbol
Description
∪
Union Symbol
Union symbol is used for taking the union of two or more sets.
∩
Intersection Symbol
Intersection symbol is used for taking the intersection of two or more sets.
A' or Ac
Compliment Symbol
Complement symbol is used for taking the complement of a set.
How to Draw a Venn Diagram?
Step 1: Identify the sets
Decide how many sets you are representing (1, 2, 3, or more).
Example: Set A = even numbers, Set B = prime numbers.
Step 2: Draw the Universal Set
Draw a rectangle to represent the universal set (U) containing all possible elements in the problem.
Step 3: Draw Circles for Each Set
Draw one circle for each set inside the rectangle.
Label each circle clearly (A, B, C…).
Make the circles overlap if the sets have common elements.
Step 4: Fill in the Elements
Intersection areas: Place elements that belong to more than one set in the overlapping regions.
Exclusive areas: Place elements that belong to only one set in the non-overlapping parts of the circle.
Outside all circles: Place elements that do not belong to any set in the rectangle outside the circles.
Example: Take a set A representing even numbers up to 10 and another set B representing natural numbers less than 5 then their interaction is represented using the Venn diagram.
Venn Diagram for Sets Operations
There are different operations that can be done on sets in order to find the possible unknown parameter, for example, if two sets have something in common, their intersection is possible. The basic operations performed on the set are,
The Union of two or more two sets represents the data of the sets without repeating the same data more than once, it is shown with the symbol ⇢∪.
n(A∪ B) = {a: a∈ A OR a∈ B}
Venn Diagram of Intersection of Sets
The intersection of two or more two sets means extracting only the amount of data that is common between/among the sets. The symbol used for the intersection⇢ ∩.
n(A∩ B)= {a: a∈ A and a∈ B}
Venn Diagram of Complement of a Set
Complementing a set means finding the value of the data present in the Universal set other than the data of the set.
n(A') = U- n(A)
Venn Diagram of Difference of Set
Suppose we take two sets, Set A and Set B then their difference is given as A - B. This difference represents all the values of set A which are not present in set B.
For example, if we take Set A = {1, 2, 3, 4, 5, 6} and set B = {2, 4, 6, 8} then A- B = {1, 3, 5}.
In the Venn diagram, we represent the A - B as the area of set A which is not intersecting with set B.
Three-Set Venn Diagram
We can represent three sets easily using the Venn Diagram. Their representation is done by three overlapping circles. Suppose we take three sets of Set A of the people who play cricket. Set B of the people who are graduates and Set C of the people who are 18 years and above of the age.
Then the Venn diagram representing the above three sets is drawn using three circles and taking their intersection wherever required.
We can represent the intersection of three sets using the Venn diagram. The below image represents the intersection of three sets.
We can find the various parameters using the above Venn diagram.
Suppose we have to find,
No of graduates who play cricket it is given by B⋂C
No of graduates who play cricket and are at least 18 years old is given by A⋂B⋂C, etc.
Example: In a class of 40 students, 18 like Mathematics, 16 like Science, and 10 like both Mathematics and Science. Then find the students who like either Mathematics or Science.
Solution:
Let A be the set of students who like Mathematics and B be the set of students who like Science, then n(A) = 18, n(B) = 16, and
n(A ⋂ B) = 10
Now to find the number of students who like either Mathematics or Science i.e. n(A U B) we use the above formula.
n(A U B) = n(A) + n(B) – n (A ⋂ B) ⇒ n(A U B) = 18 + 16 - 10 ⇒ n(A U B) = 24
Applications of Venn Diagram
Venn diagrams have various use cases such as solving various problems and representing the data in an easy-to-understand format. Various applications of Venn Diagrams are:
The relation between various sets and their operations can be easily achieved using Venn diagrams.
They are used for explaining large data sets in a very easy way.
They are used for logic building and finding the solution to complex data problems.
They are used to solve problems based on various analogies.
Analysts use Venn diagrams to represent complex data in easily understandable ways, etc.
Example 1: Set A= {1, 2, 3, 4, 5} and U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Represent A' or Ac on the Venn diagram.
Solution:
Venn Diagram for A'
Example 2: In a Group of people, 50 people either speak Hindi or English, 10 prefer speaking both Hindi and English, 20 prefer only English. How many people prefer speaking Hindi? Explain both by formula and by Venn diagram.
Solution:
According to formula,
n(H∪E) = n(H) + n(E) - n(H∩E)
Both English and Hindi speakers, n(H∩E) = 10
English speakers, n(E)= 20
Either Hindi or English, n(H∪E)= 50
50= 20+ n(H) - 10
n(H)= 50 - 10
n(H)= 40
From Venn Diagram,
Example 3: In a Class, Students like to play these games- Football, Cricket, and Volleyball. 5 students Play all 3 games, 20 play Football, 30 play Volleyball, and 40 play Cricket. 10 play both cricket and volleyball, 12 play both football and cricket, 9 play both football and volleyball. How many students are present in the class?
Example 4: Represent the above information with the help of a Venn diagram showing the amount of data present in each set.
Solution:
Above information should look something like this on Venn diagram,
Example 5: Below given Venn diagram has all the sufficient information required to show the data of all the sets possible. Observe the diagram carefully then answer the following.
What is the value of n(A∩ B∩ C)?
What is the value of n(C)?
What is the value of n(B ∩ A)?
What is the value of n(A∪ B∪ C)?
What is the value of n(B')?
Solution:
Observing the Venn diagram, the above questions can be easily answered,
Question 1: Consider two sets, A and B, where A represents fruits and B represents vegetables. Set A contains apples, bananas, and grapes, while set B contains carrots, lettuce, and apples. Draw a Venn diagram to represent these sets. How many items are only in the fruit category?
Question 2: In a small neighborhood, 10 households have dogs, 7 have cats, and 3 households have both dogs and cats. How many households have at least one kind of pet? Draw a Venn diagram to represent this situation.
Question 3: In a sports club, 120 members play tennis, 150 play badminton, and 50 play both tennis and badminton. How many members play either tennis or badminton? Create a Venn diagram to help you answer.
Question 4: In a class of 30 students, 18 students play basketball, 12 students play football, and 8 students play both basketball and football. How many students do not play either basketball or football?
Question 5: A survey of 100 people was conducted to find their preferences for three types of movies: Action, Comedy, and Drama. The survey results showed:
