Venn Diagram Questions

Venn diagram questions play a key role in mathematics, particularly in areas like set theory, logical reasoning, and data interpretation. John Venn first introduced these diagrams. They help us visually show the relationships between different sets using circles or other closed shapes. 

In Venn diagrams, each set is typically shown as a circle. The areas where these circles overlap or don’t overlap indicate how the sets relate to each other. These diagrams are very useful for solving problems involving the union of sets, the intersection of sets, the complement of a set, and subset relationships. 

The appeal of Venn diagram questions is in their clarity and usefulness. Whether in school math or competitive exams, questions about set operations, universal sets, disjoint sets, and symmetric difference can be addressed easily with Venn diagrams.

 

Table of Contents

• Basic Set Operations with Meanings
• Important Venn Diagram Formulas
• Venn Diagram Questions with Solutions
• Practice Questions on Venn Diagrams
• Real-Life Use of Venn Diagram Questions
• Tips to Solve Venn Diagram Questions
• Conclusion
• FAQs on Venn Diagram Questions

Basic Set Operations with Meanings

To solve venn diagram questions effectively, it's important to first understand the basic set operations. Here's a breakdown of the key set operations used in Venn diagrams:

 

Understanding these set operations is key to mastering venn diagram questions. They also help in solving problems involving disjoint sets (no common elements), subsets, and complement of a set.

 

Important Venn Diagram Formulas

Let’s now look at some critical formulas that simplify solving venn diagram questions:

 

Two Sets

•  n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

•  n(A - B) = n(A) - n(A ∩ B)

•  n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

 

Three Sets

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)

 

Universal Set Relations

•  A ∪ A′ = U (Universal Set)

•  A ∩ A′ = ∅ (Empty Set)

•  (A ∪ B)′ = A′ ∩ B′

•  (A ∩ B)′ = A′ ∪ B′

•  (A′)′ = A

• U′ = ∅ and ∅′ = U

These formulas allow quick evaluation of even complex venn diagram questions, especially in exams where time is limited.

 

Venn Diagram Questions with Solutions

Let’s now solve some commonly asked venn diagram questions:

Question 1
If in a class, 30 students like Maths, 25 like Science and 10 like both, how many students like only Maths?

Solution:
n(M) = 30
n(S) = 25
n(M ∩ S) = 10
n(M - S) = 30 - 10 = 20

This question used the intersection of sets and set difference.

 

Question 2
Out of 100 people, 70 like tea, 50 like coffee, and 30 like both. Find how many people like only tea.

Solution:
n(Tea) = 70
n(Coffee) = 50
n(Tea ∩ Coffee) = 30
n(Only Tea) = 70 - 30 = 40

Here, the union of sets would be 70 + 50 - 30 = 90, showing 10 people like neither.

Question 3
In a class of 50 students, 30 students play football, 25 play cricket, and 15 play both. How many students play only cricket?

Solution:
n(Cricket - Football) = 25 - 15 = 10

In this venn diagram question, we apply set difference again.

 

Question 4 (Three-Set Problem)
In a survey of 100 students:
60 like Apples (A)
50 like Bananas (B)
40 like Cherries (C)
25 like both Apples and Bananas
20 like both Bananas and Cherries
15 like both Apples and Cherries
10 like all three

Find: How many students like only Apples?

Solution:
Use the formula:
n(A only) = n(A) - [n(A ∩ B) + n(A ∩ C) - n(A ∩ B ∩ C)]
= 60 - [25 + 15 - 10] = 60 - 30 = 30

 

Question 5:

In a class of 60 students, 35 study Hindi, 30 study English, and 20 study both. Find how many students study neither subject.

Solution:
n(H ∪ E) = 35 + 30 - 20 = 45
n(Neither) = 60 - 45 = 15

 

Question 6:

Among 100 students:

• 60 study Physics

• 70 study Chemistry

• 50 study both

 Find how many study only one subject.

Solution:
Only Physics = 60 - 50 = 10
Only Chemistry = 70 - 50 = 20
Total only one subject = 10 + 20 = 30

 

Question 7:

In a survey:

• 80 people watch Movies

• 60 watch TV Shows

• 30 watch both

How many people watch only Movies or only TV Shows?

Solution:
Only Movies = 80 - 30 = 50
Only TV Shows = 60 - 30 = 30
Total = 50 + 30 = 80

 

Question 8:

In a school:

• 100 students like Maths

• 80 like Science

• 70 like English

• 40 like both Maths and Science

• 30 like both Science and English

• 20 like both English and Maths

• 10 like all three
  
  

How many students like only English?

Solution:
Use the formula:
n(Only English) = n(English) - [n(E ∩ S) + n(E ∩ M) - n(M ∩ S ∩ E)]
= 70 - (30 + 20 - 10) = 70 - 40 = 30

 

Practice Questions on Venn Diagrams

Now it’s your turn! Try solving the following venn diagram questions using the concepts of set theory and venn diagram formulas.

Practice Question 1
In a group of 80 people, 50 like coffee, 30 like tea, and 10 like both.

Find:
(i) How many like only coffee?
(ii) How many like neither?

 

Practice Question 2
Out of 90 students:
40 play Football
50 play Hockey
30 play both

Find: how many play only one game and how many play none.

 

Practice Question 3
In a school:
100 students study Maths
80 study Science
60 study English
40 study both Maths and Science
30 study both Science and English
25 study both Maths and English
10 study all three

Find: how many students study only English.

 

Practice Question 4
Among 150 people, 90 people like music, 80 like dance, and 60 like both.
(a) How many people like only music?
(b) How many people like at least one of the two activities?

Practice Question 5
In a survey of 120 people, 70 like apples, 60 like bananas, and 40 like both.
(a) How many people like only apples?
(b) How many people like neither apples nor bananas?

Practice Question 6
In a class, 100 students were asked about their participation in sports:
60 play basketball, 50 play volleyball, and 30 play both.
(a) How many play only basketball?
(b) How many students play neither sport?

Practice Question 7
In a survey, 85 people said they watch Netflix, 75 said they watch Amazon Prime, and 50 said they watch both.
(a) How many people watch only Netflix?
(b) How many people watch either Netflix or Prime but not both?

Practice Question 8
In a school, 70 students like Science, 65 like Maths, and 40 like both subjects.
(a) How many like only one subject?
(b) How many like neither?

 

Real-Life Use of Venn Diagram Questions

Venn diagram questions are more than exam tools. They are used in:

•  Business: Customer segmentation

•  Data Science: Feature comparison in datasets

•  Linguistics: Commonalities in language roots

•  Computer Science: Logical circuits and query filtering

They help visualize set operations, compare universal sets, and identify disjoint sets or subsets within complex data.

 

Tips to Solve Venn Diagram Questions

• Always start with the intersection of sets, which is at the center of the diagram. 

• Use the set formulas to avoid missing overlapping counts. Identify whether the sets are disjoint, complementary, or subsets. 

• Carefully use symmetric difference if the problem mentions “either but not both.” 

• Know your universal set to manage complements and outside elements.
  
  

Conclusion

Venn diagram questions are important in mathematics and logic. With a solid grasp of set operations, subset relationships, unions, intersections, and complements, any student can tackle these questions easily. 

Remember to:

• Use Venn diagrams for clear visuals

• Practice with formulas like:

• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

• n(A ∪ B ∪ C) for three sets

• Keep going over key concepts: universal set, disjoint sets, set difference, and symmetric difference

With regular practice and a clear understanding of how to solve Venn diagram questions, you’ll build a strong foundation in set theory and related subjects.

 

Related Links

Set Theory Symbols - Understand the fundamental symbols used in set theory, including union, intersection, subset, and complement, with examples to clarify their meanings.

Algebraic Expressions - Learn how to simplify, evaluate, and manipulate algebraic expressions with clear explanations and worked-out examples.

Subsets - Discover what subsets are, how to identify them, and explore examples that illustrate the concept in set theory clearly and simply.

 

Frequently Asked Questions on Venn Diagrams Questions

1. What are Venn diagram questions?

Ans: Venn diagram questions involve logical reasoning using overlapping circles to represent sets. They help visualize relationships such as union, intersection, and complement.

 

2. What are the 4 types of Venn diagrams?

Ans:

1. Two-set Venn diagram

2. Three-set Venn diagram

3. Universal set Venn diagram

4. Symmetrical or asymmetrical Venn diagrams
   
   

3. How to solve 3 Venn diagram questions?

Ans: To solve 3-set Venn diagram questions:

• Draw 3 overlapping circles.

• Start filling in the intersection of all three sets first.

• Use given values to fill in overlapping and non-overlapping parts.

• Use totals to find missing values.
  
  

4. What is ∩ in a Venn diagram?

Ans: The symbol ∩ means intersection. In a Venn diagram, A ∩ B represents the elements that are common to both sets A and B.

 

5. How to solve A ∪ B )'?

Ans: The expression (A ∪ B)' means the complement of A union B, i.e., all elements not in either A or B. To solve:

• Find all elements in A or B.

• Subtract those from the universal set.

 

Practice Venn diagram questions with solutions at Orchids The International School and boost your math skills today!