Mathematics Project on – Verify the Distributive Law Using Venn Diagram

Table of Contents

Introduction | Page No . 1

Set theory forms the foundational basis for most mathematical reasoning and logic. It deals with the study of sets, which are defined as collections of distinct objects. These objects can be numbers, symbols, or even other sets. One of the most effective ways to understand set theory visually is through the use of Venn diagrams, which help us analyze how different sets relate to each other using overlapping circles.

Among the many important properties of set theory, the distributive law plays a crucial role in simplifying expressions that involve unions and intersections of sets. In this project, we will explore and verify the distributive laws using Venn diagrams. We will use three non-empty sets and apply both theoretical and practical examples—including case-based and story-based situations—to strengthen our understanding of how these laws function.

Set Theory and Distributive Laws | Page No . 2

The distributive laws in set theory describe how the operations of union (∪) and intersection (∩) interact with each other. These laws allow us to rearrange and simplify set expressions in a structured and logical manner, much like algebraic distributive properties.

There are two main distributive laws in set theory:

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
This law states that the intersection of a set A with the union of sets B and C is equal to the union of the intersections of A with B and A with C.
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
This law states that the union of a set A with the intersection of sets B and C is equal to the intersection of the unions of A with B and A with C.

Understanding Venn Diagrams | Page No . 3 - 4

A Venn diagram is a visual tool used in set theory to show the relationships between two or more sets. It represents sets as circles (or other closed curves) that overlap to illustrate how elements are shared or separated. Venn diagrams help in understanding concepts like union, intersection, difference, and complement in an intuitive way.

They are named after the English logician John Venn, who introduced this form of diagram in 1880.

Basic Components

Universal Set (U): Represented by a rectangle that contains all the sets under consideration.
Sets (A, B, C, etc.): Represented by circles inside the universal set.
Elements: Placed inside the regions of the circles based on their membership.

Types of Venn Diagrams

(i) Two-Set Venn Diagram: This compares two sets, for example A and B.

A ∪ B: The total area covered by both circles.
A ∩ B: The overlapping area of both circles.
A – B: The part of A that does not overlap with B.
A’ : The area outside of circle A but still inside the universal set.

(ii) Three-Set Venn Diagram: This includes three sets: A, B, and C. It has more regions because of the different ways the circles can overlap.

The central overlapping area where all three circles meet represents elements common to A, B, and C.
Pairwise overlaps represent elements that are in two of the three sets.
The union A union B union C includes all areas covered by any of the three circles.

Problems | Page No . 5 - 9

Question 1 (Basic Algebraic Type Question)

Let us consider three sets: A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}. verify : A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Solutions

Step 1: Find B ∪ C

B = {2, 3, 4}

C = {3, 4, 5}

So, B ∪ C = {2, 3, 4, 5}

Step 2: Find A ∩ (B ∪ C)

A ∩ (B ∪ C) = {2, 3}

Step 3: Find A ∩ B

A = {1, 2, 3}

B = {2, 3, 4}

So, A ∩ B = {2, 3}

Step 4: Find A ∩ C

A = {1, 2, 3}

C = {3, 4, 5}

So, A ∩ C = {3}

Step 5: Find (A ∩ B) ∪ (A ∩ C)

We now take the union of {2, 3} and {3}

That gives us: {2, 3}

∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = {2, 3}

Question 2 (Problem Sum Type Questions)

In a class of 40 students, some students participate in different sports: 18 students play Football, 22 play Cricket, and 16 play Basketball. Among them, 8 students play both Football and Cricket, 7 play both Cricket and Basketball, 5 play both Football and Basketball, and 3 students play all three sports. Let the sets be:

A = students who play Football,
B = students who play Cricket, and
C = students who play Basketball.

Based on this data, draw a Venn diagram and answer the following:

How many students play only Football?
How many students play both Football and Cricket but not Basketball?
Using the Venn diagram, verify whether the distributive law A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Question 3 (Problem Sum Type Questions)

A survey was conducted on beverage preferences. Out of the total respondents, 60 people liked Tea, 50 liked Coffee, and 40 liked Juice. Additionally, 20 people liked both Tea and Coffee, 15 liked both Coffee and Juice, 10 liked both Tea and Juice, and 5 liked all three. Let the sets be defined as:

A = people who like Tea,
B = people who like Coffee, and
C = people who like Juice.

Draw a Venn diagram using this data. Then determine the number of people who like at least one beverage. Afterward, verify the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) based on the counts from the Venn diagram.

Question 4 (Problem Sum Type Questions)

In a college, students are grouped according to their chosen elective subjects. Records show that 25 students chose Computer Science, 30 chose Mathematics, and 20 chose Economics. Further, 10 students opted for both Computer Science and Mathematics, 8 for both Mathematics and Economics, 6 for both Computer Science and Economics, and 4 students chose all three electives. Denote the sets as:

A = students taking Computer Science,
B = students taking Mathematics, and
C = students taking Economics.

Draw a Venn diagram based on this data. Then answer: How many students opted for only one subject? Finally, verify whether the distributive law A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) holds in this case by comparing both sides of the identity using the diagram.

Question 5 (Story-Based Question)

At a village fair, there were three types of stalls: food stalls, toy stalls, and book stalls. A survey found that 120 people visited food stalls, 90 visited toy stalls, and 70 visited book stalls. Also, 50 people visited both food and toy stalls, 30 visited both toy and book stalls, 20 visited both food and book stalls, and 10 visited all three types of stalls. Define the sets as:

A = visitors to food stalls,
B = visitors to toy stalls, and
C = visitors to book stalls.

Draw a three-circle Venn diagram to represent this data. Then answer:

(a) How many visitors went to only one type of stall?

(b) How many visited exactly two types?

Based on your counts, verify whether the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) is satisfied in this real-world context.

Conclusion | Page No . 10

Through this project, we successfully explored the distributive laws of set theory using both algebraic and visual (Venn diagram) methods. By working with three non-empty sets in different contexts—ranging from simple number sets to real-life scenarios like student electives and public surveys—we have seen how union and intersection operations interact under distributive rules.

Each example confirmed that the identities

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

hold true when verified step by step using set operations and corresponding Venn diagrams.

This project not only strengthened our conceptual understanding but also demonstrated the power of set theory in problem solving and logical reasoning.

Bibliography | Page No . 11

NCERT Mathematics Textbook – Class 11: Chapter on Sets and Venn Diagrams
RD Sharma – Mathematics for Class 11: Exercises on Set Operations and Properties
S. Chand Publication – Set Theory and Logic: Comprehensive coverage of Set Laws
FlashEducation.online
Wikipedia.org – Set Theory and Venn Diagram: Overview and historical background
Khan Academy – Set Theory Video Lectures: Tutorials and examples on union, intersection, and distributive laws

Acknowledgement | Page No . 12

I would like to express my heartfelt gratitude to all those who guided and supported me in completing this mathematics project on “Using Venn Diagram to Verify the Distributive Law for Three Non-Empty Sets.”

First and foremost, I thank my mathematics teacher for their consistent encouragement, valuable feedback, and detailed explanations that made the concept of set theory both accessible and interesting.

I also acknowledge the help of my classmates and family for providing support in brainstorming real-life examples and helping me understand the problem-solving approach behind Venn diagrams.

Lastly, I would like to thank all the authors and online platforms whose resources helped me research and prepare this project thoroughly.
This project has helped me enhance my understanding of mathematical logic, structure, and visualization through Venn diagrams.