+2 MATHEMATICS//CHAPTER-1//RELATIONS AND FUNCTIONS//PART-2//PROBLEMS BASED ON EQUIVALENCE RELATIONS

Introduction

This tutorial will guide you through the concepts of equivalence relations as presented in the video "Problems Based on Equivalence Relations" by Albin Kattakada. Understanding equivalence relations is crucial in mathematics, especially in the fields of set theory and abstract algebra. This guide will break down the core concepts and provide examples to help solidify your understanding.

Step 1: Define Equivalence Relations

Equivalence relations are a special type of relation that satisfies three properties:

• Reflexivity: For any element a, the relation must include (a, a).
• Symmetry: For any elements a and b, if (a, b) is in the relation, then (b, a) must also be in the relation.
• Transitivity: For any elements a, b, and c, if (a, b) and (b, c) are in the relation, then (a, c) must also be included.

Practical Tips

• Verify each property when determining if a relation is an equivalence relation.
• Use concrete examples to illustrate each property.

Step 2: Identify Equivalence Classes

Once an equivalence relation is established, you can form equivalence classes. An equivalence class is a subset of elements that are all equivalent to each other under the relation.

• Notation: If a is an element of set A, the equivalence class of a is denoted as [a].

• Example: If we define a relation on the set of integers where two integers are equivalent if they leave the same remainder when divided by 3, then the equivalence classes would be:

  • [0] = {…, -6, -3, 0, 3, 6, …}
  • [1] = {…, -5, -2, 1, 4, 7, …}
  • [2] = {…, -4, -1, 2, 5, 8, …}

Practical Tips

• List out some elements of the set and their equivalence classes to visualize the concept better.
• Check if the elements in each class satisfy the equivalence relation.

Step 3: Solve Problems Involving Equivalence Relations

To apply your understanding, you can solve problems that require identifying equivalence relations or finding equivalence classes.

• Step-by-step approach:
  1. Identify the set and the relation.
  2. Verify the properties of equivalence relations.
  3. Determine the equivalence classes formed.
  4. Solve any specific problems posed in the exercise.

Example Problem

Let’s say we have a relation R on the set of people where (a, b) ∈ R if and only if a and b have the same birthday.

• Check for reflexivity: Each person has the same birthday as themselves.
• Check for symmetry: If person A shares a birthday with person B, then B shares it with A.
• Check for transitivity: If A shares a birthday with B, and B shares with C, then A shares with C.

Thus, R is an equivalence relation, and the equivalence classes would consist of all individuals sharing the same birthday.

Conclusion

Equivalence relations are foundational concepts in mathematics that help group elements based on shared properties. By understanding the definitions and solving problems related to equivalence relations, you can deepen your grasp of more complex mathematical topics. For further exploration, consider practicing with more examples and delving into related concepts such as partitions of sets and functions on equivalence classes.