Math Class Teaching Notes: Relations and Functions

Introduction

• Teacher: Shivani Sharma
• Class 12, CBSE Board
• For both Hindi and English Medium

Chapter: Relations and Functions

Chapter 1.1: Introduction and Overview

• Key Concepts: Set, Cartesian Product, Relation
• Important Point: Knowledge of sets and Cartesian products is necessary to understand relations

Sets

• Definition: A well-defined collection of objects
• Example: Set of natural numbers < 10 (1, 3, 5, 7, 9)
• Symbol: Sets are represented by capital letters

Cartesian Product

• Definition: The set of all possible ordered pairs
• Example: Set A = {1, 3, 5}, Set B = {2, 4}. Then A × B = {(1,2), (1,4), (3,2), (3,4), (5,2), (5,4)}

Relations

• Definition: Relation from A to B is a subset of A × B
• Example: Relation R, where the first element must be smaller than the second
• Types: Reflexive, Symmetric, Transitive
• Explained with examples

Reflexive Relation

• Definition: Every element must be related to itself (a, a) ∈ R
• Example: Set {1, 2, 3} is reflexive if (1,1), (2,2), (3,3) are all in R

Symmetric and Transitive Relation

• Symmetric: If (a, b) ∈ R, then (b, a) must also be in R
• Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
• Example: Set {1, 2, 3} is symmetric and transitive if it follows these conditions

Equivalence Relation

• Definition: Must be reflexive, symmetric, and transitive
• Example: Set {1, 2, 3} is an equivalence relation if it fulfills all three conditions

Equivalence Classes

• Definition: Partitioning a set based on an equivalence relation
• Example: Set {0, 1, 2, ..., 12}, divided based on multiples of mod 4

Conclusion

• In the next video, we will solve problems from Exercise 1.1
• Advise everyone to study hard and pay attention