Relations and Functions Overview

Jun 21, 2024

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