Relational and Functions - Class 12 Lecture Notes

Jul 12, 2024

Relational and Functions - Class 12 Lecture Notes

Introduction to Relations

  • Definition: A relation R from set A to set B is a subset of the Cartesian product AxB.
  • Real Life Analogy: Relations in families (internal relations like parent's relation with child and external relations like cousin's, in-laws, etc).

Importance in Board and SET Exams

  • Board Exams: Typically, relations and functions carry around 8 marks. Questions may include either a 5-mark or a 2-mark question.
  • Sample Papers: Specific types of relation questions are commonly included.
  • Preparation Strategy: Focus strictly on updated concepts and types discussed in lecture.

Basics Review - Class 11 Concepts

  • Ordered Pairs: Elements are ordered (a, b), where a ∈ set A and b ∈ set B.
  • Cartesian Product: AxB consists of all ordered pairs (a, b) where a ∈ A and b ∈ B.
  • Properties: A x B ≠ B x A in general.

Introduction to Relations in Class 12

  • Subset Concept: Relation R is a subset of AxB.
  • Types of Relations: Universal Relation, Identity Relation, Empty Relation, Reflexive, Symmetric, Transitive.

Detailed Discussion on Types of Relations

Reflexive Relation

  • Definition: Every element is related to itself (aRa ∀a ∈ A).
  • Check: Verify aRa for all elements.

Symmetric Relation

  • Definition: If aRb, then bRa for all a, b ∈ A.
  • Check: Verify aRb implies bRa.

Transitive Relation

  • Definition: If aRb and bRc, then aRc for all a, b, c ∈ A.
  • Check: Verify aRb and bRc implies aRc.

Equivalence Relation

  • Definition: A relation that is reflexive, symmetric, and transitive.

Example Problems

Problem 1: Proof by example

  • Given: A set {1, 2, 3, 4, 5}, defined relation R and a condition involving mod operation.
  • Approach: Prove reflexive, symmetric, and transitive properties.
  • Conclusion: Conclude with equivalence and find equivalence classes.

Functions

  • Definition: A relation where each element in domain has a unique element in codomain.
  • Types:
    • One-one (Injective): Different elements have different images.
    • Onto (Surjective): Every element in codomain has at least one pre-image.
    • Bijective: Both one-one and onto.

Checking Injective Function

  • Assume different elements have the same image, show a contradiction.

Checking Surjective Function

  • Assume an element in codomain, find a pre-image.

Example Problem: Verify function properties

  • Given: Function definitions.
  • Approach: Verify one-one by showing uniqueness of image and onto by finding pre-images.

Previous Exam Questions

2022 Example

  • Given Lists: Match relations and functions to their properties (symmetric, reflexive, injective, bijective).
  • Solution: Summary of matching results with explanations for correctness.

These notes capture the high-level key concepts and examples discussed in the class 12 lecture on relations and functions, complete with detailed problem-solving strategies for examination preparation.