Discrete Mathematics - Vol 1 | 21. Lecture -20 by Abraham | Learn Smarter
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21. Lecture -20

The chapter provides a comprehensive exploration of different types of relations in set theory, particularly focusing on symmetric, anti-symmetric, reflexive, irreflexive, and asymmetric relations. Various properties and the number of possible relations are systematically analyzed through logical reasoning and mathematical proofs. Key methods for establishing or disproving the existence of specific types of relations are highlighted through practical examples and exercises.

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Sections

  • 21.1

    Discrete Mathematics

    This section covers fundamental concepts of discrete mathematics, focusing on set theory and relations, particularly properties like symmetry, anti-symmetry, and reflexivity.

    Learning Practice

  • 21.1.1

    Lecture -20

    This section discusses the relationships between sets, specifically demonstrating properties such as inclusions and various types of relations like symmetric, anti-symmetric, and reflexive.

    Learning Practice

  • 21.1.2

    Tutorial 3

    This tutorial covers various aspects of set theory and relations, including proofs of subset relations and characterization of relations such as symmetric, anti-symmetric, and irreflexive relations.

    Learning Practice

  • 21.2

    Question 1

    This section discusses how to prove set relationships and explores properties of relations, including symmetric, anti-symmetric, irreflexive, and reflexive relations.

    Learning Practice

  • 21.3

    Question 2

    This section discusses the proof and implications of the condition that if a certain universally quantified statement holds true for sets A, B, and C, then the intersection of A and B is a subset of C.

    Learning Practice

  • 21.4

    Question 3

    The section explores the counting of various types of relations on a set of elements, focusing on properties like symmetry, anti-symmetry, irreflexivity, reflexivity, and their intersections.

    Learning Practice

  • 21.4.1

    Part A: Symmetric Relations

    This section explores symmetric relations in set theory, detailing their properties and implications.

    Learning Practice

  • 21.4.2

    Part B: Anti-Symmetric Relations

    This section explores the concept of anti-symmetric relations in set theory, providing the definitions and implications of these relations.

    Learning Practice

  • 21.4.3

    Part C: Asymmetric Relations

    This section explores the properties of asymmetric relations, including the conditions for reflexivity, symmetry, anti-symmetry, and their implications in set theory.

    Learning Practice

  • 21.4.4

    Part D: Irreflexive Relations

    This section explores irreflexive relations, detailing their properties and how they interact with other types of relations.

    Learning Practice

  • 21.4.5

    Part E: Reflexive And Symmetric Relations

    This section covers key concepts of reflexive and symmetric relations in set theory, including their definitions, properties, and illustrative examples.

    Learning Practice

  • 21.4.6

    Part F: Neither Reflexive Nor Irreflexive Relations

    This section explores the characteristics and properties of relations that are neither reflexive nor irreflexive, specifically focusing on how they can be classified.

    Learning Practice

  • 21.5

    Question 4

    This section discusses the properties of relations in set theory, particularly focusing on symmetric, anti-symmetric, reflexive, and irreflexive relations.

    Learning Practice

  • 21.5.1

    Part A: Symmetric, Anti-Symmetric And Reflexive Relations

    This section introduces and explains the concepts of symmetric, anti-symmetric, reflexive, and irreflexive relations in discrete mathematics.

    Learning Practice

  • 21.5.2

    Part B: Symmetric, Anti-Symmetric And Irreflexive Relations

    This section explores the definitions and characteristics of symmetric, anti-symmetric, and irreflexive relations within the context of discrete mathematics.

    Learning Practice

  • 21.5.3

    Part C: Symmetric And Anti-Symmetric Relations

    This section explores symmetric and anti-symmetric relations within the context of discrete mathematics, explaining definitions, properties, and counting the number of such relations for given sets.

    Learning Practice

References

ch20.pdf

Class Notes

Memorization

What we have learnt

  • Relations can be classified...
  • The number of specific type...
  • Logical reasoning and mathe...

Final Test

Revision Tests