Discrete Mathematics - Vol 2 | 1. Introduction to Tutorial 4: Part I by Abraham | Learn Smarter
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.

Typing
Memory
Math
English Adventures
Knowledge
1. Introduction to Tutorial 4: Part I

The chapter discusses various properties of equivalence relations and their interactions, particularly focusing on unions and intersections of these relations. It explores how unions may fail to maintain transitivity while intersections consistently result in equivalence relations. Additionally, the chapter covers the counting of partitions in sets and the conditions under which a poset can be classified as a total order.

Enroll to start learning

You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.

Sections

  • 1

    Discrete Mathematics

    This section discusses equivalence relations, particularly focusing on unions and intersections, providing proofs, counterexamples, and exploring their properties.

    Learning Practice

  • 1.1.1

    Introduction To Tutorial 4: Part I

    In the first part of Tutorial 4, various properties of equivalence relations are explored, specifically focusing on union and intersection operations.

    Learning Practice

  • 1.1.2

    Question 1: Equivalence Relations - Part A

    This section explores equivalence relations, discussing the properties of unions and intersections of equivalence relations on a non-empty set.

    Learning Practice

  • 1.1.4

    Question 2: Union And Composition Of Equivalence Relations

    The section discusses the properties of unions and intersections of equivalence relations, showing that while intersections are always equivalence relations, unions are not unless composition equals their union.

    Learning Practice

  • 1.1.5

    Question 3: Counting Equivalence Relations

    This section discusses the counting of equivalence relations defined on sets, introducing the function P(n) that denotes the number of equivalence relations over a set of n elements.

    Learning Practice

  • 1.1.6

    Question 4: Hasse Diagrams And Partial Orderings

    This section discusses the relationship between partial orderings and Hasse diagrams, including the count of distinct Hasse diagrams associated with a set of three elements.

    Learning Practice

  • 1.1.7

    Question 5: Minimum Element In Poset

    This section explores the concept of minimum elements in partially ordered sets (posets) and establishes conditions under which a poset is a total order.

    Learning Practice

  • 1.2

    Counterexamples And Properties Of Equivalence Relations

    This section explores the properties of equivalence relations, focusing on the union and intersection of such relations, including their respective behaviors and counterexamples.

    Learning Practice

  • 1.3

    Functions And Partitions

    This section explores equivalence relations, specifically examining their unions and intersections, and discusses conditions under which these operations yield equivalence relations.

    Learning Practice

  • 1.4

    Hasse Diagrams And Their Categories

    This section explores Hasse diagrams, their forms, and how they relate to partial ordering in discrete mathematics.

    Learning Practice

  • 1.5

    Properties Of Posets And Total Ordering

    This section explores the properties of partially ordered sets (posets) and total ordering, focusing on concepts such as equivalence relations and the significance of reflexivity, symmetry, and transitivity.

    Learning Practice

References

ch25.pdf

Class Notes

Memorization

What we have learnt

  • The union of two equivalenc...
  • The intersection of two equ...
  • Every equivalence relation ...

Final Test

Revision Tests