Discrete Mathematics - Vol 1 | 22. Lecture -22 by Abraham | Learn Smarter
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Knowledge
22. Lecture -22

The lecture introduces the concept of equivalence relations and partitions, establishing a significant relationship between the two. It defines a partition as a collection of non-empty, pairwise disjoint subsets that form the original set when united. The discussion illustrates that equivalence classes resulting from an equivalence relation correspond to partitions of the set, indicating that the quantity of equivalence relations equals the number of partitions for a set.

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Sections

  • 22.1

    Discrete Mathematics

    This section introduces equivalence relations and partitions, explaining how equivalence classes form a partition of a set and vice versa.

    Learning Practice

  • 22.1.1

    Lecture -22

    This lecture introduces equivalence relations, equivalence classes, and their relationship with set partitions in discrete mathematics.

    Learning Practice

  • 22.1.2

    Equivalence Relations And Partitions

    This section introduces equivalence relations and partitions, explaining their definitions and the relationship between them.

    Learning Practice

  • 22.2

    Introduction

    This section introduces the concepts of equivalence relations and partitions in mathematics.

    Learning Practice

  • 22.3

    Definition Of A Partition Of A Set

    A partition of a set is a collection of non-empty, pairwise disjoint subsets that together cover the entire set.

    Learning Practice

  • 22.3.1

    Requirements For Partition

    This section outlines the requirements for creating partitions of sets and explores the relationship between equivalence relations and partitions.

    Learning Practice

  • 22.3.2

    Trivial Partition

    This section introduces the concept of partitioning a set and its relationship with equivalence relations.

    Learning Practice

  • 22.4

    Relationship Between Equivalence Relations And Partitions

    This section discusses the relationship between equivalence relations and partitions of a set, showing how equivalence classes correspond to partitions.

    Learning Practice

  • 22.4.1

    Equivalence Classes As Partitions

    The section discusses the relationship between equivalence classes and partitions of a set, establishing properties that define each.

    Learning Practice

  • 22.4.2

    From Partition To Equivalence Relation

    This section discusses the concepts of partitions of a set and their relationship with equivalence relations and equivalence classes.

    Learning Practice

  • 22.4.2.1

    Construction Of The Equivalence Relation

    This section discusses equivalence relations and partitions, explaining how equivalence classes form partitions of a set.

    Learning Practice

  • 22.4.2.2

    Example With Subset Construction

    This section explores the concept of partitions of a set and their interplay with equivalence relations.

    Learning Practice

  • 22.5

    Proof Of Properties Of Equivalence Relation

    This section establishes the properties of equivalence relations and introduces their connection to set partitions.

    Learning Practice

  • 22.5.1

    Reflexivity

    This section introduces the concept of partitions of a set and the relationship between equivalence relations and their equivalence classes.

    Learning Practice

  • 22.5.2

    Symmetry

    This section discusses the fundamental concepts of equivalence relations and partitions, highlighting their definitions, properties, and the relationship between them.

    Learning Practice

  • 22.5.3

    Transitivity

    This section introduces the concept of partitions of a set and illustrates the relationship between equivalence relations and partitions.

    Learning Practice

  • 22.6

    Conclusion

    This section summarizes the key concepts of equivalence relations and partitions, highlighting their significance and interrelations.

    Learning Practice

  • 22.6.1

    Summary Of The Lecture

    This lecture introduces equivalence relations, equivalence classes, and partitions of a set, explaining their relationships and significance in discrete mathematics.

    Learning Practice

References

ch22.pdf

Class Notes

Memorization

What we have learnt

  • A partition of a set is a c...
  • Equivalence classes formed ...
  • The number of equivalence r...

Final Test

Revision Tests