Discrete Mathematics - Vol 1 | 23. Partial Ordering - part A by Abraham | Learn Smarter
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23. Partial Ordering - part A

This chapter introduces the concept of partial ordering, describing its properties and applications. It emphasizes reflexive, antisymmetric, and transitive properties that define a partial order, further illustrating these concepts through examples such as modular dependencies in software projects and mathematical relations like divisibility. Additionally, the chapter discusses total orderings and employs Hasse diagrams to represent partial orderings visually.

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Sections

  • 23.1

    Discrete Mathematics

    This section introduces partial ordering concepts in mathematics, including definitions, properties, and examples of partial and total ordering.

    Learning Practice

  • 23.1.1

    Partial Ordering

    This section introduces the concept of partial ordering, its properties, and its applications through various examples.

    Learning Practice

  • 23.2

    Introduction To Partial Ordering

    This section introduces partial ordering, its properties, and Hasse diagrams as a representation method.

    Learning Practice

  • 23.2.1

    Definition Of Partial Ordering

    This section introduces partial ordering, explaining the properties of reflexivity, antisymmetry, and transitivity that define it.

    Learning Practice

  • 23.2.2

    Properties Of Partial Ordering

    This section introduces the concept of partial ordering, explaining its properties and providing examples, including reflexivity, antisymmetry, and transitivity.

    Learning Practice

  • 23.2.3

    Example Of Partial Ordering With Software Modules

    The section explores the concept of partial ordering through examples, including an alphabetical dictionary arrangement and dependencies between software modules.

    Learning Practice

  • 23.2.4

    General Definition Of Partial Ordering

    Partial ordering defines a set with a relation that is reflexive, antisymmetric, and transitive. This section introduces these principles with applications and examples.

    Learning Practice

  • 23.2.5

    Example With Positive Integers

    This section introduces the concept of partial ordering, its properties, and illustrates examples using sets and relations.

    Learning Practice

  • 23.2.6

    Example With Subset Relationship

    This section introduces the concept of partial ordering, exemplified through the subset relationship and its mathematical properties.

    Learning Practice

  • 23.2.7

    Example With Integers And Less Than Equal To

    This section introduces the concept of partial ordering through examples, focusing on the definitions and properties of reflexive, antisymmetric, and transitive relations.

    Learning Practice

  • 23.2.8

    Abstract Notation For Relations

    This section covers partial ordering and its properties, including reflexivity, antisymmetry, and transitivity, along with examples and abstract notation.

    Learning Practice

  • 23.2.9

    Comparable And Incomparable Elements

    This section explores the concepts of partial orderings, including the definitions of comparable and incomparable elements.

    Learning Practice

  • 23.2.10

    Definition Of Total Ordering

    This section introduces total ordering, emphasizing its properties and contrast with partial ordering.

    Learning Practice

  • 23.3

    Hasse Diagrams

    This section introduces the concept of Hasse diagrams as a visual representation of partial orderings in mathematics.

    Learning Practice

  • 23.3.1

    Construction Of Hasse Diagrams

    This section introduces the concept of partial orderings and explains how to construct Hasse diagrams to visually represent these relationships.

    Learning Practice

  • 23.3.2

    Example With Less Than Equal To Relationship

    This section introduces the concept of partial ordering, focusing on the less than or equal to relationship, and explores its properties and examples.

    Learning Practice

  • 23.3.3

    Another Example With Divide Relationship

    This section introduces partial orderings through the lens of the divide relationship among positive integers.

    Learning Practice

  • 23.3.4

    Hasse Diagram For Subset Relationship

    This section discusses the concept of partial ordering, focusing on Hasse diagrams, and how they illustrate subset relationships within a set.

    Learning Practice

References

ch23 - part A.pdf

Class Notes

Memorization

What we have learnt

  • A partial ordering is defin...
  • All elements in a totally o...
  • Hasse diagrams provide a vi...

Final Test

Revision Tests