Discrete Mathematics - Vol 1 | 23. Cover of an Element in a Poset - part B by Abraham | Learn Smarter
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23. Cover of an Element in a Poset - part B

The chapter discusses the concepts of posets (partially ordered sets) and their properties. Key ideas include covers, minimal and maximal elements, as well as greatest and least elements within a poset framework. Additionally, the chapter introduces the topological sorting algorithm to find a schedule based on given task dependencies, demonstrating that these concepts are foundational in understanding order relations in mathematics.

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Sections

  • 24.1

    Cover Of An Element In A Poset

    This section discusses the concept of covers in partially ordered sets (posets), defining minimal and maximal elements along with their significance in Hasse diagrams.

    Learning Practice

  • 24.1.1

    Definitions And Examples

    This section introduces definitions related to partially ordered sets (posets), including covers, maximal and minimal elements, as well as greatest and least elements.

    Learning Practice

  • 24.1.2

    Properties Of Covers

    This section discusses the concept of covers in partially ordered sets (posets), detailing their properties and significance in understanding the structure of posets.

    Learning Practice

  • 24.2

    Maximal And Minimal Elements In A Poset

    This section discusses the concepts of maximal and minimal elements in a partially ordered set (poset), including covers and their relationships in Hasse diagrams.

    Learning Practice

  • 24.2.1

    Maximal Elements

    This section discusses the definitions and properties of maximal and minimal elements in relation to partially ordered sets (posets), along with concepts like covers and greatest/least elements.

    Learning Practice

  • 24.2.2

    Minimal Elements

    This section introduces key concepts related to minimal elements in partially ordered sets (posets) and defines covers, maximal elements, minimal elements, greatest elements, and least elements.

    Learning Practice

  • 24.2.3

    Existence Of Maximal And Minimal Elements

    This section explores the concepts of maximal and minimal elements in posets, along with their properties.

    Learning Practice

  • 24.3

    Greatest And Least Elements In A Poset

    This section outlines the concepts of greatest and least elements within a partially ordered set (poset) and introduces related terminology such as cover, maximal, and minimal elements.

    Learning Practice

  • 24.3.1

    Definitions

    This section defines key concepts related to partially ordered sets, such as covers, maximal/minimal elements, and greatest/least elements.

    Learning Practice

  • 24.3.2

    Existence And Uniqueness

    This section discusses the concepts of covers, maximal and minimal elements, and the greatest and least elements in a partially ordered set (poset).

    Learning Practice

  • 24.4

    Topological Sorting

    Topological sorting organizes tasks based on their dependencies, providing a schedule for task execution.

    Learning Practice

  • 24.4.1

    Definition And Algorithm Overview

    This section explains the concepts of partially ordered sets (posets), covers, maximal and minimal elements, and introduces the topological sorting algorithm.

    Learning Practice

  • 24.4.2

    Steps Of The Algorithm

    This section explains the concepts of covers, maximal and minimal elements in a poset, and introduces topological sorting for scheduling tasks based on a partial order.

    Learning Practice

  • 24.4.2.1

    Finding Minimal Elements

    This section explores the concepts of covers, maximal, minimal, greatest, and least elements in a partially ordered set (poset).

    Learning Practice

  • 24.4.2.2

    Constructing The Schedule

    This section discusses the concepts of partially ordered sets (posets), covers, maximal and minimal elements, greatest and least elements, and introduces topological sorting for task scheduling.

    Learning Practice

  • 24.4.3

    Proof Of Compatibility With Original Relation

    This section delves into the concepts of covers, maximal and minimal elements in a partially ordered set (poset), and introduces the idea of topological sorting.

    Learning Practice

  • 24.5

    Summary Of Key Concepts

    This section describes the concepts of covering relations, maximal and minimal elements in a poset, and introduces the notions of greatest and least elements.

    Learning Practice

  • 24.5.1

    Partial Ordering

    This section defines the concept of partially ordered sets (posets) and discusses key notions such as cover, maximal and minimal elements, and introduces concepts of greatest and least elements within these sets.

    Learning Practice

  • 24.5.2

    Total Ordering And Hasse Diagram

    This section introduces the concepts of total ordering within posets and the significance of Hasse diagrams in understanding relations between elements.

    Learning Practice

References

ch23 - part B.pdf

Class Notes

Memorization

What we have learnt

  • A cover of an element in a ...
  • Maximal and minimal element...
  • Topological sorting allows ...

Final Test

Revision Tests