Discrete Mathematics - Vol 2 | 2. Introduction by Abraham | Learn Smarter
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2. Introduction

The chapter focuses on advanced concepts in discrete mathematics, including properties of functions, equivalence relations, and combinatorial functions such as the Stirling function. It covers injective, surjective, and bijective functions in detail and provides various proof concepts relevant to these topics. Key exercises and activities enhance understanding through application of theories discussed.

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Sections

  • 2

    Discrete Mathematics

    This section covers key concepts in discrete mathematics, including surjective, injective, and bijective functions, equivalence relations, and Stirling numbers.

    Learning Practice

  • 2.1.1

    Introduction

    This section introduces critical concepts surrounding functions, including surjective, injective, and bijective functions, alongside equivalence relations and Stirling numbers.

    Learning Practice

  • 2.2

    Question 6: Surjective Function

    This section discusses the concept of surjective functions, including their properties and implications, particularly in relation to bijective functions in finite and infinite sets.

    Learning Practice

  • 2.3

    Question 7: Equivalence Relation

    This section discusses equivalence relations and their properties, focusing on partitioning a set into subsets of equal size.

    Learning Practice

  • 2.4

    Question 8: Functions From Set X To Set Y

    This section explores functions between two sets, defining the number of total, injective, and bijective functions based on their cardinalities and introducing Stirling numbers.

    Learning Practice

  • 2.4.1

    Part (A): Counting Functions

    This section discusses various types of functions, specifically counting surjective, injective, and bijective functions, and introduces the Stirling function related to combinatorial partitions.

    Learning Practice

  • 2.4.3

    Part (B): Counting Bijective Functions

    This section focuses on understanding the counting of bijective functions, particularly regarding surjective and injective mappings between two sets.

    Learning Practice

  • 2.4.4

    Part (C): Stirling Function Of Type 2

    This section introduces the Stirling function of type 2, which counts the ways to partition a set into non-empty disjoint subsets.

    Learning Practice

  • 2.5

    Question 9: Stirling Numbers

    Stirling numbers help count the ways to partition a set into non-empty disjoint subsets and define key properties for surjective functions.

    Learning Practice

  • 2.6

    Question 10: Relations And Functions

    This section explores various concepts and properties related to relations and functions, including equivalence relations, injective and surjective mappings, and the Stirling function.

    Learning Practice

  • 2.6.1

    Part (A): Symmetric And Transitive Relations

    This section discusses symmetric and transitive relations, highlighting their properties and the conditions under which they apply.

    Learning Practice

  • 2.6.2

    Part (B): Composition Of Functions

    This section discusses the composition of functions, exploring concepts such as surjectivity and injectivity, and the relationship of these attributes in finite versus infinite sets.

    Learning Practice

  • 2.6.3

    Part (C): Injectivity Of G

    This section discusses the injectivity of functions within the context of surjective and bijective mappings, particularly when handling finite and infinite sets.

    Learning Practice

  • 2.6.4

    Part (D): Surjectivity Of F

    This section focuses on the concept of surjective functions, exploring their characteristics, including conditions under which surjectivity implies bijectivity, particularly in finite and infinite sets.

    Learning Practice

References

ch26.pdf

Class Notes

Memorization

What we have learnt

  • Surjective functions need n...
  • Equivalence relations can p...
  • The Stirling function type ...

Final Test

Revision Tests