Discrete Mathematics - Vol 2 | 4. Module No # 05 by Abraham | Learn Smarter
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4. Module No # 05

The lecture discusses examples of countably infinite sets and explores their properties, particularly stressing the significance of listing elements in a systematic manner. It delves into the countability of the Cartesian product of integers, the rational numbers, and binary strings, illustrating methods for enumerating each set. The chapter concludes with results about cardinality including the union of countable sets and the Schroder-Bernstein theorem, emphasizing relationships between sets and their subsets.

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Sections

  • 4

    Module No # 05

    This module explores countably infinite sets, provides examples like the Cartesian product of integers and rational numbers, and illustrates enumeration methods for these sets.

    Learning Practice

  • 4.1.1

    Lecture No # 28

    This lecture explores examples of countably infinite sets, including the Cartesian product of integers and rational numbers, and demonstrates their enumeration techniques.

    Learning Practice

  • 4.1.2

    Examples Of Countably Infinite Sets

    This section presents various examples of countably infinite sets, proving their countability using enumeration methods.

    Learning Practice

  • 4.2

    Countable And Uncountable Sets

    This section explores the concepts of countably infinite sets, providing examples and proofs of their countability.

    Learning Practice

  • 4.2.1

    Cartesian Product Of The Integers

    The Cartesian product of integers is shown to be countably infinite through enumeration techniques.

    Learning Practice

  • 4.2.2

    Set Of Rational Numbers

    This section explores countably infinite sets, focusing on the set of rational numbers and their enumeration.

    Learning Practice

  • 4.2.3

    Set Of Binary Strings

    This section defines and explores the set of binary strings of finite length, demonstrating that it is countably infinite.

    Learning Practice

  • 4.3

    General Results About Cardinality

    This section discusses the classification of sets in terms of cardinality, particularly focusing on countable and uncountable sets along with their properties.

    Learning Practice

  • 4.3.1

    Theorem On Union Of Countable Sets

    This section presents the theorem that states the union of two countable sets is also countable.

    Learning Practice

  • 4.3.2

    Schroder-Bernstein Theorem

    The section introduces the Schroder-Bernstein theorem, explaining how to establish the equality of cardinalities between two sets using injective mappings.

    Learning Practice

  • 4.3.3

    Subsets Of Countable Sets

    This section explores countable and uncountable sets with specific examples illustrating countably infinite sets.

    Learning Practice

References

ch28.pdf

Class Notes

Memorization

What we have learnt

  • Countable sets can be eithe...
  • A systematic enumeration ca...
  • The union of two countable ...

Final Test

Revision Tests