Discrete Mathematics - Vol 2 | 9. Tutorial 5 by Abraham | Learn Smarter
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9. Tutorial 5

The chapter explores key concepts related to cardinality and infinity in set theory, focusing on the relationships between different sets, their cardinalities, and properties of infinite sets. It discusses important theorems such as the Schroder-Bernstein theorem and offers proofs for various claims about countability and uncountability of sets. In addition, the chapter provides exercises and activities to reinforce the understanding of these concepts.

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Sections

  • 9.1

    Tutorial 5

    This section discusses different cardinalities of sets, specifically comparing infinite sets and their properties.

    Learning Practice

  • 9.1.1

    Question 1

    This section demonstrates the process of proving that two distinct sets of real numbers have the same cardinality using injective mappings.

    Learning Practice

  • 9.1.2

    Question 2

    This section demonstrates the proof that there is no infinite set whose cardinality is strictly less than the cardinality of the set of positive integers.

    Learning Practice

  • 9.1.3

    Question 3

    This section focuses on proving that an infinite set A, whether countable or uncountable, contains a countably infinite subset.

    Learning Practice

  • 9.1.4

    Question 4

    This section discusses the cardinality of the union of countable sets and shows that the union remains countable.

    Learning Practice

  • 9.1.5

    Question 5

    In this section, we explore the concept of uncountable sets and provide examples that illustrate various intersection properties.

    Learning Practice

  • 9.1.6

    Question 6

    This section discusses whether specific sets are countable or uncountable, focusing on integers divisible by 5 but not by 7 and real numbers with decimal representations of only 1s.

    Learning Practice

  • 9.2

    Claim 1

    The section discusses the concept of cardinality in relation to sets and how to demonstrate equality of cardinalities through injective mappings.

    Learning Practice

  • 9.3

    Claim 2

    This section discusses the cardinality of sets, particularly focusing on proving that there is no infinite set with a cardinality less than that of positive integers.

    Learning Practice

  • 9.4

    Set Of Integers Divisible By 5 But Not By 7

    This section explores the set of integers that are divisible by 5 but not by 7, examining its properties and demonstrating its countability.

    Learning Practice

  • 9.4.1

    Set S Definition

    Set S is defined and explored through examples and theorems in relation to cardinality and injective mappings.

    Learning Practice

  • 9.4.2

    Listing Of Set S

    This section explores the concept of cardinality between two sets, particularly focusing on real numbers and their mappings.

    Learning Practice

  • 9.5

    Set Of Real Numbers With Decimal Representation Of Only 1s

    This section discusses sets of real numbers between specific intervals and demonstrates their cardinality.

    Learning Practice

  • 9.5.1

    Set S With Recurring 1s

    This section covers the concepts of cardinality, injective mappings, infinite sets, and subsets, specifically focusing on proving properties related to different types of infinite sets and their relationships.

    Learning Practice

  • 9.5.2

    Set S With Single 1

    This section explores the cardinality of two sets of real numbers and demonstrates the absence of a smaller infinite cardinality.

    Learning Practice

  • 9.5.3

    Union Of Sets S

    This section discusses the concept of cardinality in sets and demonstrates proofs related to set cardinalities using the Schroder-Bernstein theorem and union operations.

    Learning Practice

References

ch32.pdf

Class Notes

Memorization

What we have learnt

  • Two sets can have the same ...
  • There is no infinite set wh...
  • Any infinite set has a coun...

Final Test

Revision Tests