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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.

Sections

Tutorial 5

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

9.1 Section Overview

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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.

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.

9.1.3 Question 3

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

9.1.4 Question 4

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

9.1.5 Question 5

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

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.

Claim 1

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

9.2 Section Overview

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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.

9.3 Section Overview

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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.

9.4 Section Overview

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9.4.1 Set S Definition

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

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.

Set of Real Numbers with Decimal Representation of Only 1s

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

9.5 Section Overview

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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.

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.

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 Objectives

  • Two sets can have the same cardinality if there exist injective mappings between them.

  • There is no infinite set whose cardinality is strictly less than א₀, the cardinality of the set of positive integers.

  • Any infinite set has a countably infinite subset.

Key Concepts

Cardinality

A measure of the 'number of elements' in a set, which can be finite or infinite.

Injective Mapping

A function where each element of the domain maps to a unique element in the codomain, ensuring no two elements in the domain map to the same element in the codomain.

Countably Infinite

A set is countably infinite if its elements can be listed in a sequence, meaning they can be put into one-to-one correspondence with the set of natural numbers.

Schroder-Bernstein Theorem

If there exist injective functions from set A to set B and from set B to set A, then the two sets have the same cardinality.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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