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Mathematical Reasoning:
Writing and Proof (PreTeXt Edition)
Ted Sundstrom, Professor Emeritus
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Contents
Index
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Contents
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Front Matter
Colophon
Publisher's Note
Note to Students
Preface
1
Introduction to Writing Proofs in Mathematics
Statements and Conditional Statements
Constructing Direct Proofs
Chapter 1 Summary
2
Logical Reasoning
Statements and Logical Operators
Logically Equivalent Statements
Open Sentences and Sets
Quantifiers and Negations
Chapter 2 Summary
3
Constructing and Writing Proofs in Mathematics
Direct Proofs
More Methods of Proof
Proof by Contradiction
Using Cases in Proofs
The Division Algorithm and Congruence
Review of Proof Methods
Chapter 3 Summary
4
Mathematical Induction
The Principle of Mathematical Induction
Other Forms of Mathematical Induction
Induction and Recursion
Chapter 4 Summary
5
Set Theory
Sets and Operations on Sets
Proving Set Relationships
Properties of Set Operations
Cartesian Products
Indexed Families of Sets
Chapter 5 Summary
6
Functions
Introduction to Functions
More about Functions
Injections, Surjections, and Bijections
Composition of Functions
Inverse Functions
Functions Acting on Sets
Chapter 6 Summary
7
Equivalence Relations
Relations
Equivalence Relations
Equivalence Classes
Modular Arithmetic
Chapter 7 Summary
8
Topics in Number Theory
The Greatest Common Divisor
Prime Numbers and Prime Factorizations
Linear Diophantine Equations
Chapter 8 Summary
9
Finite and Infinite Sets
Finite Sets
Countable Sets
Uncountable Sets
Chapter 9 Summary
Back Matter
A
Guidelines for Writing Mathematical Proofs
B
Answers for the Progress Checks
C
Answers and Hints for Selected Exercises
D
List of Symbols
Index
Authored in PreTeXt
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Chapter
9
Finite and Infinite Sets
9.1
Finite Sets
9.2
Countable Sets
9.3
Uncountable Sets
9.4
Chapter 9 Summary
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