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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
3
Constructing and Writing Proofs in Mathematics
3.1
Direct Proofs
3.2
More Methods of Proof
3.3
Proof by Contradiction
3.4
Using Cases in Proofs
3.5
The Division Algorithm and Congruence
3.6
Review of Proof Methods
3.7
Chapter 3 Summary
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