Mathematical Reasoning, Version 3

Version 3 and Version 2.1
Note to Students
Preface
Introduction to Writing Proofs in Mathematics
	Statements and Conditional Statements
	Constructing Direct Proofs
	Chapter 1 Summary
Logical Reasoning
	Statements and Logical Operators
	Logically Equivalent Statements
	Open Sentences and Sets
	Quantifiers and Negations
	Chapter 2 Summary
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
Mathematical Induction
	The Principle of Mathematical Induction
	Other Forms of Mathematical Induction
	Induction and Recursion
	Chapter 4 Summary
Set Theory
	Sets and Operations on Sets
	Proving Set Relationships
	Properties of Set Operations
	Cartesian Products
	Indexed Families of Sets
	Chapter 5 Summary
Functions
	Introduction to Functions
	More about Functions
	Injections, Surjections, and Bijections
	Composition of Functions
	Inverse Functions
	Functions Acting on Sets
	Chapter 6 Summary
Equivalence Relations
	Relations
	Equivalence Relations
	Equivalence Classes
	Modular Arithmetic
	Chapter 7 Summary
Topics in Number Theory
	The Greatest Common Divisor
	Prime Numbers and Prime Factorizations
	Linear Diophantine Equations
	Chapter 8 Summary
Finite and Infinite Sets
	Finite Sets
	Countable Sets
	Uncountable Sets
	Chapter 9 Summary
Guidelines for Writing Mathematical Proofs
Answers for the Progress Checks
Answers and Hints for Selected Exercises
List of Symbols
Index