The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs

The Nuts and Bolts of ProofsAn Introduction to Mathematical Proofs: Fifth editionAntonella Cupillari?
Copyright
Dedication
Preface
	To the Reader
List of Symbols
	Some Facts and Properties of Numbers
		Trichotomy Property of Real Numbers
		Trichotomy Property of Real Numbers
		Selected Relations, Definitions, and Properties of Integer, Rational, and Irrational Numbers
		Selected Relations, Definitions, and Properties of Integer, Rational, and Irrational Numbers
		Well-Ordering Principle
		Well-Ordering Principle
		Division Algorithm
		Division Algorithm
	Some Facts and Properties of Functions
1. Getting Started
	Introduction and Basic Terminology
	General Suggestions
2. Basic Techniques to Prove If/Then Statements
	What Does “If/Then” Mean?
	Direct Proof
	The Negation of a Statement; AND/OR
	Proof by Contrapositive
	Proof by Contradiction
3. Special Kinds of Theorems
	“If and Only If” or Equivalence Theorems
	Use of Counterexamples
	Mathematical Induction
	Existence Theorems
	Uniqueness Theorems
	Composite Statements
		Multiple Hypotheses
		Multiple Conclusions
	Equality of Numbers
4. Some Mathematical Topics on Which to Practice Proof Techniques
	Basic Set Theory and Indexed Families
	Power Sets or Parts of a set
	Cartesian Product of Sets
	Indexed Families of Sets
	About Functions
	Composition of Functions
	A Little More About Functions and Sets
	Relations
	Most Common Properties of Relations
	More About Equivalence Relations
	A Special Relation and More Facts About Equivalence Classes
	The Basics of Groups
	Some Properties of Binary Operations
	Special Elements
	When the Properties Fit Together
	Group Sizes and Structures
	Groups (mod m) and Arithmetic (mod m)
	Permutations and Symmetric Groups
	Isomorphism and Subgroups
	An Important Theorem
	Limits
	Getting Closer
	Functions and Limits
	Cardinality and Sizes of Infinity
5. Review Exercises
	Exercises Without Solutions
		General Topics
		Basic Set Theory
		About Functions
		Relations
		The Basics of Groups
		Limits
		Cardinality and Sizes of Infinity
	Collection of “Proofs”
	Solutions for the Exercises at the End of the Sections and the Review Exercises
		Warning
	Chapter 2 Basic Techniques to Prove If/Then Statements
	Chapter 3 Special Kinds of Theorems
		Composite Statements
	Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques
		Basic Set Theory and Indexed Families
	Solutions for the Review Exercises
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W