Proof: Introduction to Higher Mathematics
Sixth Edition
by Warren W. Esty and Norah C. Esty



Table of Contents:  

Part I:  Theory of Proof: Language and Logic.
  

CHAPTER 1    1
Introduction to Proofs
        1.1.  Preview of Proof 2
        1.2.  Sets 12
        1.3.  Logic for Mathematics 30
        1.4.  Important Logical Equivalences 50
        1.5.  Negations 62
        1.6.  Tautologies and Proofs 75
        1.7.  RESULTS FROM LOGIC 88

CHAPTER 2    91
Sentences with Variables
        2.1.  Sentences with One Variable 91
        2.2.  Existence Statements and Negation 103
        2.3.  Reading Theorems and Definitions 120
        2.4.  Equivalence 139
        2.5.  Rational Numbers and Form 152

CHAPTER 3    158
Proofs
        3.1.  Inequalities 158
        3.2.  Absolute Values 171
        3.3.  Theory of Proofs 179
        3.4.  Proofs by Contradiction or Contrapositive 194
        3.5.  Mathematical Induction 200
        3.6.  Bad Proofs 211




Part II:  Practice

CHAPTER 4    
Set Theory
        4.1.  Set Theory 
        4.2.  Bounds (including suprema)

CHAPTER 5    
Functions
        5.1.  One-to-One and Onto 
        5.2.  Functions Applied to Sets 
        5.3.  Cardinality 

-----  This finishes one semester of a sophomore-level course at Montana State University. --------------
         Some of the following sections have been used in subsequent course entitled "Higher Mathematics for Secondary Teachers." 

CHAPTER 6    
Number Theory
        6.1.  Number Theory 
        6.2.  Common Divisors 
        6.3.  Prime Numbers 
        6.4.  Modular Arithmetic 
        6.5.  Cryptography 

CHAPTER 7
Group Theory
        7.1  Groups
        7.2  Subgroups, Cosets, and Lagrange's Theorem
        7.3  Isomorphism
        74.  Quotient Groups

CHAPTER 8    
Topology
        8.1.  Open and Closed Sets 
        8.2.  Interior Points and Accumulation Points 

CHAPTER 9
Calculus
        9.1.  Limits of Sequences  
        9.2.  Limits and Derivatives 

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