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 104
        2.3.  Reading Theorems and Definitions 121
        2.4.  Equivalence 141
        2.5.  Rational Numbers and Form 154

CHAPTER 3    158
Proofs
        3.1.  Inequalities 160
        3.2.  Absolute Values 173
        3.3.  Theory of Proofs 182
        3.4.  Proofs by Contradiction or Contrapositive 198
        3.5.  Mathematical Induction 204
        3.6.  Bad Proofs 215

Part II:  Practice

CHAPTER 4    
Set Theory
        4.1.  Set Theory  227
        4.2.  Bounds and suprema  238

CHAPTER 5    
Functions
        5.1.  One-to-One, Onto, and Composition  253 
        5.2.  Functions Applied to Sets  264 
        5.3.  Cardinality  271

-----  This finishes one semester of a sophomore-level course at Montana State University. If your school goes through the material faster, Chapter 6 on number theory is a great way to continue. ------------
         Some of the following sections have been used in subsequent course entitled "Higher Mathematics for Secondary Teachers." 
         Secondary teachers would benefit from knowing something about each of these topics, but do not have time in their curriculum to take a full course devoted to, say, number theory or group theory or proofs in calculus. By using this text students can be exposed to significant amounts of each topic in one semester.

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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