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

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

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. 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