front cover -- copyright page -- title page -- Contents -- Some Notes on Notation -- To the Students -- To Those Beginning the Journey into Proof Writing -- How to Use This Text -- Do the Exercises! -- Acknowledgments -- For the Professors -- To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines -- I THE AXIOMATIC METHOD -- Introduction -- The History of Numbers -- The Algebra of Numbers -- The Axiomatic Method -- Parallel Mathematical Universes -- Statements in Mathematics -- Mathematical Statements -- Mathematical Connectives -- Symbolic Logic -- Compound Statements in English -- Predicates and Quantifiers -- Supplemental Exercises -- Proofs in Mathematics -- What is Mathematics? -- Direct Proof -- Contraposition and Proof by Contradiction -- Proof by Induction -- Proof by Complete Induction -- Examples and Counterexamples -- Supplemental Exercises -- How to THINK about mathematics: A Summary -- How to COMMUNICATE mathematics: A Summary -- How to DO mathematics: A Summary -- II SET THEORY -- Basic Set Operations -- Introduction -- Subsets -- Intersections and Unions -- Intersections and Unions of Arbitrary Collections -- Differences and Complements -- Power Sets -- Russell's Paradox -- Supplemental Exercises -- Functions -- Functions as Rules -- Cartesian Products, Relations, and Functions -- Injective, Surjective, and Bijective Functions -- Compositions of Functions -- Inverse Functions and Inverse Images of Functions -- Another Approach to Compositions -- Supplemental Exercises -- Relations on a Set -- Properties of Relations -- Order Relations -- Equivalence Relations -- Supplemental Exercises -- Cardinality -- Cardinality of Sets: Introduction -- Finite Sets -- Infinite Sets -- Countable Sets -- Uncountable Sets -- Supplemental Exercises -- III NUMBER SYSTEMS -- Algebra of Number Systems.

Introduction: A Road Map -- Primary Properties of Number Systems -- Secondary Properties -- Isomorphisms and Embeddings -- Archimedean Ordered Fields -- Supplemental Exercises -- The Natural Numbers -- Introduction -- Zero, the Natural Numbers, and Addition -- Multiplication -- Supplemental Exercises -- Summary of the Properties of the Nonnegative Integers -- The Integers -- Introduction: Integers as Equivalence Classes -- A Total Ordering of the Integers -- Addition of Integers -- Multiplication of Integers -- Embedding the Natural Numbers in the Integers -- Supplemental Exercises -- Summary of the Properties of the Integers -- The Rational Numbers -- Introduction: Rationals as Equivalence Classes -- A Total Ordering of the Rationals -- Addition of Rationals -- Multiplication of Rationals -- An Ordered Field Containing the Integers -- Supplemental Exercises -- Summary of the Properties of the Rationals -- The Real Numbers -- Dedekind Cuts -- Order and Addition of Real Numbers -- Multiplication of Real Numbers -- Embedding the Rationals in the Reals -- Uniqueness of the Set of Real Numbers -- Supplemental Exercises -- Cantor's Reals -- Convergence of Sequences of Rational Numbers -- Cauchy Sequences of Rational Numbers -- Cantor's Set of Real Numbers -- The Isomorphism from Cantor's to Dedekind's Reals -- Supplemental Exercises -- The Complex Numbers -- Introduction -- Algebra of Complex Numbers -- Order on the Complex Field -- Embedding the Reals in the Complex Numbers -- Supplemental Exercises -- IV TIME SCALES -- Time Scales -- Introduction -- Preliminary Results -- The Time Scale and its Jump Operators -- Limits and Continuity -- Supplemental Exercises -- The Delta Derivative -- Delta Differentiation -- Higher Order Delta Differentiation -- Properties of the Delta Derivative -- Supplemental Exercises -- V HINTS.

Hints for (and Comments on) the Exercises -- Hints for Chapter 2 -- Hints for Chapter 3 -- Hints for Chapter 4 -- Hints for Chapter 5 -- Hints for Chapter 6 -- Hints for Chapter 7 -- Hints for Chapter 8 -- Hints for Chapter 9 -- Hints for Chapter 10 -- Hints for Chapter 11 -- Hints for Chapter 12 -- Hints for Chapter 13 -- Hints for Chapter 14 -- Hints for Chapter 15 -- Hints for Chapter 16 -- Bibliography -- Index -- About the Authors.