Robert L. Causey, The University of Texas, Austin

Logic, Sets and Recursion, Second Edition

ISBN: 0763737844
Cover: Cloth
Pages: 512
Copyright: 2006

Logic Sets, and Recursion, Second Edition provides a comprehensive introduction to logic, the basic ideas of set theory, and mathematical induction at the undergraduate level. This text paves the way for students preparing to enter more advanced math courses, furnishing them with the experience to construct rigorous proofs with ease. It is intended to fill the gap between advanced mathematical logic texts and Discrete Mathematics texts that do not treat logic with the depth required in modern computer science. The new Second Edition has incorporated a wealth of exercise sets to aid in students retention and understanding of material presented in the text.

Table of Contents

Introduction

1. Sentential Calculus
1.1 Syntax of the Sentential Calculus
1.2 Correspondence to Natural Languages
1.3 Semantics of the Sentential Calculus
1.4 Some Metatheoretical Concepts
1.5 Principles for Sentential Calculus Derivations
1.6 Adequacy and Use of Sentential Calculus Derivation Rules
1.6 1 Soundness and Completeness
1.6 2 Strategies for Constructing Proofs
1.6 3 Sentential Calculus Derivation Examples
1.7 Conjunctive Normal Form and Resolution Proofs

2. Basic Set Theory
2.1 Sets
2.1 1 Extensionality, Predicates, and Abstraction
2.1 2 Some Special Sets and Set Operations
2.2 Relations
2.2 1 General Features
2.2 2 Special Kinds of Relations
2.3 Functions
2.3 1 Basic Ideas
2.3 2 Compositions and Inverses
2.4 Relational Systems

3. Recursion and mathematical Induction
3.1 The Natural Number System
3.1 1 Introduction
3.1 2 Peano's Axioms and the Induction Principle
3.1 3 Definition by Recursion
3.2 Basic Arithmetic
3.2 1 Some Simple Functions
3.2 2 Additional Arithmetical Definitions
3.3 Extensions of Recursive Definition and Induction
3.3 1 Some Additional Applications of the Recursion Theorem
3.3 2 The Well-Ordering of the Natural Numbers
3.3 3 Course of Values Induction
3.3 4 Two Arithmetical Algorithms
3.3 5 Pitfalls of Recursion
3.4 Non-Numerical Data
3.4 1 Strings
3.4 2 A Simple Treatment of Lists
3.4 3 Sentential Calculus Expressions
3.4 4 Stacks and Queues

4. Predicate Calculus
4.1 Syntax of the Predicate Calculus
4.2 Semantical Aspects of the Predicate Calculus
4.2 1 Interpretations and Truth
4.2 2 Tautologous Sentences in Predicate Calculus
4.2 3 Tautological Consequences in Predicate Calculus
4.3 Predicate Calculus Derivations
4.3 1 Derivation Rules
4.3 2 Proof Strategies and Examples
4.3 3 Adequacy of the Predicate Calculus Rules
4.4 Application Example
4.5 Identity and Function Symbols
4.5 1 Extension of the Syntax
4.5 2 Semantics of Predicate Calculus with Identity and Function Symbols
4.5 3 Derivation Rules for Predicate Calculus with Identity and Function Symbols
4.5 4 Use of Identity in Representing Information
4.6 Formalized Theories


References
Answers to Selected Exercises
The Greek Alphabet
Glossary of Symbols
Index

Jason H. Goodfriend, PhD,
The Bureau of Transportation Statistics and George Washington University

A Gateway to Higher Mathematics

ISBN: 0763727334
Cover: Cloth
Pages: 309
Copyright: 2006

A Gateway to Higher Mathematics integrates the process of teaching students how to do proofs into the framework of displaying the development of the real number system. The text eases the students into learning how to construct proofs, while preparing students how to cope with the type of proofs encountered in the higher-level courses of abstract algebra, analysis, and number theory. After using this text, the students will not only know how to read and construct proofs, they will understand much about the basic building blocks of mathematics. The text is designed so that the professor can choose the topics to be emphasized, while leaving the remainder as a reference for the students.

Table of contents

Chapter 1. Logic and Techniques For Proofs

Chapter 2. Elementary Set Theory

Chapter 3. The Development of the Integers

Chapter 4. Properties and Applications of Integers

Chapter 5. Fields and the Rational Numbers

Chapter 6. The Development of the Real Numbers

Chapter 7. Some Additional Properties of Real Numbers
Appendix A: Proof of the Cantor-Schroder-Bernstein Theorem

Appendix B: Using the Axiom of Choice to Prove Some Results About Infinite Sets

Appendix C: Completion of the Construction of a Set Obeying the Real Number Postulates

References

Neville Robbins, San Francisco State University

Beginning Number Theory, Second Edition

ISBN: 0763737682
Cover: Cloth
Pages: 338
Copyright: 2006

Thoroughly revised and updated, the new Second Edition of Neville Robbins' Beginning Number Theory includes all the major topics covered in a classic Number Theory course and blends in numerous applications and specialized treatments of number theory, including Cryptology, Fibonacci numbers, and Computational Number Theory. The text strikes a balance between traditional and algorithmic approaches to elementary number theory and is supported with numerous exercises, applications, and case studies throughout. Computer exercises for CAS systems are also included.

Table of Contents

Chapter 1: Preliminaries
Chapter 2: Divisibility
Chapter 3: Primes
Chapter 4: Congruences
Chapter 5: Arithmetic Functions
Chapter 6: Primitive Roots and Indices
Chapter 7: Quadratic Congruences
Chapter 8: Sum of Squares
Chapter 9: Continued Fractions
Chapter 10: Nonlinear Diophantine Equations
Chapter 11: Computational Number Theory
Chapter 12: Cryptology

Saul Stahl, Ph.D., University of Kansas

Understanding Modern Mathematics

ISBN: 0763734012
Cover: Paperback
Copyright: 2006
Will Publish: 04/07/2006

Understanding Modern Mathematics is an exceptional collection of topics meant to better acquaint students with mathematics through an exposure to its applications and an analysis of its culture. The text provides an in-depth focus on such key topics as probability, statistics, voting systems, game theory, and linear programming. Two additional chapters on geometry and symmetry can be found on the text's web site, providing students the opportunity to see the 3-dimensional geometric figures in full color. The text provides students with an understanding of how these important mathematical topics are relevant in their everyday lives while emphasizing the history of mathematics . Understanding Modern Mathematics is the perfect complement to any Liberal Arts Mathematics course

Table of Contents

Chapter 1 - Probability

Chapter 2- Statistics

Chapter 3- Voting Systems

Chapter 4- Game Theory

Chapter 5- Linear Programming

Chapter 6- Symmetry: Planar Symmetries (Available Online)

Chapter 7- Spatial Symmetries (Available Online)