A Gateway To Higher Mathematics

by Goodfriend, Jason H., Ph.D.

• ISBN13: 9780763727338
• ISBN10: 0763727334
• Format: Hardcover
• Copyright: 2005-05-04
• Publisher: Jones & Bartlett

Summary

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

Acknowledgments	ix
Preface	xi
To the Student	xvii
Guide to Reading the Text	xix
Logic and Techniques for Proofs	1	(20)
Statements and Truth Tables	2	(7)
Logical Equivalence	9	(1)
Quantifiers	10	(4)
Some Techniques for Proofs	14	(7)
Elementary Set Theory	21	(50)
Introduction to Sets	22	(5)
Some Operations on Sets	27	(8)
Relations	35	(5)
Some Important Types of Relations on a Set	40	(6)
Functions	46	(16)
Mathematical Induction	62	(6)
The Axiom of Infinity	68	(3)
The Development of the Integers	71	(68)
Peano's Postulates	72	(5)
The Recursion Theorem, Addition, and Multiplication	77	(11)
An Order on the Positive Integers and the Second Principle of Mathematical Induction	88	(8)
The Uniqueness of Peano Spaces and the Beginnings of Counting	96	(11)
Introduction to Rings	107	(10)
Postulates for the Integers and Initial Properties	117	(14)
Proof of the Existence of the Integers	131	(8)
Properties and Application of Integers	139	(58)
Sequences, Cartesian Products, and the Generalized Recursion Theorem	140	(7)
Applications to Binary Operations	147	(16)
Introduction to Number Theory	163	(14)
Finite and Infinite Sets	177	(8)
Infinite Sets: Countable and Uncountable	185	(12)
Fields and the Rational Numbers	197	(26)
Introduction to Fields	198	(12)
The Rational Numbers and the Field of Quotients	210	(13)
The Development of the Real Numbers	223	(36)
Postulates for the Real Number System and Some Initial Properties	223	(10)
The Uniqueness of the Real Number System	233	(4)
Construction of a Set Obeying the Real Number Postulates	237	(11)
Initial Properties of Real Numbers	248	(11)
Some Additional Properties of Real Numbers	259	(28)
Convergent Real Sequences I	260	(5)
Convergent Real Sequences II	265	(5)
A Brief Introduction to Infinite Series	270	(4)
Representation of Real Numbers	274	(8)
Real Exponents	282	(5)
Appendix A: Proof of the Cantor-Schroder-Bernstein Theorem	287	(4)
Appendix B: Using the Axiom of Choice to Prove Some Results About Infinite Sets	291	(6)
Appendix C: Completion of the Construction of a Set Obeying the Real Number Postulates	297	(8)
References	305	(2)
Index	307