xx viii To the Instructor
definition. Its presence is crucial, for example, in the proof of Theorem 4.1.9,
the sequential criterion for limits of functions.
My students have never complained that the use of symbols impedes their
progress in any way. On the contrary, students learn them easily and put them
to use quickly in mastering complex concepts and proofs. A brief exposition of
the logical concepts and symbols used in this book is found in Appendix A.
Of crucial importance is a secure understanding of the negation of the various
logical forms, especially the negation of quantified statements.
Finally, the use of logical symbols is not an excuse for sloppy writing or
incorrect grammar. In fact, these symbols are subject to the usual rules of
grammar and syntax. Used properly, they reinforce these rules. As instructors,
we cannot condone careless use of symbols any more than careless thinking of
any kind.
WARNINGWhile this book began life as a textbook for a one-semester course, it contains
much more material than can be covered in one semester. The additional ma-
terial is of two types: enrichment topics included in Chapters 1- 7 to provide
optional extended study or projects to challenge more advanced students, and
additional chapters or sections intended to provide enough material for a sec-
ond course. In teaching a one-semester course it is important to avoid getting
bogged down in the optional material in the earlier chapters. For example , while
the entire Chapter 1 is essential to the logical development of the Real Number
system, only Sections 1.5 and 1.6 must be covered in detail; the others can be
skimmed.
The book incorporates several measures to ensure instructors choose ma-
terial appropriate for a one-semester course. Each chapter, and some sections,
begin with comments in a box, giving advice on which sections must be covered
in detail and which can be skimmed or omitted. Theorems, examples, results,
proofs, and exercises that are optional are clearly marked with an asterisk, "*·"
In addition, the following table suggests one way to cover the core material of
Chapters 1-7 in one semester.
SUGGESTED ONE-SEMESTER COURSEThe material identified in the table below^6 as "core" corresponds to my own
one-semester course. Where a number appears in both columns, the number
on the right estimates the amount of additional time that may be required to
completely cover the optional material.
- Based on the author's experience teaching a one-semester course meeting three 50-minute
periods p er week.