An Accompaniment to Higher Mathematics by George R. ExnerAn Accompaniment to Higher Mathematics by George R. Exner

An Accompaniment to Higher Mathematics

byGeorge R. Exner

Paperback | April 15, 1996

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For Students Congratulations! You are about to take a course in mathematical proof. If you are nervous about the whole thing, this book is for you (if not, please read the second and third paragraphs in the introduction for professors following this, so you won't feel left out). The rumors are true; a first course in proof may be very hard because you will have to do three things that are probably new to you: 1. Read mathematics independently. 2. Understand proofs on your own. :1. Discover and write your own proofs. This book is all about what to do if this list is threatening because you "never read your calculus book" or "can't do proofs. " Here's the good news: you must be good at mathematics or you wouldn't have gotten this far. Here's the bad news: what worked before may not work this time. Success may lie in improving or discarding many habits that were good enough once but aren't now. Let's see how we've gotten to a point at which someone could dare to imply that you have bad habits. l The typical elementary and high school mathematics education in the United States tends to teach students to have ineffective learning habits, 1 In the first paragraph, yet. xiv Introduction and we blush to admit college can be just as bad.
Title:An Accompaniment to Higher MathematicsFormat:PaperbackDimensions:200 pagesPublished:April 15, 1996Publisher:Springer-Verlag/Sci-Tech/Trade

The following ISBNs are associated with this title:

ISBN - 10:0387946179

ISBN - 13:9780387946177


Table of Contents

1 Examples.- 1.1 Propaganda.- 1.2 Basic Examples for Definitions.- 1.2.1 Exercises.- 1.3 Basic Examples for Theorems.- 1.3.1 Exercises.- 1.4 Extended Examples.- 1.4.1 Exercises.- 1.5 Notational Interlude.- 1.6 Examples Again: Standard Sources.- 1.6.1 Small Examples.- 1.6.2 Exercises.- 1.6.3 Extreme Examples.- 1.6.4 Exercises: Take Two.- 1.7 Non-examples for Definitions.- 1.7.1 Exercises.- 1.8 Non-examples for Theorems.- 1.8.1 Exercises.- 1.8.2 More to Do.- 1.8.3 Exercises.- 1.9 Summary and More Propaganda.- 1.9.1 Exercises.- 1.10 What Next?.- 2 Informal Language and Proof.- 2.1 Ordinary Language Clues.- 2.1.1 Exercises.- 2.1.2 Rules of Thumb.- 2.1.3 Exercises.- 2.1.4 Comments on the Rules.- 2.1.5 Exercises.- 2.2 Real-Life Proofs vs. Rules of Thumb.- 2.3 Proof Forms for Implication.- 2.3.1 Implication Forms: Bare Bones.- 2.3.2 Implication Forms: Subtleties.- 2.3.3 Exercises.- 2.3.4 Choosing a Form for Implication.- 2.4 Two More Proof Forms.- 2.4.1 Proof by Cases: Bare Bones.- 2.4.2 Proof by Cases: Subtleties.- 2.4.3 Proof by Induction.- 2.4.4 Proof by Induction: Subtleties.- 2.4.5 Exercises.- 2.5 The Other Shoe, and Propaganda.- 3 For mal Language and Proof.- 3.1 Propaganda.- 3.2 Formal Language: Basics.- 3.2.1 Exercises.- 3.3 Quantifiers.- 3.3.1 Statement Forms.- 3.3.2 Exercises.- 3.3.3 Quantified Statement Forms.- 3.3.4 Exercises.- 3.3.5 Theorem Statements.- 3.3.6 Exercises.- 3.3.7 Pause: Meaning, a Plea, and Practice.- 3.3.8 Matters of Proof: Quantifiers.- 3.3.9 Exercises.- 3.4 Finding Proofs from Structure.- 3.4.1 Finding Proofs.- 3.4.2 Exercises.- 3.4.3 Digression: Induction Correctly.- 3.4.4 One More Example.- 3.4.5 Exercises.- 3.5 Summary, Propaganda, and What Next?.- 4 Laboratories.- 4.1 Lab I: Sets by Example.- 4.1.1 Exercises.- 4.2 Lab II: Functions by Example.- 4.2.1 Exercises.- 4.3 Lab III: Sets and Proof.- 4.3.1 Exercises.- 4.4 Lab IV: Functions and Proof.- 4.4.1 Exercises.- 4.5 Lab V: Function of Sets.- 4.5.1 Exercises.- 4.6 Lab VI: Families of Sets.- 4.6.1 Exercises.- A Theoretical Apologia.- B Hints.- References.

From Our Editors

This text prepares undergraduate mathematics students to meet two challenges in the study of mathematics, namely, to read mathematics independently and to understand and write proofs. The book begins by teaching how to read mathematics actively, constructing examples, extreme cases, and non-examples to aid in understanding an unfamiliar theorem or definition (a technique familiar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. The book then turns to proofs, showing in detail how to discover the structure of a potential proof from the form of the theorem (especially the conclusion). It shows the logical structure behind proof farms (especially quantifier arguments), and analyzes, thoroughly, the often sketchy coding of these forms in proofs as they are ordinarily written. The common introductory material (such as sets and functions) is used for the numerous exercises, and the book concludes with a set of "Laboratories" on these topics in which the student can practice t