An Accompaniment to Higher Mathematics

ISBN-13: 9780387946177 / Angielski / Miękka / 1996 / 200 str.

George R. Exner

Opis

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.

Spis treści

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.