An Accompaniment to Higher Mathematics

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 famililar 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 forms (especially quantifier arguments), and analyzes, thoroughly, the often sketchy coding of these forms in proofs as they are ordinarily written. The common introductroy material (such as sets and functions) is used for the numerous exercises, and the book conc ludes with a set of "Laboratories" on these topics in which the student can practice the skills learned in the earlier chapters. Intended for use as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology, the book may also be used as the main text for a "transitions" course bridging the gab between calculus and higher mathematics.

Inhaltsverzeichnis:

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 o f Sets.- 4.6.1 Exercises.- A Theoretical Apologia.- B Hints.- References.