Exner An Accompaniment to Higher Mathematics

1996
ISBN: 978-1-4612-3998-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 200 Seiten, eBook

Reihe: Undergraduate Texts in Mathematics

Designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently, this is well suited as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology. The book teaches in detail how to construct examples and non-examples to help understand a new theorem or definition; it shows how to discover the outline of a proof in the form of the theorem and how logical structures determine the forms that proofs may take. Throughout, the text asks the reader to pause and work on an example or a problem before continuing, and encourages the student to engage the topic at hand and to learn from failed attempts at solving problems. The book may also be used as the main text for a "transitions" course bridging the gap between calculus and higher mathematics. The whole concludes with a set of "Laboratories" in which students can practice the skills learned in the earlier chapters on set theory and function theory.

Zielgruppe

Lower undergraduate

Autoren/Hrsg.

Exner, George R.

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 of Sets.- 4.6.1 Exercises.- A Theoretical Apologia.- B Hints.- References.