An Accompaniment to Higher Mathematics

Designed for students preparing to engage in their first struggles to understand mathematics independently, this book teaches in detail how to construct examples and non-examples to help understand a new theorem or definition, and how to discover the outline of a proof in the form of the theorem. It is intended to be used interactively, frequently asking the reader to pause and work on an example or a problem before continuing, and encouraging the student to learn from failed attempts at solving problems.

Reprint of a best-selling title The text is meant to be used interactively, frequently asking the reader to pause and work on an example or a problem before continuing Designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently Bridges the gap between calculus and higher mathematics

Inhalt
1 Examples.- 1.1 Propaganda.- 1.2 Basic Examples for Definitions.- 1.3 Basic Examples for Theorems.- 1.4 Extended Examples.- 1.5 Notational Interlude.- 1.6 Examples Again: Standard Sources.- 1.7 Non-examples for Definitions.- 1.8 Non-examples for Theorems.- 1.9 Summary and More Propaganda.- 1.10 What Next?.- 2 Informal Language and Proof.- 2.1 Ordinary Language Clues.- 2.2 Real-Life Proofs vs. Rules of Thumb.- 2.3 Proof Forms for Implication.- 2.4 Two More Proof Forms.- 2.5 The Other Shoe, and Propaganda.- 3 For mal Language and Proof.- 3.1 Propaganda.- 3.2 Formal Language: Basics.- 3.3 Quantifiers.- 3.4 Finding Proofs from Structure.- 3.5 Summary, Propaganda, and What Next?.- 4 Laboratories.- 4.1 Lab I: Sets by Example.- 4.2 Lab II: Functions by Example.- 4.3 Lab III: Sets and Proof.- 4.4 Lab IV: Functions and Proof.- 4.5 Lab V: Function of Sets.- 4.6 Lab VI: Families of Sets.- A Theoretical Apologia.- B Hints.- References.