An Accompaniment to Higher Mathematics - Tapa blanda

Sinopsis

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.

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