How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
by Kevin HoustonEdition: 1st
Format: Hardcover
Pub. Date: 2009-02-23
Publisher(s): Cambridge University Press
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Summary
Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.
Table of Contents
| Preface | |
| Study Skills For Mathematicians | |
| Sets and functions | |
| Reading mathematics | |
| Writing mathematics I | |
| Writing mathematics II | |
| How to solve problems | |
| How To Think Logically | |
| Making a statement | |
| Implications | |
| Finer points concerning implications | |
| Converse and equivalence | |
| Quantifiers - For all and There exists | |
| Complexity and negation of quantifiers | |
| Examples and counterexamples | |
| Summary of logic | |
| Definitions, Theorems and Proofs | |
| Definitions, theorems and proofs | |
| How to read a definition | |
| How to read a theorem | |
| Proof | |
| How to read a proof | |
| A study of Pythagoras' Theorem | |
| Techniques of Proof | |
| Techniques of proof I: direct method | |
| Some common mistakes | |
| Techniques of proof II: proof by cases | |
| Techniques of proof III: Contradiction | |
| Techniques of proof IV: Induction | |
| More sophisticated induction techniques | |
| Techniques of proof V: contrapositive method | |
| Mathematics That All Good Mathematicians Need | |
| Divisors | |
| The Euclidean Algorithm | |
| Modular arithmetic | |
| Injective, surjective, bijective - and a bit about infinity | |
| Equivalence relations | |
| Closing Remarks | |
| Putting it all together | |
| Generalization and specialization | |
| True understanding | |
| The biggest secret | |
| Appendices | |
| Greek alphabet | |
| Commonly used symbols and notation | |
| How to prove that | |
| Index | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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