# How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

Cambridge University Press

Edition: Illustrated, 2/12/2009

EAN 9780521719780, ISBN10: 052171978X

Paperback, 278 pages, 25.4 x 19.6 x 2 cm

Language: English

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.

Preface

Part I. Study Skills For Mathematicians

1. Sets and functions

2. Reading mathematics

3. Writing mathematics I

4. Writing mathematics II

5. How to solve problems

Part II. How To Think Logically

6. Making a statement

7. Implications

8. Finer points concerning implications

9. Converse and equivalence

10. Quantifiers Ã¢â‚¬â€œ For all and There exists

11. Complexity and negation of quantifiers

12. Examples and counterexamples

13. Summary of logic

Part III. Definitions, Theorems and Proofs

14. Definitions, theorems and proofs

15. How to read a definition

16. How to read a theorem

17. Proof

18. How to read a proof

19. A study of Pythagoras' Theorem

Part IV. Techniques of Proof

20. Techniques of proof I

direct method

21. Some common mistakes

22. Techniques of proof II

proof by cases

23. Techniques of proof III

Contradiction

24. Techniques of proof IV

Induction

25. More sophisticated induction techniques

26. Techniques of proof V

contrapositive method

Part V. Mathematics That All Good Mathematicians Need

27. Divisors

28. The Euclidean Algorithm

29. Modular arithmetic

30. Injective, surjective, bijective Ã¢â‚¬â€œ and a bit about infinity

31. Equivalence relations

Part VI. Closing Remarks

32. Putting it all together

33. Generalization and specialization

34. True understanding

35. The biggest secret

Appendices

A. Greek alphabet

B. Commonly used symbols and notation

C. How to prove that Ã¢â‚¬Â¦

Index.