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 – 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 – 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 …
Index.