Preface ..................................................... ix
I Study skills for mathematicians .............................. 1
1 Sets and functions ........................................... 3
2 Reading mathematics ......................................... 14
3 Writing mathematics I ....................................... 21
4 Writing mathematics II ...................................... 35
5 How to solve problems ....................................... 41
II How to think logically ..................................... 51
6 Making a statement .......................................... 53
7 Implications ................................................ 63
8 Finer points concerning implications ........................ 69
9 Converse and equivalence .................................... 75
10 Quantifiers - For all and There exists ...................... 80
11 Complexity and negation of quantifiers ...................... 84
12 Examples and counterexamples ................................ 90
13 Summary of logic ............................................ 96
III Definitions, theorems and proofs .......................... 97
14 Definitions, theorems and proofs ............................ 99
15 How to read a definition ................................... 103
16 How to read a theorem ...................................... 109
17 Proof ...................................................... 116
18 How to read a proof ........................................ 119
19 A study of Pythagoras' Theorem ............................. 126
IV Techniques of proof ....................................... 137
20 Techniques of proof I: Direct method ....................... 139
21 Some common mistakes ....................................... 149
22 Techniques of proof II: Proof by cases ..................... 155
23 Techniques of proof III: Contradiction ..................... 161
24 Techniques of proof IV: Induction .......................... 166
25 More sophisticated induction techniques .................... 175
26 Techniques of proof V: Contrapositive method ............... 180
V Mathematics that all good mathematicians need .............. 185
27 Divisors ................................................... 187
28 The Euclidean Algorithm .................................... 196
29 Modular arithmetic ......................................... 208
30 Injective, surjective, bijective - and a bit about
infinity ................................................... 218
31 Equivalence relations ...................................... 230
VI Closing remarks ........................................... 241
32 Putting it all together .................................... 243
33 Generalization and specialization .......................... 248
34 True understanding ......................................... 252
35 The biggest secret ......................................... 255
Appendices .................................................... 257
A Greek alphabet ............................................. 257
В Commonly used symbols and notation ......................... 258
С How to prove that .......................................... 260
Index ...................................................... 263
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