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

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. Cover......Page 1 Half-title......Page 3 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 The concept......Page 11 To instructors and lecturers – a moment of your valuable time......Page 12 I’d just like to thank.........Page 13 PART I Study skills for mathematicians......Page 15 Sets......Page 17 Natural numbers......Page 18 The empty set......Page 19 Operations on sets......Page 22 Maps and functions......Page 24 Exercises......Page 26 Summary......Page 27 CHAPTER 2 Reading mathematics......Page 28 Read with pen and paper at hand......Page 29 Identify what is important......Page 30 Read statements first – proofs later......Page 31 Reflect......Page 32 Exercises......Page 33 Summary......Page 34 CHAPTER 3 Writing mathematics I......Page 35 An example......Page 36 Write in sentences......Page 37 Use punctuation......Page 38 Keep it simple......Page 39 Explain what you are doing – keeping the reader informed......Page 40 Explain your assertions......Page 41 Words or symbols?......Page 42 Displaying results with the equals sign......Page 44 Don’t draw arrows everywhere......Page 45 Exercises......Page 46 Summary......Page 48 If you use ‘if’, then use ‘then’......Page 49 Decimal approximations......Page 50 The curse of the implication symbol......Page 51 Define your symbols and notation......Page 52 Use synonyms......Page 53 Summary......Page 54 Sample problems......Page 55 Understand all the words and symbols in the problem......Page 56 Work backwards and forwards......Page 57 Draw a picture......Page 58 Devising a plan......Page 59 Give things names......Page 60 Check the answer......Page 61 Exercises......Page 62 Summary......Page 63 PART II How to think logically......Page 65 Statements......Page 67 An important example......Page 68 Using non-mathematical examples......Page 69 Negation......Page 70 Truth tables......Page 71 Statements using ‘and’......Page 72 Statements using ‘or’......Page 73 Negation of ‘and’ and ‘or’......Page 74 Summary......Page 75 ‘If..., then...’ statements......Page 77 Hypothesis, assumption and conclusion......Page 78 False statements can imply true statements......Page 79 ‘B if A’ is the same as.........Page 80 Exercises......Page 81 Summary......Page 82 The inverse: a common mistake......Page 83 Necessary conditions......Page 85 The contrapositive......Page 86 Exercises......Page 87 Summary......Page 88 The converse......Page 89 Logical equivalence......Page 90 Another example......Page 91 Exercises......Page 92 Summary......Page 93 For all – the universal quantifier......Page 94 There exists – the existential quantifier......Page 95 Warning! The order of quantifiers is important......Page 96 Summary......Page 97 One quantifier......Page 98 Statements beginning ‘for all’......Page 99 Statements beginning ‘there exists’......Page 100 Negation of quantifiers......Page 101 Exercises......Page 102 Summary......Page 103 CHAPTER 12 Examples and counterexamples......Page 104 Reversing worked examples......Page 105 Counterexamples......Page 106 How to create examples and counterexamples......Page 107 Exercises......Page 108 Summary......Page 109 CHAPTER 13 Summary of logic......Page 110 PART III Definitions, theorems and proofs......Page 111 Definitions......Page 113 Proofs......Page 114 Fermat’s Last Theorem......Page 115 Summary......Page 116 What is a Definition?......Page 117 The ‘if and only if’ nature of mathematical Definitions......Page 118 Find standard examples......Page 119 Find extreme examples......Page 120 Exercises......Page 121 Summary......Page 122 Three theorems......Page 123 Rate the strength of the assumptions and conclusions......Page 124 Compare with earlier theorems......Page 125 Apply to trivial examples and other extreme cases......Page 126 What happens to non-examples?......Page 127 Exercises......Page 128 Summary......Page 129 Why prove statements?......Page 130 Proofs are hard to create – but there is hope......Page 131 Summary......Page 132 A simple theorem and its proof......Page 133 Identify the methods used......Page 134 Draw a picture......Page 135 Look for mistakes – try extreme cases......Page 136 Reflection......Page 137 Exercises......Page 138 Summary......Page 139 Study of the theorem......Page 140 Compare with previous theorems......Page 141 Draw a picture......Page 142 Apply the theorem to non-examples......Page 143 Proof of Pythagoras’ Theorem......Page 144 Check the text......Page 145 What about the converse?......Page 146 Exercises......Page 147 Summary......Page 149 PART IV Techniques of proof......Page 151 Examples of the direct method......Page 153 How to show that an equation holds......Page 156 If and only if proofs......Page 157 Proving that two sets are equal......Page 159 Exercises......Page 160 Summary......Page 161 Don’t assume what had to be proved......Page 163 Square root is a function so it gives a single number......Page 164 Don’t divide by zero.........Page 165 Exercises......Page 167 Summary......Page 168 Examples of cases......Page 169 The modulus function......Page 170 The importance of cases in extreme examples......Page 172 Exercises......Page 173 Summary......Page 174 Simple examples of proof by contradiction......Page 175 The irrationality of the square root of 2......Page 177 How to write a proof by contradiction......Page 178 Summary......Page 179 The Principle of Mathematical Induction......Page 180 Examples of induction......Page 182 Exercises......Page 186 Summary......Page 188 First variant......Page 189 Second variant......Page 190 Third variant......Page 191 Exercises......Page 192 Summary......Page 193 Revision of the contrapositive......Page 194 Don’t confuse contradiction and contrapositive......Page 196 Summary......Page 197 PART V Mathematics that all good mathematicians need......Page 199 Divisibility......Page 201 There exist an infinite number of primes......Page 205 Greatest common divisor......Page 206 A common mistake......Page 207 Summary......Page 208 The Division Lemma......Page 210 More general version of the Division Lemma......Page 213 The Euclidean Algorithm......Page 214 Calculating gcd......Page 215 Euclid’s Lemma......Page 217 Diophantine equations......Page 218 Exercises......Page 220 Summary......Page 221 Modular arithmetic......Page 222 The arithmetic of mod......Page 224 Fermat’s Little Theorem......Page 225 Finding remainders......Page 227 Divisibility tests......Page 228 Exercises......Page 229 Summary......Page 230 Injective functions......Page 232 Surjective functions......Page 234 Composition of functions......Page 235 Inverse functions......Page 236 Problems with trigonometrical notation......Page 237 Types of infinity – countable and uncountable......Page 238 The rationals are countable......Page 239 The reals are uncountable......Page 240 Exercises......Page 241 Summary......Page 242 Relations......Page 244 Equivalence relations......Page 245 A subtlety......Page 246 Equivalence classes......Page 247 Partitions......Page 248 Modular arithmetic......Page 252 Exercises......Page 253 Summary......Page 254 PART VI Closing remarks......Page 255 Play with examples......Page 257 Change the problem......Page 258 Ask ‘What happens if...?’......Page 259 Exercises......Page 260 Summary......Page 261 Weakening the hypotheses......Page 262 Specialization......Page 263 Exercises......Page 264 Summary......Page 265 Understanding theorems......Page 266 Understanding a major topic......Page 267 Summary......Page 268 CHAPTER 35 The biggest secret......Page 269 Summary......Page 270 APPENDIX A Greek alphabet......Page 271 APPENDIX B Commonly used symbols and notation......Page 272 How to prove that two numbers are equal......Page 274 Proving with quantifiers......Page 275 How to prove a map is bijective......Page 276 Index......Page 277 This Arsenal Of Tips And Techniques Eases New Students Into Undergraduate Mathematics, Unlocking The World Of Definitions, Theorems, And Proofs. I. Study Skills For Mathematicians -- Ii. How To Think Logically -- Iii. Definitions, Theorems And Proofs -- Iv. Techniques Of Proof -- V. Mathematics That All Good Mathematicians Need -- Vi. Closing Remarks. Kevin Houston. Includes Index. Includes Bibliographical References And Index. Cover; Half-title; Title; Copyright; Dedication; Contents; Preface; PART I Study skills for mathematicians; PART II How to think logically; PART III Definitions, theorems and proofs; PART IV Techniques of proof; PART V Mathematics that all good mathematicians need; PART VI Closing remarks; APPENDIX A Greek alphabet; APPENDIX B Commonly used symbols and notation; APPENDIX C How to prove that ... ; Index. Looking for a head start in your undergraduate degree in mathematics? This friendly companion eases beginning students into real mathematical thinking, unlocking important techniques for effective mathematics so you can communicate with clarity, solve problems, and explore the world of definitions, theorems and proofs with real confidence.