LOGIC, SETS AND THE TECHNIQUES OF MATHEMATICAL PROOFS: A COMPANION FOR HIGH SCHOOL AND COLLEGE STUDENTSAs its title indicates, this book is about logic, sets and mathematical proofs. It is a careful, patient and rigorous introduction for readers with very limited mathematical maturity. It teaches the reader not only how to read a mathematical proof, but also how to write one. To achieve this, we carefully lay out all the various proof methods encountered in mathematical discourse, give their logical justifications, and apply them to the study of topics [such as real numbers, relations, functions, sequences, fine sets, infinite sets, countable sets, uncountable sets and transfinite numbers] whose mastery is important for anyone contemplating advanced studies in mathematics. The book is completely self-contained; since the prerequisites for reading it are only a sound background in high school algebra. Though this book is meant to be a companion specifically for senior high school pupils and college undergraduate students, it will also be of immense value to anyone interested in acquiring the tools and way of thinking of the mathematician. |
Contents
| 11 | |
| 17 | |
Chapter 2 Tautologies and contradictions | 37 |
Chapter 3 Theorems and Proof methods | 49 |
Chapter 4 Sets | 62 |
Chapter 5 Operations on sets | 81 |
Chapter 6 Singlevariable sentential logic | 101 |
Chapter 7 Sentential implications and equivalences | 115 |
Chapter 12 Methods of proof by mathematical induction | 203 |
Chapter 13 Relations | 218 |
Chapter 14 Functions and absolute value | 239 |
Chapter 15 types of functions | 254 |
Chapter 16 Sequences | 273 |
Chapter 17 Fundamental and monotonic sequences | 295 |
Chapter 18 Finite and Infinite sets | 309 |
Chapter 19 Indexed Family of sets | 328 |
Chapter 8 Twovariable predicate logic | 125 |
Chapter 9 Real numbers | 139 |
Subtraction and Division | 160 |
The axiom of continuity | 176 |
Bibliography | 348 |
| 349 | |
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Common terms and phrases
Assume Axiom belong bijection called Chapter 9 clearly collection combination complete conditional contradiction converges Corollary countably infinite defined Definition denial denote distinct element empty equal equinumerous equivalent establish exactly Example Exercise exists F F F F T F fact false fixed function give given Hence hypothesis implication important induction infinite sets injective interval introduce least least upper bound mathematics means method n n s natural number nonempty Note object open sentence ordered pairs positive principle Problem proof properties proposition propositional expression prove quantified proposition rational real number relation Remark result Rx Qx satisfies sequence simply Solution Step subset Suppose symbol tautology Theorem true truth set truth values universal set variable write x x x