How to Prove It: A Structured ApproachMany mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems. This textbook will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed "scratchwork" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |
Contents
Sentential Logic | 7 |
12 Truth tables | 13 |
13 Variables and sets | 24 |
14 Operations on sets | 32 |
15 The conditional and biconditional connectives | 41 |
Quantificational Logic | 53 |
22 Equivalences involving quantifiers | 62 |
23 More operations on sets | 71 |
45 Closures | 193 |
46 Equivalence relations | 203 |
Functions | 215 |
52 Onetoone and onto | 224 |
53 Inverses of functions | 232 |
a research project | 240 |
Mathematical Induction | 245 |
62 More examples | 251 |
Proofs | 82 |
32 Proofs involving negations and conditionals | 92 |
33 Proofs involving quantifiers | 104 |
34 Proofs involving conjunctions and biconditionals | 120 |
35 Proofs involving disjunctions | 131 |
36 Existence and uniqueness proofs | 141 |
37 More examples of proofs | 149 |
Relations | 157 |
42 Relations | 165 |
43 More about relations | 174 |
44 Ordering relations | 181 |
63 Recursion | 261 |
64 Strong induction | 270 |
65 Closures again | 279 |
Infinite Sets | 284 |
72 Countable and uncountable sets | 291 |
73 The CantorSchröderBernstein theorem | 297 |
304 | |
Summary of proof techniques | 305 |
307 | |
Common terms and phrases
a₁ Analyze the logical arbitrary element argument assume assumption Chapter conclude conditional statement countable defined definition DeMorgan's law disjoint element of F equation equivalence classes equivalence relation example exercise fact false family of sets Figure formula free variable function f ƒ is one-to-one givens and goal go f induction step inductive hypothesis logical form mathematical induction mathematicians minimal element natural number notation ordered pairs P(xo partial order plug positive integer prime numbers proof by contradiction proof strategies prove a goal PV Q real number recursive reexpress Scratch second given Section set of P(x Similarly smallest element Solution Theorem stand statement P(x strong induction subset Suppose F symbols symmetric closure total order transitive closure truth set truth table truth values try to prove Uiel universe of discourse Vx(x well-ordering principle words write