Proof and the Art of Mathematics
ISBN: 9780262539791 | Copyright 2020
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| Contents (pg. vii) | |
| Preface (pg. xiii) | |
| A Note to the Instructor (pg. xvii) | |
| A Note to the Student (pg. xxi) | |
| About the Author (pg. xxv) | |
| 1. A Classical Beginning (pg. 1) | |
| 1.1 The number 2 is irrational (pg. 2) | |
| 1.2 Lowest terms (pg. 4) | |
| 1.3 A geometric proof (pg. 5) | |
| 1.4 Generalizations to other roots (pg. 6) | |
| Mathematical Habits (pg. 7) | |
| Exercises (pg. 8) | |
| 2. Multiple Proofs (pg. 9) | |
| 2.1 n2-n is even (pg. 10) | |
| 2.2 One theorem, seven proofs (pg. 10) | |
| 2.3 Different proofs suggest different generalizations (pg. 12) | |
| Mathematical Habits (pg. 13) | |
| Exercises (pg. 14) | |
| Credits (pg. 14) | |
| 3. Number Theory (pg. 15) | |
| 3.1 Prime numbers (pg. 15) | |
| 3.2 The fundamental theorem of arithmetic (pg. 16) | |
| 3.3 Euclidean division algorithm (pg. 19) | |
| 3.4 Fundamental theorem of arithmetic, uniqueness (pg. 21) | |
| 3.5 Infinitely many primes (pg. 21) | |
| Mathematical Habits (pg. 24) | |
| Exercises (pg. 25) | |
| 4. Mathematical Induction (pg. 27) | |
| 4.1 The least-number principle (pg. 27) | |
| 4.2 Common induction (pg. 28) | |
| 4.3 Several proofs using induction (pg. 29) | |
| 4.4 Proving the induction principle (pg. 32) | |
| 4.5 Strong induction (pg. 33) | |
| 4.6 Buckets of Fish via nested induction (pg. 34) | |
| 4.7 Every number is interesting (pg. 37) | |
| Mathematical Habits (pg. 37) | |
| Exercises (pg. 38) | |
| Credits (pg. 39) | |
| 5. Discrete Mathematics (pg. 41) | |
| 5.1 More pointed at than pointing (pg. 41) | |
| 5.2 Chocolate bar problem (pg. 43) | |
| 5.3 Tiling problems (pg. 44) | |
| 5.4 Escape! (pg. 47) | |
| 5.5 Representing integers as a sum (pg. 49) | |
| 5.6 Permutations and combinations (pg. 50) | |
| 5.7 The pigeon-hole principle (pg. 52) | |
| 5.8 The zigzag theorem (pg. 53) | |
| Mathematical Habits (pg. 55) | |
| Exercises (pg. 55) | |
| Credits (pg. 56) | |
| 6. Proofs without Words (pg. 57) | |
| 6.1 A geometric sum (pg. 57) | |
| 6.2 Binomial square (pg. 58) | |
| 6.3 Criticism of the "without words'' aspect (pg. 58) | |
| 6.4 Triangular choices (pg. 59) | |
| 6.5 Further identities (pg. 60) | |
| 6.6 Sum of odd numbers (pg. 60) | |
| 6.7 A Fibonacci identity (pg. 61) | |
| 6.8 A sum of cubes (pg. 61) | |
| 6.9 Another infinite series (pg. 62) | |
| 6.10 Area of a circle (pg. 62) | |
| 6.11 Tiling with dominoes (pg. 63) | |
| 6.12 How to lie with pictures (pg. 66) | |
| Mathematical Habits (pg. 68) | |
| Exercises (pg. 69) | |
| Credits (pg. 70) | |
| 7. Theory of Games (pg. 71) | |
| 7.1 Twenty-One (pg. 71) | |
| 7.2 Buckets of Fish (pg. 73) | |
| 7.3 The game of Nim (pg. 74) | |
| 7.4 The Gold Coin game (pg. 79) | |
| 7.5 Chomp (pg. 81) | |
| 7.6 Games of perfect information (pg. 83) | |
| 7.7 The fundamental theorem of finite games (pg. 85) | |
| Mathematical Habits (pg. 89) | |
| Exercises (pg. 89) | |
| Credits (pg. 90) | |
| 8. Pick's Theorem (pg. 91) | |
| 8.1 Figures in the integer lattice (pg. 91) | |
| 8.2 Pick's theorem for rectangles (pg. 92) | |
| 8.3 Pick's theorem for triangles (pg. 93) | |
| 8.4 Amalgamation (pg. 95) | |
| 8.5 Triangulations (pg. 97) | |
| 8.6 Proof of Pick's theorem, general case (pg. 98) | |
| Mathematical Habits (pg. 98) | |
| Exercises (pg. 99) | |
| Credits (pg. 100) | |
| 9. Lattice-Point Polygons (pg. 101) | |
| 9.1 Regular polygons in the integer lattice (pg. 101) | |
| 9.2 Hexagonal and triangular lattices (pg. 104) | |
| 9.3 Generalizing to arbitrary lattices (pg. 106) | |
| Mathematical Habits (pg. 107) | |
| Exercises (pg. 108) | |
| Credits (pg. 110) | |
| 10. Polygonal Dissection Congruence Theorem (pg. 111) | |
| 10.1 The polygonal dissection congruence theorem (pg. 111) | |
| 10.2 Triangles to parallelograms (pg. 112) | |
| 10.3 Parallelograms to rectangles (pg. 113) | |
| 10.4 Rectangles to squares (pg. 113) | |
| 10.5 Combining squares (pg. 114) | |
| 10.6 Full proof of the dissection congruence theorem (pg. 115) | |
| 10.7 Scissors congruence (pg. 115) | |
| Mathematical Habits (pg. 117) | |
| Exercises (pg. 118) | |
| Credits (pg. 119) | |
| 11. Functions and Relations (pg. 121) | |
| 11.1 Relations (pg. 121) | |
| 11.2 Equivalence relations (pg. 122) | |
| 11.3 Equivalence classes and partitions (pg. 125) | |
| 11.4 Closures of a relation (pg. 127) | |
| 11.5 Functions (pg. 128) | |
| Mathematical Habits (pg. 129) | |
| Exercises (pg. 130) | |
| 12. Graph Theory (pg. 133) | |
| 12.1 The bridges of Königsberg (pg. 133) | |
| 12.2 Circuits and paths in a graph (pg. 134) | |
| 12.3 The five-room puzzle (pg. 137) | |
| 12.4 The Euler characteristic (pg. 138) | |
| Mathematical Habits (pg. 139) | |
| Exercises (pg. 140) | |
| Credits (pg. 142) | |
| 13. Infinity (pg. 143) | |
| 13.1 Hilbert's Grand Hotel (pg. 143) | |
| Hilbert's bus (pg. 144) | |
| Hilbert's train (pg. 144) | |
| Hilbert's half marathon (pg. 145) | |
| Cantor's cruise ship (pg. 146) | |
| 13.2 Countability (pg. 146) | |
| 13.3 Uncountability of the real numbers (pg. 150) | |
| Alternative proof of Cantor's theorem (pg. 152) | |
| Cranks (pg. 153) | |
| 13.4 Transcendental numbers (pg. 154) | |
| 13.5 Equinumerosity (pg. 156) | |
| 13.6 The Shröder-Cantor-Bernstein theorem (pg. 157) | |
| 13.7 The real plane and real line are equinumerous (pg. 159) | |
| Mathematical Habits (pg. 160) | |
| Exercises (pg. 160) | |
| Credits (pg. 161) | |
| 14. Order Theory (pg. 163) | |
| 14.1 Partial orders (pg. 163) | |
| 14.2 Minimal versus least elements (pg. 164) | |
| 14.3 Linear orders (pg. 166) | |
| 14.4 Isomorphisms of orders (pg. 167) | |
| 14.5 The rational line is universal (pg. 168) | |
| 14.6 The eventual domination order (pg. 170) | |
| Mathematical Habits (pg. 171) | |
| Exercises (pg. 171) | |
| 15. Real Analysis (pg. 173) | |
| 15.1 Definition of continuity (pg. 173) | |
| 15.2 Sums and products of continuous functions (pg. 175) | |
| 15.3 Continuous at exactly one point (pg. 177) | |
| 15.4 The least-upper-bound principle (pg. 178) | |
| 15.5 The intermediate-value theorem (pg. 178) | |
| 15.6 The Heine-Borel theorem (pg. 179) | |
| 15.7 The Bolzano-Weierstrass theorem (pg. 181) | |
| 15.8 The principle of continuous induction (pg. 182) | |
| Mathematical Habits (pg. 185) | |
| Exercises (pg. 185) | |
| Credits (pg. 187) | |
| Answers to Selected Exercises (pg. 189) | |
| Bibliography (pg. 199) | |
| Index of Mathematical Habits (pg. 201) | |
| Notation Index (pg. 203) | |
| Subject Index (pg. 205) | |
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