Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

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390 Index for a function, 44 injection, 45, 46, 50, 117 counting, 123 integer lattice, 73 integers, set of, 26, 27 intersection of sets, 27, 31 inverse image, 48, 50 comparison to inverse function, 48 of a subset, 50 isomorphic intuitive definition, 235 isomorphism class, 237 iteration, 169, 171 k-permutation of n elements, 83 Königsberg, Seven Bridges of, 231, 278 K n , 239 knights and knaves, 4, 198 Kruskal’s algorithm, 254 lattice path, 73 correspondence with bit string, 74 length of, 73 lattice, integer, See integer lattice law of logic, 208 leaf, 243, 249 length of a bit string, 72 logical connectives, 5 logical equivalence, 201 logically valid, See law of logic magic chocolate bunnies, 195 marriage problem, See matching matching, 283 matching condition, 285 mathematical induction, See induction maximum degree, 271 minimum spanning tree, 254 mod, 312 modular arithmetic, 313 modus ponens, 205 molecular statement, 4 monochromatic, 273 multigraph, 239, 242 multiplicative principle, 57 multiset relation to multigraph, 239 natural numbers, set of, 27 negation, 6 neighbors of vertices, 284, 285 non-planar graph, 260 K 3,3 , 261 K 5 , 260 Petersen graph, 266 not (logical connective), 6 truth condition for, 6 NP-complete, 280 number theory, 307 octahedron, 264 one-to-one function, See injection onto function, See surjection operations on sets, 31 or (logical connective), 6 inclusive vs exclusive, 6 truth condition for, 6 parent (in a rooted tree), 251 partial sums, See sequence of partial sums partition, 311 Pascal’s triangle, 77, 146, 166 patterns in, 89 path, 243 Euler, 277 Hamilton, 277, 279 type of graph, 241 perfect graph, 271 perfect matching, See matching permutation, 81 of k elements chosen from n, See k-permutation of n elements
Index 391 vs combination, 84, 123, 128, 129 Petersen graph, 266 PIE, See principle of inclusion/exclusion pigeonhole principle, 219 planar graph, 243, 258 chromatic number of, 269 non-planar graph, 260 K 3,3 , 261 K 5 , 260 Petersen graph, 266 planar region, See face (planar graph) planar representation, 258 Platonic solid, See regular polyhedron polyhedron, 262 regular, 262 polynomial fitting, 160 power set, 27, 29 powers of 2, 141 predicate, 15 premises, 197 Prim’s algorithm, 254 prime numbers, 187, 214 principle of inclusion/exclusion, 64, 111 for 4 or more sets, 113 product notation, 144 product principle, See multiplicative principle proof by cases, 221 by contradiction, 218 by contrapositive, 216 by indcution, 177 combinatorial, 89, 95 proper vertex coloring, 243 proposition, 198 propositional variable, 5 puzzle chocolate bar, 185 knights and knaves, 4, 198 seven bridges, 231 Tower of Hanoi, 135 Pythagorean theorem, 7 Pythagorean triple, 319 quantifier for all, 15 implicit, 16 there exists, 15 racetrack principle, 184 Ramsey theory, 273 range of a function, 39, 50 rational numbers, set of, 27 real numbers, set of, 27 recurrence relation, 138 for a function, 44 for number of bit strings, 72 for number of lattice paths, 74 generating function, 302 solving, 167, 172, 175 recursive definition, 138 reference, self, See self reference region (graph), See face (planar graph) regular polyhedron, 262 remainder class, 310 residue class, See remainder class rook paths, 70 root (in a tree), 251 rooted tree, 248, 251 rule of four, 40 search breadth first, 253 depth first, 253 self reference, See reference, self sentence (compared to statement), 5 sentential variable, See propositional variable sequence, 136
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Discrete Mathematics An Open Introd
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Oscar Levin School of Mathematical
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Acknowledgements This book would no
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Preface This text aims to give an i
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How to use this book In addition to
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Contents Acknowledgements Preface H
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Contents xiii Proof by Cases . . .
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Chapter 0 Introduction and Prelimin
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0.1. What is Discrete Mathematics?
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0.2. Mathematical Statements 5 •
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0.2. Mathematical Statements 7 Impl
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0.2. Mathematical Statements 9 3. I
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0.2. Mathematical Statements 11 The
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0.2. Mathematical Statements 13 It
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0.2. Mathematical Statements 15 Inv
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0.2. Mathematical Statements 17 and
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0.2. Mathematical Statements 19 7.
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0.2. Mathematical Statements 21 12.
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0.2. Mathematical Statements 23 (b)
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0.3. Sets 25 say that the set B is
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0.3. Sets 27 We already have a lot
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0.3. Sets 29 Example 0.3.3 Let A {
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0.3. Sets 31 Operations On Sets Is
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0.3. Sets 33 You might notice that
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0.3. Sets 35 Exercises 1. Let A {1
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0.3. Sets 37 17. Let A {a, b, c, d
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0.4. Functions 39 0.4 Functions A f
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0.4. Functions 41 It would be absol
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0.4. Functions 43 We will also be i
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0.4. Functions 45 2. We are told th
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0.4. Functions 47 2. g : {1, 2, 3}
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0.4. Functions 49 Example 0.4.8 Con
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0.4. Functions 51 Exercises 1. Cons
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0.4. Functions 53 10. Suppose f : N
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0.4. Functions 55 (c) f is bĳectiv
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Chapter 1 Counting One of the first
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1.1. Additive and Multiplicative Pr
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1.2. Binomial Coefficients 71 Of th
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1.2. Binomial Coefficients 73 probl
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1.2. Binomial Coefficients 75 In fa
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1.2. Binomial Coefficients 77 Remem
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1.2. Binomial Coefficients 79 3. Le
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1.3. Combinations and Permutations
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1.4. Combinatorial Proofs 89 Invest
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1.4. Combinatorial Proofs 91 Soluti
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1.4. Combinatorial Proofs 93 Exampl
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1.4. Combinatorial Proofs 95 More P
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1.4. Combinatorial Proofs 97 Since
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1.4. Combinatorial Proofs 99 to (n,
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1.4. Combinatorial Proofs 101 8. Co
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1.5. Stars and Bars 103 Investigate
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1.5. Stars and Bars 105 some stars
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1.5. Stars and Bars 107 Example 1.5
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1.5. Stars and Bars 109 (b) How man
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1.6. Advanced Counting Using PIE 11
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1.7. Chapter Summary 127 Investigat
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1.7. Chapter Summary 129 (c) The pr
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1.7. Chapter Summary 131 (b) How ma
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1.7. Chapter Summary 133 one could
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Chapter 2 Sequences Investigate! Th
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2.1. Describing Sequences 137 While
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2.1. Describing Sequences 139 You m
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2.1. Describing Sequences 141 5. (
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2.1. Describing Sequences 143 and s
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2.1. Describing Sequences 145 5. Th
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2.1. Describing Sequences 147 18. W
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2.2. Arithmetic and Geometric Seque
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2.3. Polynomial Fitting 161 sequenc
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2.3. Polynomial Fitting 163 a 2 7
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2.3. Polynomial Fitting 165 5. Make
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2.4. Solving Recurrence Relations 1
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2.5. Induction 177 2.5 Induction Ma
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2.5. Induction 179 2. Prove that if
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2.5. Induction 181 the next is easi
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2.5. Induction 183 But we want to k
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2.5. Induction 185 Investigate! Str
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2.5. Induction 187 a − 1 breaks;
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2.5. Induction 189 8. Zombie Euler
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23. Use induction to prove that 2.5
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2.6. Chapter Summary 193 Investigat
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2.6. Chapter Summary 195 (c) Find a
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Chapter 3 Symbolic Logic and Proofs
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3.1. Propositional Logic 199 Truth
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3.1. Propositional Logic 201 statem
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3.1. Propositional Logic 203 propos
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3.1. Propositional Logic 205 How do
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3.1. Propositional Logic 207 Beyond
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3.1. Propositional Logic 209 Exerci
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3.1. Propositional Logic 211 ∴ P
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3.2. Proofs 213 Investigate! 3.2 Pr
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3.2. Proofs 215 5. N is not divisib
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3.2. Proofs 217 to prove directly,
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3.2. Proofs 219 Example 3.2.7 Prove
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3.2. Proofs 221 In fact, we can qui
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3.2. Proofs 223 Exercises 1. Consid
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3.2. Proofs 225 14. Prove that ther
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3.3. Chapter Summary 227 3.3 Chapte
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3.3. Chapter Summary 229 6. Conside
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Chapter 4 Graph Theory Investigate!
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4.1. Definitions 233 Investigate! 4
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4.1. Definitions 235 we have edges
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4.1. Definitions 237 Alternatively,
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4.1. Definitions 239 2-element subs
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4.1. Definitions 241 the vertices w
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4.1. Definitions 243 Path A path is
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4.1. Definitions 245 9. For each of
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4.2. Trees 247 Investigate! 4.2 Tre
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4.2. Trees 249 Assume T is a tree,
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4.2. Trees 251 one. Let T ′ be th
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4.2. Trees 253 are adjacent (they a
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4.2. Trees 255 Exercises 1. Which o
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4.2. Trees 257 a d b c f e 14. Give
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4.3. Planar Graphs 259 WARNING: you
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4.3. Planar Graphs 261 which says t
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4.3. Planar Graphs 263 sphere to li
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4.3. Planar Graphs 265 which will b
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4.4. Coloring 267 Investigate! 4.4
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4.4. Coloring 269 representations o
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4.4. Coloring 271 KQEJ KQEA KQEB P
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4.4. Coloring 273 We must color the
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4.4. Coloring 275 6. Prove the chro
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4.5. Euler Paths and Circuits 277 I
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4.5. Euler Paths and Circuits 279 W
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4.5. Euler Paths and Circuits 281 (
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4.6. Matching in Bipartite Graphs 2
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4.7. Chapter Summary 289 4.7 Chapte
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4.7. Chapter Summary 291 (b) For wh
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4.7. Chapter Summary 293 (d) Every
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Chapter 5 Additional Topics 5.1 Gen
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5.1. Generating Functions 297 shift
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5.1. Generating Functions 299 numbe
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5.1. Generating Functions 301 relat
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5.1. Generating Functions 303 and t
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5.1. Generating Functions 305 7. Us
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5.2. Introduction to Number Theory
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Appendix A Selected Hints 0.2 Exerc
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Selected Hints 327 1.6 Exercises 1.
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Selected Hints 329 2.5.27. For the
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Selected Hints 331 3.2.16. Your fri
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Selected Hints 333 4.4.10. The chro
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Appendix B Selected Solutions 0.2.1
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Selected Solutions 337 (d) Equivale
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Page 359 and 360: Selected Solutions 343 Proof. Let x
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Page 369 and 370: Selected Solutions 353 (p) Neither.
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Page 393 and 394: Selected Solutions 377 4.6 Exercise
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Page 399 and 400: Selected Solutions 383 5.2.6. (a) 3
Page 401 and 402: Appendix C List of Symbols Symbol D
Page 403 and 404: Index additive principle, 57 adjace
Page 405: Index 389 Goldbach conjecture, 227
Page 409 and 410: Index 393 set difference, 34 vertex
Page 411: Colophon This book was authored in
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