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

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264 4. Graph Theory of edges is also kv/2. Putting this together gives which says e 3 f 2 k k(2 + f /2) , 2 6 f 4 + f . Both k and f must be positive integers. Note that 6 f 4+ f is an increasing function for positive f , bounded above by a horizontal asymptote at k 6. Thus the only possible values for k are 3, 4, and 5. Each of these are possible. To get k 3, we need f 4 (this is the tetrahedron). For k 4 we take f 8 (the octahedron). For k 5 take f 20 (the icosahedron). Thus there are exactly three regular polyhedra with triangles for faces. Case 2: Each face is a square. Now we have e 4 f /2 2 f . Using Euler’s formula we get v 2 + f , and counting edges using the degree k of each vertex gives us e 2 f k(2 + f ) . 2 Solving for k gives k 4 f 2 + f 8 f 4 + 2 f . This is again an increasing function, but this time the horizontal asymptote is at k 4, so the only possible value that k could take is 3. This produces 6 faces, and we have a cube. There is only one regular polyhedron with square faces. Case 3: Each face is a pentagon. We perform the same calculation as above, this time getting e 5 f /2 so v 2 + 3 f /2. Then so e 5 f 2 k k(2 + 3 f /2) , 2 10 f 4 + 3 f . Now the horizontal asymptote is at 10 3 . This is less than 4, so we can only hope of making k 3. We can do so by using 12 pentagons, getting the dodecahedron. This is the only regular polyhedron with pentagons as faces. Case 4: Each face is an n-gon with n ≥ 6. Following the same procedure as above, we deduce that k 2n f 4 + (n − 2) f ,
4.3. Planar Graphs 265 which will be increasing to a horizontal asymptote of 2n n−2 . When n 6, this asymptote is at k 3. <strong>An</strong>y larger value of n will give an even smaller asymptote. Therefore no regular polyhedra exist with faces larger than pentagons.8 qed Exercises 1. Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain. 2. The graph G has 6 vertices with degrees 2, 2, 3, 4, 4, 5. How many edges does G have? Could G be planar? If so, how many faces would it have. If not, explain. 3. Is it possible for a connected graph with 7 vertices and 10 edges to be drawn so that no edges cross and create 4 faces? Explain. 4. Is it possible for a graph with 10 vertices and edges to be a connected planar graph? Explain. 5. Is there a connected planar graph with an odd number of faces where every vertex has degree 6? Prove your answer. 6. I’m thinking of a polyhedron containing 12 faces. Seven are triangles and four are quadralaterals. The polyhedron has 11 vertices including those around the mystery face. How many sides does the last face have? 7. Consider some classic polyhedrons. (a) <strong>An</strong> octahedron is a regular polyhedron made up of 8 equilateral triangles (it sort of looks like two pyramids with their bases glued together). Draw a planar graph representation of an octahedron. How many vertices, edges and faces does an octahedron (and your graph) have? (b) The traditional design of a soccer ball is in fact a (spherical projection of a) truncated icosahedron. This consists of 12 regular pentagons and 20 regular hexagons. No two pentagons are adjacent (so the edges of each pentagon are shared only by hexagons). How many vertices, edges, and faces does a truncated icosahedron have? Explain how you arrived at your answers. Bonus: draw the planar graph representation of the truncated icosahedron. 8Notice that you can tile the plane with hexagons. This is an infinite planar graph; each vertex has degree 3. These infinitely many hexagons correspond to the limit as f → ∞ to make k 3.
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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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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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Selected Solutions 339 (b) We get t
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Selected Solutions 341 (d) Such an
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Selected Solutions 343 Proof. Let x
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Selected Solutions 345 (c) 2 6 −
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Selected Solutions 347 (c) To build
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Selected Solutions 349 the box. The
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Selected Solutions 351 (c) ( 16) [(
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Selected Solutions 353 (p) Neither.
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Selected Solutions 355 x + y n. So
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Selected Solutions 357 2.1.4. (a) T
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Selected Solutions 359 (b) 6n + 1,
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Selected Solutions 361 2.4.12. We h
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Selected Solutions 363 F 2k+3 −
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Selected Solutions 365 2.6.7. (a) 4
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Selected Solutions 367 3.1.16. (a)
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Selected Solutions 369 P Q R P →
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Selected Solutions 371 (b) The conv
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Selected Solutions 373 (c) Not poss
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Selected Solutions 375 number of fa
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Selected Solutions 377 4.6 Exercise
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Selected Solutions 379 (b) Now addi
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Selected Solutions 381 4.7.21. (a)
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Selected Solutions 383 5.2.6. (a) 3
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Appendix C List of Symbols Symbol D
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Index additive principle, 57 adjace
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Index 389 Goldbach conjecture, 227
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Index 391 vs combination, 84, 123,
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Index 393 set difference, 34 vertex
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Colophon This book was authored in
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