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66 Counting we can choose any one of the six letters A,B,C,D,F or G). Thus there are 6 · 6 · 6 · 6 = 6 4 = 1296 lists that do not have an E. Therefore the final answer to our question is that there are 2401 − 1296 = 1105 lists with repetition allowed that contain at least one E. Perhaps you wondered if Part (d) of Example 3.2 could be solved with a setup similar to that of Part (c). Let’s try doing it that way. We want to count the length-4 lists (with repetition allowed) that contain at least one E. The following diagram is adapted from Part (c), the only difference being that there are now seven choices in each slot because we are allowed to repeat any of the seven letters. Type 1 Type 2 Type 3 Type 4 ( E , , , ) ( , E , , ) ( , , E , ) ( , , , E ) 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices 7 choices This gives a total of 7 3 + 7 3 + 7 3 + 7 3 = 1372 lists, an answer that is substantially larger than the (correct) value of 1105 that we got in our solution to Part (d) above. It is not hard to see what went wrong. The list (E, E, A,B) is of type 1 and type 2, so it got counted twice. Similarly (E, E,C, E) is of type 1, 3 and 4, so it got counted three times. In fact, you can find many similar lists that were counted multiple times. In solving counting problems, we must always be careful to avoid this kind of double-counting or triple-counting, or worse. Exercises for Section 3.1 Note: A calculator may be helpful for some of the exercises in this chapter. This is the only chapter for which a calculator may be helpful. (As for the exercises in the other chapters, a calculator makes them harder.) 1. Consider lists made from the letters T,H,E,O,R,Y, with repetition allowed. (a) How many length-4 lists are there? (b) How many length-4 lists are there that begin with T? (c) How many length-4 lists are there that do not begin with T? 2. Airports are identified with 3-letter codes. For example, the Richmond, Virginia airport has the code RIC, and Portland, Oregon has PDX. How many different 3-letter codes are possible? 3. How many lists of length 3 can be made from the symbols A,B,C,D,E,F if...
Counting Lists 67 (a) ... repetition is allowed. (b) ... repetition is not allowed. (c) ... repetition is not allowed and the list must contain the letter A. (d) ... repetition is allowed and the list must contain the letter A. 4. Five cards are dealt off of a standard 52-card deck and lined up in a row. How many such line-ups are there in which all 5 cards are of the same suit? 5. Five cards are dealt off of a standard 52-card deck and lined up in a row. How many such line-ups are there in which all 5 cards are of the same color (i.e., all black or all red)? 6. Five cards are dealt off of a standard 52-card deck and lined up in a row. How many such line-ups are there in which exactly one of the 5 cards is a queen? 7. This problems involves 8-digit binary strings such as 10011011 or 00001010 (i.e., 8-digit numbers composed of 0’s and 1’s). (a) How many such strings are there? (b) How many such strings end in 0? (c) How many such strings have the property that their second and fourth digits are 1’s? (d) How many such strings have the property that their second or fourth digits are 1’s? 8. This problem concerns lists made from the symbols A,B,C,D,E. (a) How many such length-5 lists have at least one letter repeated? (b) How many such length-6 lists have at least one letter repeated? 9. This problem concerns 4-letter codes made from the letters A,B,C,D,...,Z. (a) How many such codes can be made? (b) How many such codes have no two consecutive letters the same? 10. This problem concerns lists made from the letters A,B,C,D,E,F,G,H,I,J. (a) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin with a vowel? (b) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin and end with a vowel? (c) How many length-5 lists can be made from these letters if repetition is not allowed and the list must contain exactly one A? 11. This problem concerns lists of length 6 made from the letters A,B,C,D,E,F,G,H. How many such lists are possible if repetition is not allowed and the list contains two consecutive vowels? 12. Consider the lists of length six made with the symbols P, R, O, F, S, where repetition is allowed. (For example, the following is such a list: (P,R,O,O,F,S).) How many such lists can be made if the list must end in an S and the symbol O is used more than once?
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Book of Proof Richard Hammack Virgi
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To my students
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v II How to Prove Conditional State
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Preface In writing this book I have
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ix The book is organized into four
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Part I Fundamentals
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4 Sets Some sets are so significant
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6 Sets These last three examples hi
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8 Sets 1.2 The Cartesian Product Gi
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10 Sets We can also take Cartesian
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12 Sets This idea of “making” a
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14 Sets This is a subset C ⊆ R 2
Page 26 and 27: 16 Sets y y y x x x (a) (b) (c) Fig
Page 28 and 29: 18 Sets B A ∪ B A ∩ B A − B A
Page 30 and 31: 20 Sets Example 1.7 If P is the set
Page 32 and 33: 22 Sets We can also think of A ∩
Page 34 and 35: 24 Sets 1.8 Indexed Sets When a mat
Page 36 and 37: 26 Sets Example 1.11 Here the sets
Page 38 and 39: 28 Sets 1.9 Sets that Are Number Sy
Page 40 and 41: 30 Sets Russell’s paradox arises
Page 42 and 43: 32 Logic In proving theorems, we ap
Page 44 and 45: 34 Logic R(f , g) : The function f
Page 46 and 47: 36 Logic 2.2 And, Or, Not The word
Page 48 and 49: 38 Logic To conclude this section,
Page 50 and 51: 40 Logic You can think of P ⇒ Q a
Page 52 and 53: 42 Logic that it’s impossible tha
Page 54 and 55: 44 Logic P if and only if Q. P is a
Page 56 and 57: 46 Logic Notice that when we plug i
Page 58 and 59: 48 Logic There are two pairs of log
Page 60 and 61: 50 Logic Likewise, a statement such
Page 62 and 63: 52 Logic 2.8 More on Conditional St
Page 64 and 65: 54 Logic Example 2.9 Consider Goldb
Page 66 and 67: 56 Logic Example 2.10 Consider nega
Page 68 and 69: 58 Logic Example 2.13 Negate the fo
Page 70 and 71: 60 Logic 2.12 An Important Note It
Page 72 and 73: 62 Counting Occasionally we may get
Page 74 and 75: 64 Counting To answer this question
Page 78 and 79: 68 Counting 3.2 Factorials In worki
Page 80 and 81: 70 Counting We close this section w
Page 82 and 83: 72 Counting Definition 3.2 If n and
Page 84 and 85: 74 Counting Fact 3.3 ( ( ) n n! n I
Page 86 and 87: 76 Counting 3.4 Pascal’s Triangle
Page 88 and 89: 78 Counting coefficients 1 2 1. Sim
Page 90 and 91: 80 Counting We seek the number of 3
Page 92 and 93: 82 Counting 5. How many 7-digit bin
Page 95 and 96: CHAPTER 4 Direct Proof It is time t
Page 97 and 98: Definitions 87 4.2 Definitions A pr
Page 99 and 100: Definitions 89 it necessary to defi
Page 101 and 102: Direct Proof 91 So the setup for di
Page 103 and 104: Direct Proof 93 Here is another exa
Page 105 and 106: Direct Proof 95 This proposition te
Page 107 and 108: Treating Similar Cases 97 Propositi
Page 109 and 110: Treating Similar Cases 99 19. Suppo
Page 111 and 112: Contrapositive Proof 101 Since P
Page 113 and 114: Congruence of Integers 103 Proposit
Page 115 and 116: Mathematical Writing 105 Propositio
Page 117 and 118: Mathematical Writing 107 Since a |
Page 119 and 120: CHAPTER 6 Proof by Contradiction We
Page 121 and 122: Proving Statements with Contradicti
Page 123 and 124: Proving Conditional Statements by C
Page 125 and 126: Some Words of Advice 115 for intege
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Part III More on Proof
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120 Proving Non-Conditional Stateme
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130 Proofs Involving Sets Generally
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132 Proofs Involving Sets In practi
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134 Proofs Involving Sets Example 8
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136 Proofs Involving Sets Thus, sin
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138 Proofs Involving Sets Though it
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140 Proofs Involving Sets If we add
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142 Proofs Involving Sets so that d
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CHAPTER 9 Disproof E ver since Chap
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146 Disproof deciding whether a sta
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148 Disproof Example 9.2 Conjecture
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150 Disproof 9.3 Disproof by Contra
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CHAPTER 10 Mathematical Induction T
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154 Mathematical Induction This pic
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156 Mathematical Induction To round
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158 Mathematical Induction Proposit
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160 Mathematical Induction Strong i
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164 Mathematical Induction We next
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168 Mathematical Induction 13. For
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Part IV Relations, Functions and Ca
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174 Relations The set R encodes the
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176 Relations Exercises for Section
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178 Relations Example 11.7 Here A =
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180 Relations Example 11.8 Prove th
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182 Relations 11.2 Equivalence Rela
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184 Relations We close this section
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186 Relations 11.3 Equivalence Clas
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188 Relations Notationally, the uni
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190 Relations In a similar vein, [2
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192 Relations Exercises for Section
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CHAPTER 12 Functions You know from
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196 Functions value a to the output
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198 Functions To answer this, first
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200 Functions For more concrete exa
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202 Functions To see that g is surj
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204 Functions Although the underlyi
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206 Functions 12.4 Composition You
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208 Functions Proof. First suppose
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210 Functions A B a 1 a 1 b 2 b 2 c
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212 Functions We can check our work
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214 Functions If you continue your
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216 Cardinality of Sets On the othe
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218 Cardinality of Sets There is a
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220 Cardinality of Sets And saying
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222 Cardinality of Sets Here is a s
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224 Cardinality of Sets Beginning a
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226 Cardinality of Sets Exercises f
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228 Cardinality of Sets f : A → P
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230 Cardinality of Sets 13.4 The Ca
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232 Cardinality of Sets and the whi
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234 Cardinality of Sets Here are so
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236 Cardinality of Sets On the face
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Solutions Chapter 1 Exercises Secti
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240 Solutions Sketch the following
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242 Solutions 5. Sketch the sets X
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244 Solutions Section 1.8 1. Suppos
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246 Solutions Section 2.4 Without c
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248 Solutions 7. P ⇒ Q = (P∧
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250 Solutions Section 2.10 Negate t
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252 Solutions (b) How many such cod
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254 Solutions 5. How many 7-digit b
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256 Solutions 15. If n ∈ Z, then
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258 Solutions 7. Proposition Suppos
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260 Solutions 27. If a ≡ 0 (mod 4
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262 Solutions 7. If a, b ∈ Z, the
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264 Solutions This expresses a 3 +
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266 Solutions 19. If n ∈ N, then
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268 Solutions Chapter 8 Exercises 1
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270 Solutions 19. Prove that {9 n :
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272 Solutions 3. If n ∈ Z and n 5
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274 Solutions 25. For all a, b, c
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276 Solutions 5. If n ∈ N, then 2
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278 Solutions (2) Now assume the st
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280 Solutions 23. Use induction to
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282 Solutions Thus, with n + 1 line
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284 Solutions a b a b a b a b a b a
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286 Solutions Therefore 3x − 5z i
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288 Solutions Chapter 12 Exercises
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290 Solutions 13. Consider the func
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292 Solutions Section 12.5 Exercise
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294 Solutions 13. Let f : A → B b
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296 Solutions 11. Partition N into
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298 Solutions 7. Prove or disprove:
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300 Index notation, 197 one-to-one,
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