Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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Note: Generating functions can be used to solve many counting problems. Examples: • How many solutions are there to the constrained problem a+b = 9 for 3;a;5 and 4;b;6? There are 3 total. The number of solutions with the constraints is the coefficient of x 9 in (x 3 +x 4 +x 5 )(x 4 +x 5 +x 6 ). We choose x a and x b from the two factors, respectively, so that a+b = 9. By inspection, there are only 3 choices for a and b. • How many ways can 8 CPUs be distributed in 3 servers if each server gets 2-4 CPUs each? The generating function is f(x) = (x 2 +x 3 +x 4 ) 3 . We need the coefficient of x 8 in f(x). Expansion of f(x) gives us 6 ways. Note: Maple or Mathematica is really useful in the examples above. 151 Note: Generating functions are useful in solving recurrence relations, too. Example: a k = 3a k41 , k > 0 with a 0 = 2. Let f(x) = " a k x k be the generating " function for {a k }. Then xf(x) = # a k!1 x k=1 k . Using the recurrence relation directly, we have ! k=0 ! k=0 f(x) – 3xf(x) = ! a k x k " 3! a k"1 x k=1 k ! = a 0 + (a k ! 3a k!1 )x k = a 0 = 2 " Hence, f(x) 4 3xf(x) = (143x)f(x) = 2 or f(x) = 2 / (143x). Using the identity for (14ax) 41 , we see that ! k=0 k=1 f(x) = " 2!3 k x k or a k = 2!3 k . ! 152
Example: a n = 8a n41 + 10 n41 with a 0 = 1, which gives us a 1 = 9. Find a n in closed form. First multiply the recurrence relation by x n to give us a n x n + 8a n!1 x n + 10 n-1 x n ! . If f(x) = " a k x k , then f(x) 4 1 = ! ! ! k=1 ! k=1 k=0 a k x k = (8a k-1 x k +10 k-1 x k ) = 8xf(x) + x/(1410x) Hence, f(x) = 1!9x (1!8x)(1!10x) = 1 " 1 2 1!8x + 1!10x 1 $ $ ' # ! k=0 ! % ' ' & = 1 8 k +10 k 2 # &x k or a n = .5(8 k +10 k ). " $ % 153 Note: It is possible to prove many identities using generating functions. Exclusion-Inclusion Theorem: Given sets A i , 1;i;n, the number of elements in the union is n i=1 A i n i=1 " ! = ! A i ! 1!i
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Discrete Mathematics University of
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Definition: A proposition is a stat
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We can compound logical operators t
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Theorem: p" (q!r) ' (p"q)!(p"r). Pr
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Propositional logic is pretty limit
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Rules of Inference are used instead
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Sets, Functions, Sequences, and Sum
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Definition: n sets A i are disjoint
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Definition: The composition of n fu
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Some other common summations with c
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Notation: Timing, as a function of
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! $ procedure binary_search(x, " a
Page 25 and 26: Definition: If a,b,Z and a-0, we sa
Page 27 and 28: Theorem: There are infinitely many
Page 29 and 30: Algorithm: Addition of integers pro
Page 31 and 32: Euclidean Algorithm: Compute gcd(a,
Page 33 and 34: Sun Tzu’s Puzzle: The a k ,{2, 1,
Page 35 and 36: • Sometimes P(k) is for a (possib
Page 37 and 38: Example: Every postage amount < $.1
Page 39 and 40: Definition: A recursive algorithm s
Page 41 and 42: Merge sort is a balanced binary tre
Page 43 and 44: Examples: • Consider 3 students i
Page 45 and 46: Theorem: The number of r-combinatio
Page 47 and 48: If we allow repetitions in the perm
Page 49 and 50: Algorithm: Generating the next r-co
Page 51 and 52: Theorem: Let E and F be events in a
Page 53 and 54: Note: For bit strings of length 4,
Page 55 and 56: Theorem: If X i , 1;i;n, are random
Page 57 and 58: Definition: The random variables X
Page 59 and 60: Example: We have 2 boxes. The first
Page 61 and 62: Let E be the event that an incoming
Page 63 and 64: Advanced Counting Principles Defini
Page 65 and 66: Examples: Typical ones include •
Page 67 and 68: Now comes the second case for n = 2
Page 69 and 70: Theorem: Assume {b i },{c i },R. Su
Page 71 and 72: Example (funny integer multiplicati
Page 73 and 74: • In step 4, we have to take care
Page 75: Definition: The extended binomial c
Page 79 and 80: Relations Definition: A relation on
Page 81 and 82: Representation: The relation R from
Page 83 and 84: ! # 0 0 0 0 # • M R4 = # 1 0 0 0
Page 85 and 86: Definition: A partition of a set S
Page 87 and 88: Examples of Simple Graphs: • A co
Page 89 and 90: Examples: v 1 v 2 • and v 4 v 3 v
Page 91 and 92: Theorem: Let G = (V,E) be a graph w
Page 93 and 94: Definition: A coloring of a simple
Page 95 and 96: Definition: A m-ary tree is a roote
Page 97 and 98: Definition: A decision tree is a ro
Page 99 and 100: Note: Game trees are another highly
Page 101 and 102: Definition (Postorder Traversal): L
Page 103 and 104: Definition: Let G = (V,E) be a simp
Page 105 and 106: Definition: A minimum spanning tree
Page 107 and 108: Examples: Most circuits are of the
Page 109: Definition: An implicant is sum ter
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Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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