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

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Example: Infinite ladder with reachable rungs. For mathematical or strong induction, we need to verify the following: Step Mathematical Strong Basis We can reach the first rung. Induction If we can reach an arbitrary (k,N, if we can reach all k rung k, then we can reach rungs, then we can reach rung k+1. rung k+1. We cannot prove that you can climb an infinite ladder using mathematical induction. Using strong induction, however, you can prove this result using a trick: since you can prove that you can climb to rungs 1, 2, …, k, it follows that you can climb 2 rungs arbitrarily, which gets you from rung k-1 to rung k+1. Rule of thumb: Always use mathematical induction if P(k)%P(k+1) % (k,N. Only resort to strong induction when that fails. 71 Fundamental Theorem of Arithmetic: Every n,N (n>1) is the product of primes. Proof: Let P(n) be the proposition that n can be written as the product of primes. (Basis) P(2) is true: 2 = 2, the product of 1 prime. (Induction) Assume P(j) is true (j;k. We must verify that P(k+1) is true. Case 1: k+1 is a prime. Hence, P(k+1) is true. Case 2: k+1 is a composite. Hence k+1 = a•b, where 2;a;b
Example: Every postage amount < $.12 can be formed using $.04 and $.05 stamp combinations only. We can prove this using modified strong induction. (Basis) Consider 4 specific cases: Postage Number of $.04’s Number of $.05’s $.12 3 0 $.13 2 1 $.14 1 2 $.15 0 3 Hence, P(j) is true for 12;j;15. (Induction) Assume P(j) is true for 12;j;k and k2. • h(0) = 1, h(n) = nh(n-1) = n! • Fibonacci numbers: f 0 = 0, f 1 = 1, f n = f n-1 + f n-2 , n>1. n 0 1 2 3 4 f(n) 1 6 16 36 76 g(n) 12 1 -10 -21 -32 h(n) 1 1 2 6 24 f n 0 1 1 2 3 74
Page 1 and 2: Discrete Mathematics University of
Page 3 and 4: Definition: A proposition is a stat
Page 5 and 6: We can compound logical operators t
Page 7 and 8: Theorem: p" (q!r) ' (p"q)!(p"r). Pr
Page 9 and 10: Propositional logic is pretty limit
Page 11 and 12: Rules of Inference are used instead
Page 13 and 14: Sets, Functions, Sequences, and Sum
Page 15 and 16: Definition: n sets A i are disjoint
Page 17 and 18: Definition: The composition of n fu
Page 19 and 20: Some other common summations with c
Page 21 and 22: Notation: Timing, as a function of
Page 23 and 24: ! $ 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: • Sometimes P(k) is for a (possib
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 and 76: Definition: The extended binomial c
Page 77 and 78: Example: a n = 8a n41 + 10 n41 with
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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