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

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! procedure insertion_sort( " a # i for j := 2 to n i := 1 while a j > a i i := i + 1 t := a j for k := 0 to j-i-1 a j-k := a j-k-1 a i := t ! { " a # i n $ % &i=1 n $ % &i=1 is in increasing order} : reals, n>1) This is not a very efficient sorting algorithm either. However, it is easy to see that at the j th step that the j th element is put into the correct spot. The worst case time is O(n 2 ). In fact, insertion_sort is trivially slower than bubble_sort. 47 Number theory is a rich field of mathematics. We will study four aspects briefly: 1. Integers and division 2. Primes and greatest common denominators 3. Integers and algorithms 4. Applications of number theory Most of the theorems quoted in this part of the textbook require knowledge of mathematical induction to rigorously prove, a topic covered in detail in the next chapter. ! 48
Definition: If a,b,Z and a-0, we say that a divides b if )c,Z(b=ac), denoted by a | b. When a divides b, we denote a as a factor of b and b as a multiple of a. When a does not divide b, we denote this as a/|b. Theorem: Let a,b,c,Z. Then 1. If a | b and a | c, then a | (b+c). 2. If a | b, then a | (bc). 3. If a | b and b | c, then a | c. Proof: Since a | b, )s,Z(b=as). 1. Since a | c it follows that ) t,Z(c=at). Hence, b+c = as + at = a(s+t). Therefore, a | (b+c). 2. bc = (as)c = a(sc). Therefore, a | (bc). 3. Since b | c it follows that ) t,Z(c=bt). c = bt = (as)t = a(st), Therefore, a | c. Corollary: Let a,b,c,Z. If a | b and b | c, then a | (mb+nc) for all m,n,Z. 49 Theorem (Division Algorithm): Let a,d,Z(d > 0). Then )!q,r,Z(a = dq+r). Definition: In the division algorithm, a is the dividend, d is the divisor, q is the quotient, and r is the remainder. We write q = a div d and r = a mod d. Examples: • Consider 101 divided by 9: 101 = 11$9 + 2. • Consider -11 divided by 3: -11 = 3(-4) + 1. Definition: Let a,b,m,Z(m > 0). Then a is congruent to b modulo m if m | (a-b), denoted a ' b (mod m). The set of integers congruent to an integer a modulo m is called the congruence class of a modulo m. Theorem: Let a,b,m,Z(m > 0). Then a ' b (mod m) if and only if a mod m = b mod m. 50
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: ! $ procedure binary_search(x, " a
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 and 76:
Definition: The extended binomial c
Page 77 and 78:
Example: a n = 8a n41 + 10 n41 with
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Relations Definition: A relation on
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Representation: The relation R from
Page 83 and 84:
! # 0 0 0 0 # • M R4 = # 1 0 0 0
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Definition: A partition of a set S
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Examples of Simple Graphs: • A co
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Examples: v 1 v 2 • and v 4 v 3 v
Page 91 and 92:
Theorem: Let G = (V,E) be a graph w
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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
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Note: Game trees are another highly
Page 101 and 102:
Definition (Postorder Traversal): L
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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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