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

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Definition: Let R be an equivalence relation on a set A. The set of all elements that are related to an element a,A is called the equivalence class of a and is denoted by [a] R . When R is obvious, it is just [a]. If b,[a] R , b is called a representative of this equivalence class. Example: Let A = Z. Define aRb if and only if either a = b or a = 4b. There are two cases for the equivalence class: • [0] = {0} • [a] = {a, 4a} if a-0. 167 Theorem: Let R be an equivalence relation on a set A. For a,b,A, the following are equivalent: 1. aRb 2. [a] = [b] 3. [a] 3 [b] - /. Proof: 1 E 2 E 3 E 1. • 1 E 2: Assume aRb. Suppose c,[a]. Then aRc. Due to symmetry, we know that bRa. Knowing that bRa and aRc, by transitivity, bRc. Hence, c,[b]. A similar argument shows that if c,[b], then c,[a]. Hence, [a] = [b]. • Assume that [a] = [b]. Since a,A and R is reflexive, [a] 3 [b] - /. • Assume [a] 3 [b] - /. So there is a c,[a] and c,[b], too. So, aRc and bRc. By symmetry, cRb. By transitivity, aRc and cRb, so aRb. Lemma: For any equivalence relation R on a set A, [a] R =A a!A ! . Proof: For all a,A, a,[a] R . 168
Definition: A partition of a set S is a collection of disjoint sets whose union is A. Theorem: Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition {A i | i,I} of the set S, there is an equivalence relation R that has the sets A i , i,I, as its equivalence classes. 169 Graphs Definition: A graph G = (V,E) consists of a nonempty set of vertices V and a set of edges E. Each edge has either one or two vertices as endpoints. An edge connects its endpoints. Note: We will only study finite graphs (|V| < .). Categorizations: • A simple graph has edges that connects two different vertices and no two edges connect the same vertex. • A multigraph has multiple edges connecting the same vertices. • A loop is a set of edges from a vertex back to itself. • A pseudograph is a graph in which the edges do not have a direction associated with them. • An undirected graph is a graph in which the edges do not have direction. • A mixed graph has both directed and undirected edges. 170
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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
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Definition: If a,b,Z and a-0, we sa
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Theorem: There are infinitely many
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Algorithm: Addition of integers pro
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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
Page 79 and 80: Relations Definition: A relation on
Page 81 and 82: Representation: The relation R from
Page 83: ! # 0 0 0 0 # • M R4 = # 1 0 0 0
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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