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

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Definitions: The floor and ceiling functions are defined by Examples: • 7x8 = largest integer smaller or equal to x. • 9x: = smallest integer larger or equal to x. • 72.998 = 2, 92.99: = 3 • 7-2.998 = -3, 9-2.99: = -2 Definition: A sequence is a function from either N or a subset of N to a set A whose elements a i are the terms of the sequence. Definitions: A geometric progression is a sequence of the form {ar i , i=0, 1, …}. An arithmetic progression is a sequence of the form {a+id, i=0, 1,…}. Translation: f(a,r,i) = ar i and f(a,d,i) = a + id are the corresponding functions. 35 There are a number of interesting summations that have closed form solutions. Theorem: If a,r,R, then "(n+1)a, if r=1, n $ ! ar i = i=0 # ar n+1 -a $ % r-1 , otherwise. Proof: If r = 1, then we are left summing a n+1 times. Hence, the r = 1 case is n trivial. Suppose r - 1. Let S = ! ar i . Then n i=0 i=0 rS = r! ar i Substitution S formula. n+1 ! ar i Simplifying. " # $ ! i=1 n i=0 ar i S+(ar n+1 -a) ( ) Removing n+1 term and adding 0 term. % & ' + arn+1 -a Substituting S for formula Solve for S in rS = S+(ar n+1 -a) to get the desired formula. qed 36
Some other common summations with closed form solutions are Sum Closed Form Solution n i n(n+1) ! i=1 2 n ! i i=1 2 n(n+1)(2n+1) 6 n ! i i=1 3 n 2 (n+1) 2 4 ! ! x i=0 i , |x|
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: Definition: The composition of n fu
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 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
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Theorem: Assume {b i },{c i },R. Su
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Example (funny integer multiplicati
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• In step 4, we have to take care
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Definition: The extended binomial c
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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
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! # 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
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Theorem: Let G = (V,E) be a graph w
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Definition: A coloring of a simple
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Definition: A m-ary tree is a roote
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Definition: A decision tree is a ro
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Note: Game trees are another highly
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Definition (Postorder Traversal): L
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Definition: Let G = (V,E) be a simp
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Definition: A minimum spanning tree
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Examples: Most circuits are of the
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Definition: An implicant is sum ter
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Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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