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

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Theorem: If f i (x) is O(g i (x)), for 1;i;n, then n i=1 ! f i (x) is O(max{|g 1 (x)|, |g 2 (x)|, …, |g n (x)|}). Proof: Let g(x) = max{|g 1 (x)|, |g 2 (x)|, …, |g n (x)|} and C i the constants associated with O(g i (x)). Then n i=1 n i=1 n i=1 ! f i (x) ; ! C i g i (x) ; ! C i g(x) = |g(x)| ! C i = C|g(x)|. Theorem: If f i (x) is O(g i (x)), for 1;i;n, then ! f i (x) is O( ! g i (x) ). Proof: Let g(x) = |g 1 (x)|$|g 2 (x)|$…|g n (x)| and C i the constants associated with O(g i (x)). Then n i=1 n i=1 n i=1 ! f i (x) ; ! C i g i (x) ; C! g i (x) . n i=1 n i=1 n i=1 39 Definition: Let f and g be functions from either Z or R to R. Then f(x) is =(g(x)) if there are constants C and k such that |f(x)| < C|g(x)| whenever x>k. Definition: Let f and g be functions from either Z or R to R. Then f(x) is >(g(x)) if f(x) = O(g(x)) and f(x) = =(g(x)). In this case, we say that f(x) is of order g(x). Comment: f(x) = O(g(x)) notation is great in the limit, but does not always provide the right bounds for all values of x. =, denoted Big Omega, is used to provide lower bounds. >, denoted Big Theta, is used to provide both lower and upper bounds. n i=0 Example: f(x) = ! a i x i with a n -0 is of order x n . 40
Notation: Timing, as a function of the number of elements falls into the field of Complexity. Complexity Terminology >(1) Constant >(log(n)) Logarithmic >(n) Linear >(nlog(n)) nlog(n) >(n k ) Polynomial >(n k log(n)) Polylog >(k n ), where k>1 Exponential >(n!) Factorial Notation: Problems are tractable if they can be solved in polynomial time and are intractable otherwise. 41 Algorithms, Integers, and Matrices Definition: An algorithm is a finite set of precise instructions for solving a problem. Computational algorithms should have these properties: • Input: Values from a specified set. • Output: Results using the input from a specified set. • Definiteness: The steps in the algorithm are precise. • Correctness: The output produced from the input is the right solution. • Finiteness: The results are produced using a finite number of steps. • Effectiveness: Each step must be performable and in a finite amount of time. • Generality: The procedure should accept all input from the input set, not just special cases. 42
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: Some other common summations with c
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
Page 69 and 70: 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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