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# lecturenotes - Discrete Mathematics Lecture Notes...

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Discrete Mathematics Lecture Notes Incomplete Preliminary Version Instructor: L´ aszl´o Babai Last revision: June 22, 2003 Last update: October 24, 2003

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Contents 1 Logic 1 1.1 Quantiﬁer notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Asymptotic Notation 5 2.1 Limit of sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Asymptotic Equality and Inequality . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Little-oh notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Big-Oh, Omega, Theta notation ( O , Ω, Θ) . . . . . . . . . . . . . . . . . . . . . 10 2.5 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Convex Functions and Jensen’s Inequality 15 4 Basic Number Theory 19 4.1 Introductory Problems: g.c.d., congruences, multiplicative inverse, Chinese Re- mainder Theorem, Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . 19 4.2 Gcd, congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.6 Lattices and diophantine approximation . . . . . . . . . . . . . . . . . . . . . . 35 iii

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iv CONTENTS 5 Counting 37 5.1 Binomial coefcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Recurrences, generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Graphs and Digraphs 43 6.1 Graph Theory Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Digraph Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7 Finite Probability Spaces 63 7.1 ±inite Probability Spaces and Events . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Random Variables and Expected Value . . . . . . . . . . . . . . . . . . . . . . . 68 7.3 Standard deviation and Chebyshev’s Inequality . . . . . . . . . . . . . . . . . . 70 7.4 Independence oF random variables . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.5 Cherno²’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8 Finite Markov Chains 79 8.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Last update: October 24, 2003
List of Figures 3.1 Deﬁnition of convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.1 The complete graph K 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2 The complete bipartite graph K 3 , 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.3 P 5 , the path of length 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4 C 5 , the cycle of length 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.5 The trees on 6 vertices (complete list). . . . . . . . . . . . . . . . . . . . . . . . 47 6.6 The 4 × 10 grid, with a shortest path between opposite corners highlighted. . . 49 6.7 Graph of knight moves on a 4 × 4 chessboard . . . . . . . . . . . . . . . . . . . 51 6.8 The Petersen graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.9 Is this graph isomorphic to Petersen’s? . . . . . . . . . . . . . . . . . . . . . . . 52 6.10 K 4 drawn two diFerent ways. Only one is a plane graph. . . . . . . . . . . . . . 53 6.11 The numbers indicate the number of sides of each region of this plane graph. . 54 8.1 NEED CAPTION! AND RE±. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 A graph with transition probabilities. ±IX THIS! . . . . . . . . . . . . . . . . . 84 8.3 Transition graph for a Markov chain. . . . . . . . . . . . . . . . . . . . . . . . . 89 8.4 The transition graph for a Markov chain. . . . . . . . . . . . . . . . . . . . . . 90 v

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vi LIST OF FIGURES Last update: October 24, 2003
Chapter 1 Logic 1.1 Quantiﬁer notation Quantiﬁer notation: - “universal quantiﬁer,” - “existential quatiﬁer.” ( x ) is read as “for all x ( x ) is read as “there exists x such that ( x, statement( x )) is read as “for all x such that statement( x ) holds, ... Example. ( x 6 = 0)( y )( xy = 1) says that every x other than zero has a multiplicative inverse. The validity of this statement depends on the universe over which the variables range. The statement holds (is true) over R (real numbers) and Q (rational numbers) but does not hold over Z (integers) or N (nonnegative integers). It holds over Z m (the set of residue classes modulo m ) if m is prime but not if m is composite. (Why?) 1.2 Problems Several of the problems below will refer to the divisibility relation between integers.

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## This note was uploaded on 10/10/2009 for the course CMSC 37701 taught by Professor Xu during the Fall '09 term at UChicago.

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lecturenotes - Discrete Mathematics Lecture Notes...

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