“mcs” — 2013/1/10 — 0:28 — page i — #1 Mathematics for Computer Science revised Thursday 10th January, 2013, 00:28 Eric Lehman Google Inc F Thomson Leighton Department of Mathematics and the Computer Science and AI Laboratory, Massachussetts Institute of Technology; Akamai Technologies Albert R Meyer Department of Electrical Engineering and Computer Science and the Computer Science and AI Laboratory, Massachussetts Institute of Technology Creative Commons 2011, Eric Lehman, F Tom Leighton, Albert R Meyer “mcs” — 2013/1/10 — 0:28 — page ii — #2 “mcs” — 2013/1/10 — 0:28 — page iii — #3 Contents I Proofs Introduction What is a Proof? 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 The Well Ordering Principle 25 2.1 2.2 2.3 2.4 26 Propositions from Propositions 40 Propositional Logic in Computer Programs Equivalence and Validity 46 The Algebra of Propositions 48 The SAT Problem 53 Predicate Formulas 54 Mathematical Data Types 75 4.1 4.2 4.3 4.4 4.5 Well Ordering Proofs 25 Template for Well Ordering Proofs Factoring into Primes 28 Well Ordered Sets 29 Logical Formulas 39 3.1 3.2 3.3 3.4 3.5 3.6 Propositions Predicates The Axiomatic Method Our Axioms Proving an Implication 11 Proving an “If and Only If” 13 Proof by Cases 15 Proof by Contradiction 16 Good Proofs in Practice 17 References 19 Sets 75 Sequences 79 Functions 79 Binary Relations Finite Cardinality 82 86 Induction 101 5.1 Ordinary Induction 101 43 “mcs” — 2013/1/10 — 0:28 — page iv — #4 iv Contents 5.2 5.3 5.4 Recursive Definitions and Structural Induction 153 Strings of Matched Brackets 157 Recursive Functions on Nonnegative Integers 160 Arithmetic Expressions 163 Induction in Computer Science 168 Infinite Sets 181 7.1 7.2 7.3 7.4 Infinite Cardinality 182 The Halting Problem 187 The Logic of Sets 191 Does All This Really Work? 194 II Structures Introduction 207 Number Theory 209 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 115 Recursive Data Types 153 6.1 6.2 6.3 6.4 6.5 Strong Induction 110 Strong Induction vs Induction vs Well Ordering State Machines 116 Divisibility 209 The Greatest Common Divisor 214 Prime Mysteries 220 The Fundamental Theorem of Arithmetic 223 Alan Turing 225 Modular Arithmetic 229 Remainder Arithmetic 231 Turing’s Code (Version 2.0) 234 Multiplicative Inverses and Cancelling 236 Euler’s Theorem 240 RSA Public Key Encryption 247 What has SAT got to with it? 250 References 250 Directed graphs & Partial Orders 277 9.1 9.2 9.3 9.4 Digraphs & Vertex Degrees 279 Adjacency Matrices 283 Walk Relations 286 Directed Acyclic Graphs & Partial Orders 287 “mcs” — 2013/1/10 — 0:28 — page v — #5 v Contents 9.5 9.6 9.7 9.8 9.9 9.10 9.11 Weak Partial Orders 290 Representing Partial Orders by Set Containment Path-Total Orders 293 Product Orders 294 Scheduling 295 Equivalence Relations 301 Summary of Relational Properties 303 292 10 Communication Networks 329 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 Complete Binary Tree 329 Routing Problems 329 Network Diameter 330 Switch Count 331 Network Latency 332 Congestion 332 2-D Array 333 Butterfly 335 Beneˇ Network 337 s 11 Simple Graphs 349 11.1 Vertex Adjacency and Degrees 349 11.2 Sexual Demographics in America 351 11.3 Some Common Graphs 353 11.4 Isomorphism 355 11.5 Bipartite Graphs & Matchings 357 11.6 The Stable Marriage Problem 362 11.7 Coloring 369 11.8 Simple Walks 373 11.9 Connectivity 375 11.10 Odd Cycles and 2-Colorability 378 11.11 Forests & Trees 380 11.12 References 388 12 Planar Graphs 417 12.1 12.2 12.3 12.4 12.5 12.6 12.7 Drawing Graphs in the Plane 417 Definitions of Planar Graphs 417 Euler’s Formula 428 Bounding the Number of Edges in a Planar Graph Returning to K5 and K3;3 430 Coloring Planar Graphs 431 Classifying Polyhedra 433 429 “mcs” — 2013/1/10 — 0:28 — page vi — #6 vi Contents 12.8 Another Characterization for Planar Graphs 436 III Counting Introduction 445 13 Sums and Asymptotics 447 13.1 13.2 13.3 13.4 13.5 13.6 13.7 The Value of an Annuity 448 Sums of Powers 454 Approximating Sums 456 Hanging Out Over the Edge 460 Products 467 Double Trouble 469 Asymptotic Notation 472 14 Cardinality Rules 491 14.1 Counting One Thing by Counting Another 14.2 Counting Sequences 492 14.3 The Generalized Product Rule 495 14.4 The Division Rule 499 14.5 Counting Subsets 502 14.6 Sequences with Repetitions 504 14.7 Counting Practice: Poker Hands 507 14.8 The Pigeonhole Principle 512 14.9 Inclusion-Exclusion 521 14.10 Combinatorial Proofs 527 14.11 References 531 15 Generating Functions 563 15.1 15.2 15.3 15.4 15.5 15.6 Infinite Series 563 Counting with Generating Functions Partial Fractions 570 Solving Linear Recurrences 573 Formal Power Series 578 References 582 IV Probability Introduction 597 16 Events and Probability Spaces 599 564 491 “mcs” — 2013/1/10 — 0:28 — page vii — #7 vii Contents 16.1 16.2 16.3 16.4 16.5 Let’s Make a Deal 599 The Four Step Method 600 Strange Dice 609 The Birthday Principle 617 Set Theory and Probability 619 17 Conditional Probability 629 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 Monty Hall Confusion 629 Definition and Notation 630 The Four-Step Method for Conditional Probability Why Tree Diagrams Work 634 The Law of Total Probability 641 Simpson’s Paradox 642 Independence 645 Mutual Independence 646 18 Random Variables 669 18.1 18.2 18.3 18.4 18.5 Random Variable Examples 669 Independence 671 Distribution Functions 672 Great Expectations 680 Linearity of Expectation 692 19 Deviation from the Mean 717 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 Why the Mean? 717 Markov’s Theorem 718 Chebyshev’s Theorem 720 Properties of Variance 724 Estimation by Random Sampling 729 Confidence versus Probability 734 Sums of Random Variables 735 Really Great Expectations 745 20 Random Walks 767 20.1 Gambler’s Ruin 767 20.2 Random Walks on Graphs V Recurrences Introduction 793 21 Recurrences 795 777 632 “mcs” — 2013/1/10 — 0:28 — page viii — #8 viii Contents 21.1 21.2 21.3 21.4 21.5 The Towers of Hanoi 795 Merge Sort 798 Linear Recurrences 802 Divide-and-Conquer Recurrences A Feel for Recurrences 816 Bibliography 823 Glossary of Symbols 827 Index 830 809 “mcs” — 2013/1/10 — 0:28 — page — #9 I Proofs “mcs” — 2013/1/10 — 0:28 — page — #10 “mcs” — 2013/1/10 — 0:28 — page 826 — #834 “mcs” — 2013/1/10 — 0:28 — page 827 — #835 Glossary of Symbols symbol WWD Ô ^ _ ! ! :P; P ! ˚  6 [ S i 2I Si \ T i 2I Si ; A pow.A/ A B Sn Z N; Z ZC ; NC Z Q R C brc dre meaning is defined to be end of proof symbol not equal and, AND or, OR implies, if , then , IMPLIES state transition not P , NOT.p/ iff, equivalent, IFF xor, exclusive-or, XOR exists for all is a member of, is in is a (possibly =) subset of is not a (possibly =) subset of is a proper (not =) subset of is not a proper (not =) subset of set union union of sets Si where i ranges over set I of indices set intersection intersection of sets Si where i ranges over set I of indices the empty set, f g complement of set A set difference powerset of set, A Cartesian product of sets A and B Cartesian product of n copies of set S integers nonnegative integers positive integers negative integers rational numbers real numbers complex numbers the floor of r: the greatest integer Ä r the ceiling of r: the least integer r “mcs” — 2013/1/10 — 0:28 — page 828 — #836 828 Glossary of Symbols symbol R.X / R R X / surj inj bij ŒÄ in Œ in ŒÄ out Œ out ŒD out; D in A A! rev.s/ s t #c s/ mjn gcd lcm k; n/ Œk; n/ k; n Œk; n P Q i 2I ri i 2I ri qcnt.n; d / rem n; d / Á mod n/ 6Á Zn Cn ; n k m Zn / gcd1fng n/ ord.k; n/ hu ! vi IdA meaning image of set X under binary relation R inverse of binary relation R inverse image of set X under relation R A surj B iff 9f W A ! B: f is a surjective function A inj B iff 9R W A ! B: R is an injective relation A bij B iff 9f W A ! B: f is a bijection injective property of a relation surjective property of a relation function property of a relation total property of a relation bijection relation the empty string/list the finite strings over alphabet A the infinite strings over alphabet A the reversal of string s concatenation of strings s; t; append.s; t / number of occurrences of character c in string s integer m divides integer n; m is a factor of n greatest common divisor least common multiple fi j k < i < ng fi j k Ä i < ng fi j k < i Ä ng fi j k Ä i Ä ng sum of numbers ri where i ranges over set I of indices product of numbers ri where i ranges over set I of indices quotient of n divided by d remainder of n divided by d congruence modulo n not congruent the ring of integers modulo n addition and multiplication operations in Zn mth power of k in Zn the set of numbers in Œ0; n/ relatively prime to n Euler’s totient function WWDj gcd1fngj the order of k in Zn directed edge from vertex u to vertex v identity relation on set A: aIdA a0 iff a D a0 “mcs” — 2013/1/10 — 0:28 — page 829 — #837 829 Glossary of Symbols symbol R RC hu—vi E.G/ V G/ Cn Ln Kn Hn L.G/ R.G/ Kn;m G/ Hn nŠ n m o./ O./ ‚./ / !./ PrŒA Pr A j B S IA PDF CDF ExŒR ExŒR j A Ex2 ŒR VarŒR Var2 ŒR R meaning path relation of relation R; reflexive transitive closure of R positive path relation of R; transitive closure of R undirected edge connecting vertices u Ô v the edges of graph G the vertices of graph G the length-n undirected cycle the length-n line graph the n-vertex complete graph the n-dimensional hypercube the “left” vertices of bipartite graph G the “right” vertices of bipartite graph G the complete bipartite graph with n left and m right vertices chromatic number of simple graph G P the nth Harmonic number nD1 1= i i asymptotic equality n factorial WWDn n 1/ WWDnŠ=mŠ n m/Š; the binomial coefficient asymptotic notation “little oh” asymptotic notation “big oh” asymptotic notation “Theta” asymptotic notation “big Omega” asymptotic notation “little omega” probability of event A conditional probability of A given B sample space indicator variable for event A probability density function cumulative distribution function expectation of random variable R conditional expectation of R given event A abbreviation for ExŒR/2 variance of R the square of the variance of R standard deviation of R “mcs” — 2013/1/10 — 0:28 — page 830 — #838 Index , set difference, 76 k1 ; k2 ; : : : ; km /-split of A, 504 Cn , 354, 375 IE , indicator for event E, 670 K3;3 , 417 K5 , 417 big omega, 477 ‚./, 474 Zn , 234 bij, 87 G/, 370 C, 76 ;, 76 WWD, Á mod n/, 229 Ex2 ŒR, 725 8, 2, inj, 87 Z, 76 Z , 76 \, 76 , 79 N, 6, 76 A, 77 ZC , pow.A/, 77 Q, 76 R, 76 RC , 76 , 472 (asymptotic equality), 466 , 76 Â, 76 surj, 87 [, 76 k-combinations, 506 k-edge connected, 376 k-to-1 function, 499 k-way independent, 649, 672 n C 1-bit adder, 132 r-permutation, 537 IQ, 718, 724 icr , 384 2-D Array, 344 2-Layer Array, 344 2-dimensional array, 333 5-choosable, 108 acyclic, 287 adjacency matrix, 283 adjacent, 350 Adleman, 247 Akra-Bazzi formula, 811 Akra-Bazzi Theorem, 813, 819 alphabet, 154 annuity, 447, 448 antecedents, 10 antichain, 300, 318 antisymmetric, 291, 303 antisymmetry, 291 a posteriori, 639 arrows, 277 assignment statement, 126 asymmetric, 290, 316 asymmetry, 290 asymptotically equal, 466 asymptotically smaller, 472 asymptotic relations, 484 average, 680, 717 average degree, 352, 412 axiomatic method, Axiom of Choice, 194 axioms, 4, “mcs” — 2013/1/10 — 0:28 — page 831 — #839 831 INDEX Banach-Tarski, 194 base case, 104 basis step, 104 Bayes’ Rule, 640 Beneˇ nets, 337 s Bernoulli distribution, 675 Bernoulli variable, 725 Bernoulli variables, 670 biased, 767 bijection, 543 Bijection Rule, 491 bijective, 84 binary predicate, 58 binary relation, 82 Binary relations, 82 binary trees, 170 Binet’s formula, 138 binomial, 506 binomial coefficient, 507 binomial coefficients, 538, 540 binomial distribution, 675, 679, 728 Binomial Theorem, 506 bin packing, 736 bipartite graph, 357, 361, 396, 430 degree-constrained, 361 birthday principle, 618 blocks, 299 bogus proofs, 19 Bookkeeper Rule, 583 Boole’s inequality, 620 Boolean variables, 40 Borel-Cantelli Lemma, 762 bottleneck, 361 boundary conditions, 803 bridge, 426 Brin, Sergey, 277, 777 buildup error, 378 busy, 709 butterfly, 335 butterfly net, 347 cancellable, 238 Cancellation, 237 Cantor’s paradise, 182, 195 cardinality, 86 carry bit, 61 Cartesian product, 79, 91 CDO, 761 ceiling, 827 chain, 298, 318 chain of “iff”, 14 characteristic equation, 806 characters, 154 Chebyshev’s bound, 752 Chebyshev’s Theorem, 721, 733 Chebyshev Bound, 760 Chebyshev bound, 751 Chernoff Bound, 736, 737 Chinese Appetizer problem, 719 Chinese Remainder Theorem, 265 Choice axiom, 193 chromatic number, 370 Church-Turing thesis, 226 closed forms, 447 closed walk, 281, 374 CML, 346, 347 CNF, 49 codomain, 79, 82 Cohen, 194 collateralized debt obligation, 761 collusion, 689, 691 colorable, 370 coloring, 370 solid, 386 combinatorial proof, 445, 529, 561, 562 common divisor, 214 communication nets, 277 commutative ring, 234, 259, 581 “mcs” — 2013/1/10 — 0:28 — page 832 — #840 832 INDEX compilation, 187 complement, 77 Complement Rule, 620 complete binary tree, 329 complete bipartite graph, 417 complete digraph, 307 complete graph, 353, 417 components, 79 composing, 81 composite, 220 composition, 82, 286, 305 concatenation, 154, 155, 282 conclusion, 10, 42 conditional expectation, 683 conditional probability, 629 confidence, 757 confidence level, 735, 756 congestion, 332, 347 congestion for min-latency, 346, 347 congestion of the network, 333 congruence, 229 congruent, 229 congruent modulo n, 229 Conjectured Inefficiency of Factoring, 220 conjunctive form, 49 conjunctive normal form, 49, 52 connected, 375, 377 k-edge, 376, 377 edge, 376 connected components, 376 connects, 350 consequent, 10 consistent, 195 continuous faces, 421 Continuum Hypothesis, 194 contrapositive, 13, 47 converse, 47 convex function, 742 Convolution, 567 convolution, 566 Convolution Counting Principle, 583 coprime, 236 corollary, countable, 184, 185, 195, 196 countably infinite, 185 counter model, 59 coupon collector problem, 696 cover, 306, 360 covering edge, 306 critical path, 298, 299, 300 cumulative distribution function, 673 cut edge, 377 cycle, 281, 370, 374 of length n, 354 cycle of a graph, 375 DAG, 207, 306 de Bruijn sequences, 312 degree, 350 degree d linear recurrence, 578 degree d linear-recursive, 550 degree-constrained, 361, 519, 548 degree sequence, 543 DeMorgan’s Laws, 51 depth, 299 derived variables, 128 describable, 201 deviation from the mean, 717 diagonal argument, 187 diameter, 330 Die Hard, 212, 213 Difference Rule, 620 digraphs, 277 directed acyclic graph (DAG), 287 directed edge, 279 directed graph, 279 Directed graphs, 277 directed graphs, 207 “mcs” — 2013/1/10 — 0:28 — page 833 — #841 833 INDEX discrete faces, 424 disjoint, 77 disjunctive form, 48 disjunctive normal form, 49, 52 distance between vertices, 282 Distributive Law, 78 distributive law, 49 divide-and-conquer, 811 divides, 209 divisibility relation, 279 divisible, 210 Division Rule, 499 Division Theorem, 211 divisor, 210 DNF, 49 domain, 57, 79, 82 domain of discourse, 57, 553 Dongles, 426 dot product, 752 double letter, 188 Double or nothing, 612 double summations, 469 drawing, 417 edge connected, 376 edge cover, 360 edges, 279, 350 efficient solution, 53 elements, 75 Elkies, ellipsis, 27 empty graph, 353, 371 empty relation, 312, 314, 316, 322, 326 empty sequence, 79 empty string, 70 end of chain, 298 endpoints, 350 end vertex, 279 Enigma, 235 equivalence class, 302, 326 equivalence classes, 326 equivalence relation, 301, 304, 326, 327 equivalence relations, 230 equivalent, 44 erasable, 177 Euclid, 8, 210, 251 Euclid’s Algorithm, 215 Euler, 6, 251 formula, 428 Euler’s function, 240 Euler’s constant, 464 Euler’s formula, 435 Euler’s Theorem, 240 Euler’s theorem, 268, 269 Euler circuit, 310 Euler tour, 310 evaluation function, 164 event, 603, 619 events, 669 exclusive-or, 41 execution, 120 existential, 55 expectation, 680 expected absolute deviation, 708, 752, 752 expected return, 687 expected value, 598, 680, 682, 717 exponential growth, 53 exponentially, 50, 53 extends F , 385 Extensionality, 192 face-down four-card trick, 549 factor, 210 factorial function, 448 factorials, 538, 540 fair, 687 “mcs” — 2013/1/10 — 0:28 — page 834 — #842 834 INDEX Fast Exponentiation, 126 father, 533 Fermat’s Last Theorem, Fermat’s Little Theorem, 246 Fermat’s theorem, 264 Fibonacci number, 136, 137 Fibonacci numbers, 32 Fibonacci recurrence, 803 Fifteen Puzzle, 146 floor, 827 Floyd’s Invariant Principle, 116 flush, 551 Foundation, 193 Four Step Method, 625 four-step method, 651 Frege, 194 Frege, Gotlob, 191 function, 79, 84 Fundamental Theorem of Arithmetic, 223 Gă del, 194, 195 o Gale, 368 Gauss, 228, 229 gcd, 214 general binomial density function, 679 Generalized Pigeonhole Principle, 514 Generalized Product Rule, 496 generating function, 582, 589 Generating Functions, 563 geometric distribution, 686, 687 geometric series, 563 geometric sum, 447 going broke, 767 Goldbach’s Conjecture, 56, 57 Goldbach Conjecture, 221 golden ratio, 216, 253 good count, 179, 590, 590 Google, 767 google, 36 graph bipartite, 357 coloring problem, 370 matching, 360 perfect, 360 shortest path, 285 valid coloring, 370 graph coloring, 370 graph of R, 82 gray edge, 386 greatest common divisor, 214 greatest common divisors, 209 grid, 333 grows unboundedly, 22 guess-and-verify, 793 half-adder, 61 Hall’s Matching Theorem, 358 Hall’s Theorem, 361, 548 Hall’s theorem, 396 halt, 187 Halting Problem, 187 halting problem, 188 Hamiltonian Cycle Problem, 310, 409 Handshake Lemma, 353 Hardy, 209, 226 Harmonic number, 462 Hat-Check problem, 719 head, 279 homogeneous linear recurrence, 803 homogeneous solution, 807 hypercube, 406 hypothesis, 42 identity relation, 316, 326 image, 81, 85 implications, 11 incident, 350 Inclusion-Exclusion, 522, 524 “mcs” — 2013/1/10 — 0:28 — page 835 — #843 835 INDEX inclusion-exclusion for probabilities, 620 Inclusion-Exclusion Rule, 522 increasing subsequence, 324 in-degree, 279 independence, 645 independent, 727 independent random variables, 671 indicator random variable, 670 indicator variable, 682, 725, 750 indicator variables, 672 indirect proof, 16 Induction, 101 induction hypothesis, 104 inductive step, 104 inference rules, 10 infinite, 181 Infinity axiom, 192 infix notation, 82 inhomogeneous linear recurrence, 807 injective, 84 integer linear combination, 211 intended profit, 767 interest rate, 479 interpreters, 187 intersection, 76 interval, 212 Invariant, 212 invariant, 116 inverse, 85, 93 inverse image, 85 irrational, 13 irreducible, 256 irreflexive, 289, 303, 316 irreflexivity, 289 isomorphic, 292, 315, 440 King Chicken Theorem, 309 known-plaintext attack, 239 latency, 332 latency for min-congestion, 346, 347 Latin square, 394 lattice basis reduction, 516 Law of Large Numbers, 733 leaf, 381 least common multiple, 252 lemma, length-n cycle, 354 length-n walk relation, 287 length of a walk, 374 Let’s Make a Deal, 625 letters, 154 linear combination, 211 Linearity of Expectation, 692, 693 linear orders, 293 literal, 712 LMC, 346, 347 load balancing, 735, 739 logical deductions, logical formulas, lower bound, 29 lowest terms, 25 Mapping Rules, 491, 513 Markov’s bound, 752 Markov’s Theorem, 718, 747 Markov Bound, 760 Markov bound, 741 matched string, 157 matching, 358, 360 matching birthdays, 731 matching condition, 359 mathematical proof, matrix multiplication, 474 maximal, 296 maximum, 296 maximum dilation, 789 mean, 14, 680 mean square deviation, 721 “mcs” — 2013/1/10 — 0:28 — page 836 — #844 836 INDEX Menger, 377 merge, 281 Merge Sort, 798 merging vertices, 436 minimal, 204, 296, 296 minimum, 296 minimum weight spanning tree, 384 minor, 436 modus ponens, 10 Monty Hall Problem, 599 multigraphs, 351 multinomial coefficient, 504 multinomials, 507 Multinomial Theorem, 561 multiple, 210 multiplicative, 266 multiplicative inverse, 236, 581 Multiplicative Inverses and Cancelling, 236 multisets, 75 Murphy’s Law, 743, 760 mutual independence, 727 mutually independent, 646, 665, 672, 731, 737 mutually recursive, 589 MySQL, 96 neighbors, 361, 390 network latency, 332 node, 279, 350 nodes, 351 nonconstant polynomial, 22 nonconstructive proof, 516 nondecreasing, 456 nondeterministic polynomial time, 54 nonincreasing, 457 non-unique factorization, 256 norm, 256, 752 not primes, 22 numbered tree, 533 numbered trees, 543 number of processors, 299 Number theory, 209 o(), asymptotically smaller, 472 O(), big oh, 473 o(), little oh, 472 one-sided Chebyshev bound, 752 optimal spouse, 367 order, 242, 803 order over Zn , 242 ordinary induction, 102 outcome, 601, 619 out-degree, 279 outside face, 421 overhang, 460 packet, 329 Page, Larry, 277, 777 page rank, 778, 780 Pairing, 192 pairwise disjoint, 203 pairwise independence, 727 pairwise independent, 649, 728, 731 Pairwise Independent Additivity, 728 Pairwise Independent Sampling, 732, 755 parallel schedule, 299 parallel time, 300 parity, 147 partial correctness, 125 partial fractions, 571 partial functions, 81 partial order, 315 particular solution, 807 partition, 299, 326, 357 partitions, 302 Pascal’s Identity, 529 path, 703 path-total, 304 “mcs” — 2013/1/10 — 0:28 — page 837 — #845 837 INDEX perfect graph, 360 perfect number, 210, 251 permutation, 442, 498, 539 Perturbation Method, 449 perturbation method, 563 pessimal spouse, 367 Pick-4, 738 pigeonhole principle, 445 planar drawing, 417 planar embedding, 423, 424, 440 planar graph, 421 planar graphs, 373 planar subgraph, 431 plug-and-chug, 793 pointwise, 81 Polyhedra, 433 polyhedron, 433 polynomial growth, 53 polynomial time, 53, 220, 357 population size, 734 positive walk relation, 286 potential, 140 power set, 77, 90, 186 Power Set axiom, 193 Power sets, 186 predicate, prefix notation, 82 pre-MST, 385 preserved, 231 preserved invariant, 120 preserved under isomorphism, 356 prime, 5, 220 prime factorization, 252 Prime Factorization Theorem, 28 Prime Number Theorem, 221, 241 private key, 247 probability density function, 673 probability density function,, 672 probability function, 619, 657 probability of an event, 619 probability space, 619 Product Rule, 493, 635 product rule, 664 Product Rule for generating functions, 566 proof, proof by contradiction, 16 proposition, 4, propositional variables, 40 public key, 247 public key cryptography, 247 Pulverizer, 252 Pythagoreans, 433 quotient, 212 Rabin cryptosystem, 272 randomized, 597 randomized algorithm, 679 random sample, 757 random sampling, 756 random variable, 669 random variables, 670 random walk, 703, 779 Random Walks, 767 range, 81 rank, 539 rational, 13, 17 rational functions, 175 reachable, 120 reciprocal, 581 recognizable, 188 recurrence, 793 Recursive data types, 153 recursive definitions, 153 reflexive, 286, 303 regular polyhedron, 433 relational databases, 96 relation on a set, 82 “mcs” — 2013/1/10 — 0:28 — page 838 — #846 838 INDEX relatively prime, 236 relaxed, 709 remainder, 212 Replacement axiom, 193 reversal, 169 reverse-linear, 550 Riemann Hypothesis, 241, 241 ring axioms, 259 ring of formal power series, 581 ring of integers modulo n, 234 ripple-carry, 61 ripple-carry circuit, 132 Rivest, 247 root mean square, 723 round-robin tournament, 308 routing, 330 routing problem, 330 RSA, 247, 271 RSA public key crypto-system, 209 Rubin, Herman, 740 ruined, 767 Russell, 191, 194 Russell’s Paradox, 191, 193 sample space, 601, 619 sampling, 756 SAT, 53 satisfiable, 48, 53, 65, 712 SAT-solvers, 54 scheduled at step k, 299 Schră der-Bernstein, 183, 196 o secret key, 227 self-loop, 351 self-loops, 281 sequence, 79 set, 75 covering, 360 set difference, 76, 89 Shamir, 247 Shapley, 368 simple graph, 350 Simple graphs, 349 simple graphs, 207 Simpson’s Paradox, 643 sink, 785 smallest counterexample, 27 solid coloring, 386 solves, 330 sound, 10 spanning subgraph, 383 spanning tree, 383 Square Multiple Rule, 726 St Petersburg Paradox, 746 St Petersburg paradox, 765 St Petersburg Paradox, 746 stable distributions, 785 stable matching, 363 stable stack, 460 standard deviation, 723, 724, 727 start vertex, 279 state graph, 116 state machines, 29 stationary distribution, 780 Stirling’s formula, 703 strictly bigger, 186 strictly decreasing, 128, 456 Strictly increasing, 128 strictly increasing, 456 strict partial order, 289, 304 string procedure, 188 Strong Induction, 110 strongly connected, 789 Structural induction, 155 structural induction, 153, 174 structure, 207 subsequence, 324 subset, 76 “mcs” — 2013/1/10 — 0:28 — page 839 — #847 839 INDEX subset relation, 315 substitution function, 165 suit, 539 summation notation, 27 Sum Rule, 494, 620 surjective, 84 switches, 329 symbols, 154 symmetric, 207, 303, 349, 789 tail, 279 tails, 679 tails of the distribution, 679 terminals, 329 terms, 79 theorems, topological sort, 296 total, 84 total expectation, 685 total function, 81 totient function, 240 tournament digraph, 307, 308, 706 Towers of Hanoi, 588, 795 trail, 311 transition, 116 transition relation, 116 transitive, 286, 304, 316, 614 tree diagram, 601, 651 truth tables, 40 Turing, 225, 226, 239 Turing’s code, 226, 234, 239 Twin Prime Conjecture, 220 type-checking, 187, 189 unbiased binomial distribution, 679, 708 unbiased game, 767 unbounded Gambler’s ruin, 776 uncountable, 200, 202 undirected, 349 undirected edge, 350 uniform, 612, 621, 675 uniform distribution, 675 union, 76 Union axiom, 192 Union Bound, 621 unique factorization, 252 unique factorizations, 255 Unique Factorization Theorem, 223 universal, 55 unlucky, 709 upper bound, 29 valid, 48 valid coloring, 370 value of an annuity, 450 variance, 721, 730, 750 Venn diagram, 665 vertex, 279, 350 vertex connected, 377 vertices, 279, 350 virtual machines, 187 walk, 409 walk counting matrix, 284 walk in a digraph, 280 walk in a simple graph, 373 walk relation, 286 Weak Law of Large Numbers, 733, 756 weakly connected, 311 weakly decreasing, 128, 139, 223, 457 weakly increasing, 128, 456 weak partial order, 304 well founded, 203 well ordered, 29 Well Ordering, 111 Well Ordering Principle, 25, 103, 115 width, 402 winnings, 687 “mcs” — 2013/1/10 — 0:28 — page 840 — #848 840 INDEX wrap, 591 Zermelo, 194 Zermelo-Fraenkel, Zermelo-Fraenkel Set Theory, 192 ZFC, 9, 192, 194, 195 ZFC axioms, 194 ... of numbers of the form rn, where r is a positive real number and n N Well ordering commonly comes up in Computer Science as a method for proving that computations won’t run forever The idea is... proposition for each possible set of truth values for the variables For example, the truth table for the proposition “P AND Q” has four lines, since there are four settings of truth values for the... before we start into mathematics, we need to investigate the problem of how to talk about mathematics To get around the ambiguity of English, mathematicians have devised a special language for