“mcs”一2017/6/5一19：42一page i一#1 Mathematics for Computer Science revised Monday 5th June,2017,19:42 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 2017.Eric Lehman,F Tom Leighton.Albert R Meyer.This work is available under the terms of the Creative Commons Attribution-ShareAlike 3.0 license
“mcs” — 2017/6/5 — 19:42 — page i — #1 Mathematics for Computer Science revised Monday 5th June, 2017, 19:42 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 2017, Eric Lehman, F Tom Leighton, Albert R Meyer. This work is available under the terms of the Creative Commons Attribution-ShareAlike 3.0 license
“mcs”一2017/6/5一19：42一page ifⅲi一#3 Contents Proofs Introduction 3 0.1 References 4 1 What is a Proof?5 1.1 Propositions 5 P 1.2 Predicates 1.3 The Axiomatic Method 8 1.4 Our Axioms 9 1.5 Proving an Implication 11 1.6 Proving an“If and Only If'”l3 1.7 Proof by Cases 15 1.8 Proof by Contradiction 16 1.9 Good Proofs in Practice 17 1.10 References 19 2 The Well Ordering Principle 29 2.1 Well Ordering Proofs 29 2.2 Template for WOP Proofs 30 2.3 Factoring into Primes 32 2.4 Well Ordered Sets 33 3 Logical Formulas 47 3.1 Propositions from Propositions 48 3.2 Propositional Logic in Computer Programs 52 3.3 Equivalence and Validity 54 3.4 The Algebra of Propositions 57 3.5 The SAT Problem 62 3.6 Predicate Formulas 63 3.7 References 68 4 Mathematical Data Types 97 4.1 Sets 97 4.2 Sequences 102 4.3 Functions 103 4.4 Binary Relations 105 4.5 Finite Cardinality 109
“mcs” — 2017/6/5 — 19:42 — page iii — #3 Contents I Proofs Introduction 3 0.1 References 4 1 What is a Proof? 5 1.1 Propositions 5 1.2 Predicates 8 1.3 The Axiomatic Method 8 1.4 Our Axioms 9 1.5 Proving an Implication 11 1.6 Proving an “If and Only If” 13 1.7 Proof by Cases 15 1.8 Proof by Contradiction 16 1.9 Good Proofs in Practice 17 1.10 References 19 2 The Well Ordering Principle 29 2.1 Well Ordering Proofs 29 2.2 Template for WOP Proofs 30 2.3 Factoring into Primes 32 2.4 Well Ordered Sets 33 3 Logical Formulas 47 3.1 Propositions from Propositions 48 3.2 Propositional Logic in Computer Programs 52 3.3 Equivalence and Validity 54 3.4 The Algebra of Propositions 57 3.5 The SAT Problem 62 3.6 Predicate Formulas 63 3.7 References 68 4 Mathematical Data Types 97 4.1 Sets 97 4.2 Sequences 102 4.3 Functions 103 4.4 Binary Relations 105 4.5 Finite Cardinality 109
“mcs”一2017/6/5一19：42一page iv一#4 公 Contents 5 Induction 131 5.1 Ordinary Induction 131 5.2 Strong Induction 140 5.3 Strong Induction vs.Induction vs.Well Ordering 147 6 State Machines 167 6.1 States and Transitions 167 6.2 The Invariant Principle 168 6.3 Partial Correctness Termination 176 6.4 The Stable Marriage Problem 181 7 Recursive Data Types 211 7.1 Recursive Definitions and Structural Induction 211 7.2 Strings of Matched Brackets 215 7.3 Recursive Functions on Nonnegative Integers 219 7.4 Arithmetic Expressions 221 7.5 Games as a Recursive Data Type 226 7.6 Induction in Computer Science 230 8 Infinite Sets 257 8.1 Infinite Cardinality 258 8.2 The Halting Problem 267 8.3 The Logic of Sets 271 8.4 Does All This Really Work?275 II Structures Introduction 299 9 Number Theory 301 9.1 Divisibility 301 9.2 The Greatest Common Divisor 306 9.3 Prime Mysteries 313 9.4 The Fundamental Theorem of Arithmetic 315 9.5 Alan Turing 318 9.6 Modular Arithmetic 322 9.7 Remainder Arithmetic 324 9.8 Turing's Code (Version 2.0)327 9.9 Multiplicative Inverses and Cancelling 329 9.10 Euler's Theorem 333 9.11 RSA Public Key Encryption 338
“mcs” — 2017/6/5 — 19:42 — page iv — #4 Contentsiv 5 Induction 131 5.1 Ordinary Induction 131 5.2 Strong Induction 140 5.3 Strong Induction vs. Induction vs. Well Ordering 147 6 State Machines 167 6.1 States and Transitions 167 6.2 The Invariant Principle 168 6.3 Partial Correctness & Termination 176 6.4 The Stable Marriage Problem 181 7 Recursive Data Types 211 7.1 Recursive Definitions and Structural Induction 211 7.2 Strings of Matched Brackets 215 7.3 Recursive Functions on Nonnegative Integers 219 7.4 Arithmetic Expressions 221 7.5 Games as a Recursive Data Type 226 7.6 Induction in Computer Science 230 8 Infinite Sets 257 8.1 Infinite Cardinality 258 8.2 The Halting Problem 267 8.3 The Logic of Sets 271 8.4 Does All This Really Work? 275 II Structures Introduction 299 9 Number Theory 301 9.1 Divisibility 301 9.2 The Greatest Common Divisor 306 9.3 Prime Mysteries 313 9.4 The Fundamental Theorem of Arithmetic 315 9.5 Alan Turing 318 9.6 Modular Arithmetic 322 9.7 Remainder Arithmetic 324 9.8 Turing’s Code (Version 2.0) 327 9.9 Multiplicative Inverses and Cancelling 329 9.10 Euler’s Theorem 333 9.11 RSA Public Key Encryption 338
“mcs”一2017/6/5一19：42-page v一#5 Contents 9.12 What has SAT got to do with it?340 9.13 References 341 10 Directed graphs Partial Orders 381 10.1 Vertex Degrees 383 10.2 Walks and Paths 384 10.3 Adjacency Matrices 387 10.4 Walk Relations 390 10.5 Directed Acyclic Graphs Scheduling 391 10.6 Partial Orders 399 10.7 Representing Partial Orders by Set Containment 403 10.8 Linear Orders 404 10.9 Product Orders 404 10.10 Equivalence Relations 405 10.11 Summary of Relational Properties 407 10.12 References 409 11 Communication Networks 441 11.1 Routing 441 11.2 Routing Measures 442 11.3 Network Designs 445 12 Simple Graphs 461 12.1 Vertex Adjacency and Degrees 461 12.2 Sexual Demographics in America 463 12.3 Some Common Graphs 465 12.4 Isomorphism 466 12.5 Bipartite Graphs Matchings 469 12.6 Coloring 474 12.7 Walks in Simple Graphs 478 12.8 Connectivity 480 12.9 Special Walks and Tours 483 12.10 k-connected Graphs 485 12.11 Forests Trees 487 12.12 References 494 13 Planar Graphs 533 13.1 Drawing Graphs in the Plane 533 13.2 Definitions of Planar Graphs 533 13.3 Euler's Formula 544 13.4 Bounding the Number of Edges in a Planar Graph 545 13.5 Returning to K's and K3,3 546
“mcs” — 2017/6/5 — 19:42 — page v — #5 Contentsv 9.12 What has SAT got to do with it? 340 9.13 References 341 10 Directed graphs & Partial Orders 381 10.1 Vertex Degrees 383 10.2 Walks and Paths 384 10.3 Adjacency Matrices 387 10.4 Walk Relations 390 10.5 Directed Acyclic Graphs & Scheduling 391 10.6 Partial Orders 399 10.7 Representing Partial Orders by Set Containment 403 10.8 Linear Orders 404 10.9 Product Orders 404 10.10 Equivalence Relations 405 10.11 Summary of Relational Properties 407 10.12 References 409 11 Communication Networks 441 11.1 Routing 441 11.2 Routing Measures 442 11.3 Network Designs 445 12 Simple Graphs 461 12.1 Vertex Adjacency and Degrees 461 12.2 Sexual Demographics in America 463 12.3 Some Common Graphs 465 12.4 Isomorphism 466 12.5 Bipartite Graphs & Matchings 469 12.6 Coloring 474 12.7 Walks in Simple Graphs 478 12.8 Connectivity 480 12.9 Special Walks and Tours 483 12.10 k-connected Graphs 485 12.11 Forests & Trees 487 12.12 References 494 13 Planar Graphs 533 13.1 Drawing Graphs in the Plane 533 13.2 Definitions of Planar Graphs 533 13.3 Euler’s Formula 544 13.4 Bounding the Number of Edges in a Planar Graph 545 13.5 Returning to K5 and K3;3 546
“mcs”一2017/6/5一19：42一page vi-#6 vi Contents 13.6 Coloring Planar Graphs 547 13.7 Classifying Polyhedra 549 13.8 Another Characterization for Planar Graphs 552 III Counting Introduction 561 14 Sums and Asymptotics 563 14.1 The Value of an Annuity 564 14.2 Sums of Powers 570 14.3 Approximating Sums 572 14.4 Hanging Out Over the Edge 576 14.5 Products 582 14.6 Double Trouble 585 14.7 Asymptotic Notation 588 15 Cardinality Rules 613 15.1 Counting One Thing by Counting Another 613 15.2 Counting Sequences 614 15.3 The Generalized Product Rule 617 15.4 The Division Rule 621 15.5 Counting Subsets 623 15.6 Sequences with Repetitions 625 15.7 Counting Practice:Poker Hands 629 15.8 The Pigeonhole Principle 634 15.9 Inclusion-Exclusion 643 15.10 Combinatorial Proofs 649 15.11 References 653 16 Generating Functions 693 16.1 Infinite Series 693 16.2 Counting with Generating Functions 695 16.3 Partial Fractions 701 16.4 Solving Linear Recurrences 704 16.5 Formal Power Series 709 16.6 References 712
“mcs” — 2017/6/5 — 19:42 — page vi — #6 Contentsvi 13.6 Coloring Planar Graphs 547 13.7 Classifying Polyhedra 549 13.8 Another Characterization for Planar Graphs 552 III Counting Introduction 561 14 Sums and Asymptotics 563 14.1 The Value of an Annuity 564 14.2 Sums of Powers 570 14.3 Approximating Sums 572 14.4 Hanging Out Over the Edge 576 14.5 Products 582 14.6 Double Trouble 585 14.7 Asymptotic Notation 588 15 Cardinality Rules 613 15.1 Counting One Thing by Counting Another 613 15.2 Counting Sequences 614 15.3 The Generalized Product Rule 617 15.4 The Division Rule 621 15.5 Counting Subsets 623 15.6 Sequences with Repetitions 625 15.7 Counting Practice: Poker Hands 629 15.8 The Pigeonhole Principle 634 15.9 Inclusion-Exclusion 643 15.10 Combinatorial Proofs 649 15.11 References 653 16 Generating Functions 693 16.1 Infinite Series 693 16.2 Counting with Generating Functions 695 16.3 Partial Fractions 701 16.4 Solving Linear Recurrences 704 16.5 Formal Power Series 709 16.6 References 712
“mcs”-2017/6/5一19：42一page vii一#7 vii Contents IV Probability Introduction 729 17 Events and Probability Spaces 731 17.1 Let's Make a Deal 731 17.2 The Four Step Method 732 17.3 Strange Dice 741 17.4 The Birthday Principle 748 17.5 Set Theory and Probability 750 17.6 References 754 18 Conditional Probability 763 18.1 Monty Hall Confusion 763 18.2 Definition and Notation 764 18.3 The Four-Step Method for Conditional Probability 766 18.4 Why Tree Diagrams Work 768 18.5 The Law of Total Probability 776 18.6 Simpson's Paradox 778 18.7 Independence 780 18.8 Mutual Independence 782 18.9 Probability versus Confidence 786 19 Random Variables 815 19.1 Random Variable Examples 815 19.2 Independence 817 19.3 Distribution Functions 819 19.4 Great Expectations 827 19.5 Linearity of Expectation 839 20 Deviation from the Mean 871 20.1 Markov's Theorem 871 20.2 Chebyshev's Theorem 874 20.3 Properties of Variance 878 20.4 Estimation by Random Sampling 884 20.5 Confidence in an Estimation 888 20.6 Sums of Random Variables 889 20.7 Really Great Expectations 899 21 Random Walks 923 21.1 Gambler's Ruin 923
“mcs” — 2017/6/5 — 19:42 — page vii — #7 Contentsvii IV Probability Introduction 729 17 Events and Probability Spaces 731 17.1 Let’s Make a Deal 731 17.2 The Four Step Method 732 17.3 Strange Dice 741 17.4 The Birthday Principle 748 17.5 Set Theory and Probability 750 17.6 References 754 18 Conditional Probability 763 18.1 Monty Hall Confusion 763 18.2 Definition and Notation 764 18.3 The Four-Step Method for Conditional Probability 766 18.4 Why Tree Diagrams Work 768 18.5 The Law of Total Probability 776 18.6 Simpson’s Paradox 778 18.7 Independence 780 18.8 Mutual Independence 782 18.9 Probability versus Confidence 786 19 Random Variables 815 19.1 Random Variable Examples 815 19.2 Independence 817 19.3 Distribution Functions 819 19.4 Great Expectations 827 19.5 Linearity of Expectation 839 20 Deviation from the Mean 871 20.1 Markov’s Theorem 871 20.2 Chebyshev’s Theorem 874 20.3 Properties of Variance 878 20.4 Estimation by Random Sampling 884 20.5 Confidence in an Estimation 888 20.6 Sums of Random Variables 889 20.7 Really Great Expectations 899 21 Random Walks 923 21.1 Gambler’s Ruin 923
“mcs”一2017/6/5一19：42一page viii一#8 vii Contents 21.2 Random Walks on Graphs 933 V Recurrences Introduction 951 22 Recurrences 953 22.1 The Towers of Hanoi 953 22.2 Merge Sort 956 22.3 Linear Recurrences 960 22.4 Divide-and-Conquer Recurrences 967 22.5 A Feel for Recurrences 974 Bibliography 981 Glossary of Symbols 985 Index 989
“mcs” — 2017/6/5 — 19:42 — page viii — #8 Contentsviii 21.2 Random Walks on Graphs 933 V Recurrences Introduction 951 22 Recurrences 953 22.1 The Towers of Hanoi 953 22.2 Merge Sort 956 22.3 Linear Recurrences 960 22.4 Divide-and-Conquer Recurrences 967 22.5 A Feel for Recurrences 974 Bibliography 981 Glossary of Symbols 985 Index 989
“mcs”一2017/6/5一19：42一page1一#9 I Proofs
“mcs” — 2017/6/5 — 19:42 — page 1 — #9 I Proofs
“mcs”-2017/6/5一19：42一page3一#11 Introduction This text explains how to use mathematical models and methods to analyze prob- lems that arise in computer science.Proofs play a central role in this work because the authors share a belief with most mathematicians that proofs are essential for genuine understanding.Proofs also play a growing role in computer science;they are used to certify that software and hardware will always behave correctly,some- thing that no amount of testing can do. Simply put,a proof is a method of establishing truth.Like beauty,"truth"some- times depends on the eye of the beholder,and it should not be surprising that what constitutes a proof differs among fields.For example,in the judicial system,legal truth is decided by a jury based on the allowable evidence presented at trial.In the business world,authoritative truth is specified by a trusted person or organization, or maybe just your boss.In fields such as physics or biology,scientific truth is confirmed by experiment.In statistics,probable truth is established by statistical analysis of sample data. Philosophical proof involves careful exposition and persuasion typically based on a series of small,plausible arguments.The best example begins with"Cogito ergo sum,"a Latin sentence that translates as"I think,therefore I am."This phrase comes from the beginning of a 17th century essay by the mathematician/philosopher, Rene Descartes,and it is one of the most famous quotes in the world:do a web search for it,and you will be flooded with hits. Deducing your existence from the fact that you're thinking about your existence is a pretty cool and persuasive-sounding idea.However,with just a few more lines IActually,only scientific falsehood can be demonstrated by an experiment-when the experiment fails to behave as predicted.But no amount of experiment can confirm that the next experiment won't fail.For this reason,scientists rarely speak of truth,but rather of theories that accurately predict past, and anticipated future,experiments
“mcs” — 2017/6/5 — 19:42 — page 3 — #11 Introduction This text explains how to use mathematical models and methods to analyze problems that arise in computer science. Proofs play a central role in this work because the authors share a belief with most mathematicians that proofs are essential for genuine understanding. Proofs also play a growing role in computer science; they are used to certify that software and hardware will always behave correctly, something that no amount of testing can do. Simply put, a proof is a method of establishing truth. Like beauty, “truth” sometimes depends on the eye of the beholder, and it should not be surprising that what constitutes a proof differs among fields. For example, in the judicial system, legal truth is decided by a jury based on the allowable evidence presented at trial. In the business world, authoritative truth is specified by a trusted person or organization, or maybe just your boss. In fields such as physics or biology, scientific truth is confirmed by experiment.1 In statistics, probable truth is established by statistical analysis of sample data. Philosophical proof involves careful exposition and persuasion typically based on a series of small, plausible arguments. The best example begins with “Cogito ergo sum,” a Latin sentence that translates as “I think, therefore I am.” This phrase comes from the beginning of a 17th century essay by the mathematician/philosopher, Rene Descartes, and it is one of the most famous quotes in the world: do a web ´ search for it, and you will be flooded with hits. Deducing your existence from the fact that you’re thinking about your existence is a pretty cool and persuasive-sounding idea. However, with just a few more lines 1Actually, only scientific falsehood can be demonstrated by an experiment—when the experiment fails to behave as predicted. But no amount of experiment can confirm that the next experiment won’t fail. For this reason, scientists rarely speak of truth, but rather of theories that accurately predict past, and anticipated future, experiments
“mcs”-2017/6/5一19：42一page4一#12 0.1.References of argument in this vein,Descartes goes on to conclude that there is an infinitely beneficent God.Whether or not you believe in an infinitely beneficent God,you'll probably agree that any very short"proof"of God's infinite beneficence is bound to be far-fetched.So even in masterful hands,this approach is not reliable. Mathematics has its own specific notion of "proof." Definition.A mathematical proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms. The three key ideas in this definition are highlighted:proposition,logical deduc- tion,and axiom.Chapter 1 examines these three ideas along with some basic ways of organizing proofs.Chapter 2 introduces the Well Ordering Principle,a basic method of proof;later,Chapter 5 introduces the closely related proof method of induction. If you're going to prove a proposition,you'd better have a precise understand- ing of what the proposition means.To avoid ambiguity and uncertain definitions in ordinary language,mathematicians use language very precisely,and they often express propositions using logical formulas;these are the subject of Chapter 3. The first three Chapters assume the reader is familiar with a few mathematical concepts like sets and functions.Chapters 4 and 8 offer a more careful look at such mathematical data types,examining in particular properties and methods for proving things about infinite sets.Chapter 7 goes on to examine recursively defined data types. 0.1 References [141,[49],[1]
“mcs” — 2017/6/5 — 19:42 — page 4 — #12 0.1. References4 of argument in this vein, Descartes goes on to conclude that there is an infinitely beneficent God. Whether or not you believe in an infinitely beneficent God, you’ll probably agree that any very short “proof” of God’s infinite beneficence is bound to be far-fetched. So even in masterful hands, this approach is not reliable. Mathematics has its own specific notion of “proof.” Definition. A mathematical proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms. The three key ideas in this definition are highlighted: proposition, logical deduction, and axiom. Chapter 1 examines these three ideas along with some basic ways of organizing proofs. Chapter 2 introduces the Well Ordering Principle, a basic method of proof; later, Chapter 5 introduces the closely related proof method of induction. If you’re going to prove a proposition, you’d better have a precise understanding of what the proposition means. To avoid ambiguity and uncertain definitions in ordinary language, mathematicians use language very precisely, and they often express propositions using logical formulas; these are the subject of Chapter 3. The first three Chapters assume the reader is familiar with a few mathematical concepts like sets and functions. Chapters 4 and 8 offer a more careful look at such mathematical data types, examining in particular properties and methods for proving things about infinite sets. Chapter 7 goes on to examine recursively defined data types. 0.1 References [14], [49], [1]