Mathematics for Computer Science (MIT 6.042J)

919 Pages·2017·5.924 MB·English

by  it-ebooks

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Mathematics for Computer Science
revised Monday 18th May, 2015, 01:43
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
2015,EricLehman,FTomLeighton,AlbertRMeyer.ThisworkisavailableunderthetermsoftheCreativeCommons
Attribution-NonCommercial-ShareAlike 3.0 license.
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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 27
2.1 Well Ordering Proofs 27
2.2 Template for Well Ordering Proofs 28
2.3 Factoring into Primes 30
2.4 Well Ordered Sets 31
3 Logical Formulas 41
3.1 Propositions from Propositions 42
3.2 Propositional Logic in Computer Programs 45
3.3 Equivalence and Validity 48
3.4 The Algebra of Propositions 50
3.5 The SAT Problem 55
3.6 Predicate Formulas 56
3.7 References 61
4 Mathematical Data Types 81
4.1 Sets 81
4.2 Sequences 86
4.3 Functions 87
4.4 Binary Relations 89
4.5 Finite Cardinality 93
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iv Contents
5 Induction 115
5.1 Ordinary Induction 115
5.2 Strong Induction 124
5.3 Strong Induction vs. Induction vs. Well Ordering 129
5.4 State Machines 130
6 Recursive Data Types 173
6.1 Recursive Definitions and Structural Induction 173
6.2 Strings of Matched Brackets 177
6.3 Recursive Functions on Nonnegative Integers 180
6.4 Arithmetic Expressions 183
6.5 Induction in Computer Science 188
7 Infinite Sets 205
7.1 Infinite Cardinality 206
7.2 The Halting Problem 215
7.3 The Logic of Sets 219
7.4 Does All This Really Work? 222
II Structures
Introduction 241
8 Number Theory 243
8.1 Divisibility 243
8.2 The Greatest Common Divisor 248
8.3 Prime Mysteries 254
8.4 The Fundamental Theorem of Arithmetic 257
8.5 Alan Turing 259
8.6 Modular Arithmetic 263
8.7 Remainder Arithmetic 265
8.8 Turing’s Code (Version 2.0) 268
8.9 Multiplicative Inverses and Cancelling 270
8.10 Euler’s Theorem 274
8.11 RSA Public Key Encryption 279
8.12 What has SAT got to do with it? 281
8.13 References 282
9 Directed graphs & Partial Orders 317
9.1 Vertex Degrees 319
9.2 Walks and Paths 320
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v Contents
9.3 Adjacency Matrices 323
9.4 Walk Relations 326
9.5 Directed Acyclic Graphs & Scheduling 327
9.6 Partial Orders 335
9.7 Representing Partial Orders by Set Containment 339
9.8 Linear Orders 340
9.9 Product Orders 340
9.10 Equivalence Relations 341
9.11 Summary of Relational Properties 343
10 Communication Networks 373
10.1 Complete Binary Tree 373
10.2 Routing Problems 373
10.3 Network Diameter 374
10.4 Switch Count 375
10.5 Network Latency 376
10.6 Congestion 376
10.7 2-D Array 377
10.8 Butterfly 379
10.9 Benesˇ Network 381
11 Simple Graphs 393
11.1 Vertex Adjacency and Degrees 393
11.2 Sexual Demographics in America 395
11.3 Some Common Graphs 397
11.4 Isomorphism 399
11.5 Bipartite Graphs & Matchings 401
11.6 The Stable Marriage Problem 406
11.7 Coloring 413
11.8 Simple Walks 417
11.9 Connectivity 419
11.10 Forests & Trees 424
11.11 References 433
12 Planar Graphs 473
12.1 Drawing Graphs in the Plane 473
12.2 Definitions of Planar Graphs 473
12.3 Euler’s Formula 484
12.4 Bounding the Number of Edges in a Planar Graph 485
12.5 Returning to K5 and K3;3 486
12.6 Coloring Planar Graphs 487
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12.7 Classifying Polyhedra 489
12.8 Another Characterization for Planar Graphs 492
III Counting
Introduction 501
12.9 References 502
13 Sums and Asymptotics 503
13.1 The Value of an Annuity 504
13.2 Sums of Powers 510
13.3 Approximating Sums 512
13.4 Hanging Out Over the Edge 516
13.5 Products 522
13.6 Double Trouble 525
13.7 Asymptotic Notation 528
14 Cardinality Rules 551
14.1 Counting One Thing by Counting Another 551
14.2 Counting Sequences 552
14.3 The Generalized Product Rule 555
14.4 The Division Rule 559
14.5 Counting Subsets 562
14.6 Sequences with Repetitions 564
14.7 Counting Practice: Poker Hands 567
14.8 The Pigeonhole Principle 572
14.9 Inclusion-Exclusion 581
14.10 Combinatorial Proofs 587
14.11 References 591
15 Generating Functions 627
15.1 Infinite Series 627
15.2 Counting with Generating Functions 629
15.3 Partial Fractions 635
15.4 Solving Linear Recurrences 638
15.5 Formal Power Series 643
15.6 References 646
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vii Contents
IV Probability
Introduction 665
16 Events and Probability Spaces 667
16.1 Let’s Make a Deal 667
16.2 The Four Step Method 668
16.3 Strange Dice 677
16.4 The Birthday Principle 684
16.5 Set Theory and Probability 686
16.6 References 690
17 Conditional Probability 697
17.1 Monty Hall Confusion 697
17.2 Definition and Notation 698
17.3 The Four-Step Method for Conditional Probability 700
17.4 Why Tree Diagrams Work 702
17.5 The Law of Total Probability 710
17.6 Simpson’s Paradox 712
17.7 Independence 714
17.8 Mutual Independence 716
18 Random Variables 739
18.1 Random Variable Examples 739
18.2 Independence 741
18.3 Distribution Functions 742
18.4 Great Expectations 751
18.5 Linearity of Expectation 762
19 Deviation from the Mean 789
19.1 Markov’s Theorem 789
19.2 Chebyshev’s Theorem 792
19.3 Properties of Variance 796
19.4 Estimation by Random Sampling 800
19.5 Confidence versus Probability 806
19.6 Sums of Random Variables 807
19.7 Really Great Expectations 816
20 Random Walks 839
20.1 Gambler’s Ruin 839
20.2 Random Walks on Graphs 849
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viii Contents
V Recurrences
Introduction 865
21 Recurrences 867
21.1 The Towers of Hanoi 867
21.2 Merge Sort 870
21.3 Linear Recurrences 874
21.4 Divide-and-Conquer Recurrences 881
21.5 A Feel for Recurrences 888
Bibliography 895
Glossary of Symbols 899
Index 902
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I Proofs
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