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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.

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Contents I

Proofs Introduction 3 0.1

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Well Ordering Proofs 29 Template for WOP Proofs 30 Factoring into Primes 32 Well Ordered Sets 33

Logical Formulas 47 3.1 3.2 3.3 3.4 3.5 3.6 3.7

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Propositions 5 Predicates 8 The Axiomatic Method 8 Our Axioms 9 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

The Well Ordering Principle 29 2.1 2.2 2.3 2.4

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What is a Proof? 5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

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References

Propositions from Propositions 48 Propositional Logic in Computer Programs Equivalence and Validity 54 The Algebra of Propositions 57 The SAT Problem 62 Predicate Formulas 63 References 68

Mathematical Data Types 97 4.1 4.2 4.3 4.4 4.5

Sets 97 Sequences 102 Functions 103 Binary Relations 105 Finite Cardinality 109

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Contents

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Induction 131 5.1 5.2 5.3

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States and Transitions 167 The Invariant Principle 168 Partial Correctness & Termination 176 The Stable Marriage Problem 181

Recursive Data Types 211 7.1 7.2 7.3 7.4 7.5 7.6

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Recursive Definitions and Structural Induction 211 Strings of Matched Brackets 215 Recursive Functions on Nonnegative Integers 219 Arithmetic Expressions 221 Games as a Recursive Data Type 226 Induction in Computer Science 230

Infinite Sets 257 8.1 8.2 8.3 8.4

Infinite Cardinality 258 The Halting Problem 267 The Logic of Sets 271 Does All This Really Work?

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II Structures Introduction 299 9

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State Machines 167 6.1 6.2 6.3 6.4

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Ordinary Induction 131 Strong Induction 140 Strong Induction vs. Induction vs. Well Ordering

Number Theory 301 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Divisibility 301 The Greatest Common Divisor 306 Prime Mysteries 313 The Fundamental Theorem of Arithmetic 315 Alan Turing 318 Modular Arithmetic 322 Remainder Arithmetic 324 Turing’s Code (Version 2.0) 327 Multiplicative Inverses and Cancelling 329 Euler’s Theorem 333 RSA Public Key Encryption 338

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Contents

9.12 What has SAT got to do with it? 9.13 References 341

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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 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

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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