Mathematics for Computer Science (Eric Lehman / F Thomson Leighton / Albert R Meyer 著)

I Proofs
1 What is a Proof?
2 The Well Ordering Principle
3 Logical Formulas
4 Mathematical Data Types
5 Induction
6 State Machines
7 Recursive Data Types
8 Infinite Sets
II Structures
9 Number Theory
10 Directed graphs & Partial Orders
11 Communication Networks
12 Simple Graphs
13 Planar Graphs
III Counting
14 Sums and Asymptotics
15 Cardinality Rules
16 Generating Functions
IV Probability
17 Events and Probability Spaces
18 Conditional Probability
19 Random Variables
20 Deviation from the Mean
21 Random Walks
V Recurrences
22 Recurrences

I Proofs
Introduction
0.1 References

1 What is a Proof?

1.1 Propositions
1.2 Predicates
1.3 The Axiomatic Method
1.4 Our Axioms
1.5 Proving an Implication
1.6 Proving an “If and Only If”
1.7 Proof by Cases
1.8 Proof by Contradiction
1.9 Good Proofs in Practice
1.10 References

2 The Well Ordering Principle

2.1 Well Ordering Proofs
2.2 Template for Well Ordering Proofs
2.3 Factoring into Primes
2.4 Well Ordered Sets

3 Logical Formulas

3.1 Propositions from Propositions
3.2 Propositional Logic in Computer Programs
3.3 Equivalence and Validity
3.4 The Algebra of Propositions
3.5 The SAT Problem
3.6 Predicate Formulas
3.7 References

4 Mathematical Data Types

4.1 Sets
4.2 Sequences
4.3 Functions
4.4 Binary Relations
4.5 Finite Cardinality

5 Induction

5.1 Ordinary Induction
5.2 Strong Induction
5.3 Strong Induction vs. Induction vs. Well Ordering

6 State Machines

6.1 States and Transitions
6.2 The Invariant Principle
6.3 Partial Correctness & Termination
6.4 The Stable Marriage Problem

7 Recursive Data Types

7.1 Recursive Definitions and Structural Induction
7.2 Strings of Matched Brackets
7.3 Recursive Functions on Nonnegative Integers
7.4 Arithmetic Expressions
7.5 Induction in Computer Science

8 Infinite Sets

8.1 Infinite Cardinality
8.2 The Halting Problem
8.3 The Logic of Sets
8.4 Does All This Really Work?
II Structures
Introduction

9 Number Theory

9.1 Divisibility
9.2 The Greatest Common Divisor
9.3 Prime Mysteries
9.4 The Fundamental Theorem of Arithmetic
9.5 Alan Turing
9.6 Modular Arithmetic
9.7 Remainder Arithmetic
9.8 Turing’s Code (Version 2.0)
9.9 Multiplicative Inverses and Cancelling
9.10 Euler’s Theorem
9.11 RSA Public Key Encryption
9.12 What has SAT got to do with it?
9.13 References

10 Directed graphs & Partial Orders

10.1 Vertex Degrees
10.2 Walks and Paths
10.3 Adjacency Matrices
10.4 Walk Relations
10.5 Directed Acyclic Graphs & Scheduling
10.6 Partial Orders
10.7 Representing Partial Orders by Set Containment
10.8 Linear Orders
10.9 Product Orders
10.10 Equivalence Relations
10.11 Summary of Relational Properties

11 Communication Networks

11.1 Routing
11.2 Routing Measures
11.3 Network Designs

12 Simple Graphs

12.1 Vertex Adjacency and Degrees
12.2 Sexual Demographics in America
12.3 Some Common Graphs
12.4 Isomorphism
12.5 Bipartite Graphs & Matchings
12.6 Coloring
12.7 Simple Walks
12.8 Connectivity
12.9 Forests & Trees
12.10 References

13 Planar Graphs

13.1 Drawing Graphs in the Plane
13.2 Definitions of Planar Graphs
13.3 Euler’s Formula
13.4 Bounding the Number of Edges in a Planar Graph
13.5 Returning to K5 and K3;3
13.6 Coloring Planar Graphs
13.7 Classifying Polyhedra
13.8 Another Characterization for Planar Graphs
III Counting
Introduction

14 Sums and Asymptotics

14.1 The Value of an Annuity
14.2 Sums of Powers
14.3 Approximating Sums
14.4 Hanging Out Over the Edge
14.5 Products
14.6 Double Trouble
14.7 Asymptotic Notation

15 Cardinality Rules

15.1 Counting One Thing by Counting Another
15.2 Counting Sequences
15.3 The Generalized Product Rule
15.4 The Division Rule
15.5 Counting Subsets
15.6 Sequences with Repetitions
15.7 Counting Practice: Poker Hands
15.8 The Pigeonhole Principle
15.9 Inclusion-Exclusion
15.10 Combinatorial Proofs
15.11 References

16 Generating Functions

16.1 Infinite Series
16.2 Counting with Generating Functions
16.3 Partial Fractions
16.4 Solving Linear Recurrences
16.5 Formal Power Series
16.6 References
IV Probability
Introduction

17 Events and Probability Spaces

17.1 Let’s Make a Deal
17.2 The Four Step Method
17.3 Strange Dice
17.4 The Birthday Principle
17.5 Set Theory and Probability
17.6 References

18 Conditional Probability

18.1 Monty Hall Confusion
18.2 Definition and Notation
18.3 The Four-Step Method for Conditional Probability
18.4 Why Tree Diagrams Work
18.5 The Law of Total Probability
18.6 Simpson’s Paradox
18.7 Independence
18.8 Mutual Independence
18.9 Probability versus Confidence

19 Random Variables

19.1 Random Variable Examples
19.2 Independence
19.3 Distribution Functions
19.4 Great Expectations
19.5 Linearity of Expectation

20 Deviation from the Mean

20.1 Markov’s Theorem
20.2 Chebyshev’s Theorem
20.3 Properties of Variance
20.4 Estimation by Random Sampling
20.5 Confidence in an Estimation
20.6 Sums of Random Variables
20.7 Really Great Expectations

21 Random Walks

21.1 Gambler’s Ruin
21.2 Random Walks on Graphs
V Recurrences
Introduction

22 Recurrences

22.1 The Towers of Hanoi
22.2 Merge Sort
22.3 Linear Recurrences
22.4 Divide-and-Conquer Recurrences
22.5 A Feel for Recurrences