Table of Contents

Chapter 1 Foundations 1.1 Your mathematical biography 1.2 Reasoning about items collectively 1.3 Set notation 1.4 Set operations 1.5 Verifying facts about sets 1.6 Values, expressions, and types in ML 1.7 Characters and strings 1.8 Cardinality, disjointness, and partitions 1.9 Cartesian products 1.10 Making your own types 1.11 Making your own operations 1.12 Recursive functions 1.13 Statements and exceptions 1.14 Extended example: A cumulative song 1.15 Special topic: Comparison with object-oriented programming Chapter 2 List 2.1 Lists 2.2 Functions on lists 2.3 Datatypes that use lists 2.4 Powersets 2.5 Case expressions and option types 2.6 Extended example: A language processor 2.7 Special topic: Lists vs. tuples vs. arrays Chapter 3 Proposition 3.1 Forms 3.2 Symbols 3.3 Boolean values 3.4 Logical equivalence 3.5 Conditional propositions 3.6 Conditionals and natural language 3.7 Conditional expressions 3.8 Arguments 3.9 Using argument forms for deduction 3.10 Predicates 3.11 Quantification 3.12 Multiple quantifi cation 3.13 Quantifi cation and algorithms 3.14 Quantifi cation and arguments 3.15 Extended example: Verifying arguments automatically 3.16 Special topic: Quantifi cation and natural language Chapter 4 Proof 4.1 General outline 4.2 Subset proofs 4.3 Set equality 4.4 Set emptiness 4.5 Conditional proofs 4.6 Integers 4.7 Biconditionals 4.8 Warnings 4.9 Case study: Powersets 4.10 From theorems to algorithms 4.11 Extended example: Solving games 4.12 Special topic: Russell's paradox Chapter 5 Relation 5.1 Defi nition 5.2 Representation 5.3 Image, inverse, and composition 5.4 Properties of relations 5.5 Equivalence relations 5.6 Computing transitivity 5.7 Transitive closure 5.8 Partial orders 5.9 Comparability and topological sort 5.10 Extended example: Unifi cation and resolution 5.11 Special topic: Representing relations Chapter 6 Self Reference 6.1 Peano numbers 6.2 Trees 6.3 Mutual recursion 6.4 Structural induction 6.5 Mathematical induction 6.6 Mathematical induction on sets 6.7 Program correctness 6.8 Sorting 6.9 Iteration 6.10 Loop invariants 6.11 From theorems to algorithms, revisited 6.12 Extended example: Huff man encoding 6.13 Special topic: Recursion vs. iteration Chapter 7 Function 7.1 Defi nition 7.2 Function equality 7.3 Functions as fi rst-class values 7.4 Images and inverse images 7.5 Map 7.6 Function properties 7.7 Inverse functions 7.8 Function composition 7.9 Cardinality 7.10 Counting 7.11 Permutations and combinations 7.12 Currying 7.13 Fixed-point iteration 7.14 Extended example: Modeling mathematical functions 7.15 Special topic: Countability Chapter 8 Graph 8.1 Defi nition and terms 8.2 Propositions on graphs 8.3 Strolling about a graph 8.4 Isomorphisms 8.5 A garden of graphs 8.6 Representing graphs 8.7 Extended example: Graph algorithms 8.8 Special topic: Graph coloring Chapter 9 Complexity Class 9.1 Recurrence relations 9.2 Complexity of algorithms 9.3 Analyzing sorting algorithms 9.4 Alternative examples of analyzing algorithms 9.5 Big-oh complexity classes 9.6 Big-theta and family 9.7 Properties of complexity classes 9.8 Tables 9.9 Memoization 9.10 Extended example: The Knapsack Problem 9.11 Special topic: P vs NP Chapter 10 Lattice 10.1 Defi nition and terms 10.2 Propositions on lattices 10.3 Isomorphisms 10.4 Modular and distributive lattices 10.5 Implementing lattice operations 10.6 Boolean algebras 10.7 Special topic: Digital logic circuits Chapter 11 Group 11.1 Preliminary terms 11.2 Defi nition 11.3 Isomorphisms 11.4 Subgroups 11.5 A garden of groups 11.6 Extended example: RSA encryption Chapter 12 Automaton 12.1 Alphabets and languages 12.2 Deterministic fi nite automata 12.3 Nondeterminism 12.4 Regular expressions 12.5 Language model equivalence and limitations 12.6 Context-free grammars 12.7 Push-down automata 12.8 The lambda calculus 12.9 Hierarchies of computational models 12.10 Special topic: Computability Appendix A Patterns for proofs