# 1.1 Logical Form and Logical Equivalence 1

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1 Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic notation; Statement forms and their truth tables; Exclusive or; Logical equivalence; De Morgan's laws; Tautologies and contradictions; Summary of logical equivalences 1.2 Conditional Statements 17 Conditional statements; The negation of a conditional statement; Contrapositive; Converse and inverse; Only if and the biconditional; Necessary and sufficient conditions 1.3 Valid and Invalid Arguments 28 Arguments, argument forms and validity; Modus ponens and modus tollens; Other argument forms; Converse and inverse errors; Relation of the validity or invalidity of an argument and the truth or falsity of its conclusion; contradiction rule; Application to solving logical puzzles 1.4 Application: Digital Logic Circuits 41 Relation between switching circuits and Boolean expressions; Black boxes and gates; Circuits, input/output tables, and Boolean expressions; Simplifying circuits 1.5 Application: Number Systems and Circuits for Addition 57 Binary representation of numbers; Binary addition and subtraction; Circuits for computer addition; Two's complements and the computer representation of negative integers; Hexadecimal notation Chapter 2 The Logic of Quantified Statements Predicates and Quantified Statements I 75 Predicates; Set notation; Universal and existential statements; Translating between formal and informal language; Universal conditional statements; Equiyalent forms of universal and existential statements; Implicit v

2 vi Contents quantification; Negations of universal and existential statements; Negations of universal conditional statements; Vacuous truth of universal statements 2.2 Predicates and Quantified Statements II 89 Alternate forms for universal conditional statements; Statements containing both "for all" and "there exists;" Relation among '\I, 3, /\, and V; The use of predicates in Prolog 2.3 Arguments with Quantified Statements 99 Valid argument forms and arguments; Rule of universal instantiation; Universal modus ponens and universal modus tollens; Proving validity; Using diagrams to test for validity; Converse and inverse errors Chapter 3 Elementary Number Theory and Methods of Proof Direct Proof and Counterexample I: Introduction 113 Introduction to the basic techniques of direct proof and disproof by counterexample; Properties of even and odd integers and prime and composite numbers 3.2 Direct Proof and Counterexample II: Rational Numbers Exploring the definition and properties of rational numbers Direct Proof and Counterexample III: Divisibility 131 Definition of divisibility; Examples and properties; The unique factorization theorem 3.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem 140 Discussion of the quotient-remainder theorem and examples; div and mod; Alternate representations of integers and applications in number theory 3.5 Direct Proof and Counterexample V: Floor and Ceiling Definition and basic properties of the floor and ceiling of a number; The floor ofn/2 3.6 Indirect Argument: Contradiction and Contraposition Proof by contradiction; There is no greatest integer; The sum of a rational number and an irrational number; Proof by contraposition; When the square of an integer is even

3 Contents vii 3.7 Two Classical Theorems 161 The irrationality of V2; The infinitude of the primes 3.8 Application: Algorithms 166 Notation for algorithms; Trace tables; The division algorithm; The Euclidean algorithm Chapter 4 Sequences and Mathematical Induction Sequences 180 Terminology of sequences; Explicit formula for a sequence; Examples; Finding an explicit formula from initial terms; Summation notation; Telescoping sums; Transforming a sum by a change of variable; Product notation; Properties of summations and products; Factorial notation; One-dimensional arrays; Algorithm to change from decimal to binary notation 4.2 Mathematical Induction I 194 Principle of mathematical induction; Sum of the first n integers; Sum of a geometric sequence 4.3 Mathematical Induction II 205 Comparison of mathematical induction and inductive reasoning; Proving divisibility properties; Proving inequalities 4.4 Strong Mathematical Induction and the Well-Ordering Principle 212 Explanation and examples including proof that every integer greater than 1 is divisible by a prime, that a sequence has a certain property, that any parenthesization of a product of n distinct factors results in n - 1 multiplications, and that every positive integer has a unique binary representation; The well-ordering principle; Proof of the quotient-remainder theorem 4.5 Application: Correctness of Algorithms 221 Meaning of program correctness; General format; pre-conditions and post-conditions; Loop invariants and the loop invariant theorem; Correctness of a loop to compute a product; Correctness of the division algorithm and the Euclidean algorithm

4 viii Contents Chapter 5 Set Theory 5.1 Basic Definitions of Set Theory 231 Definition of subset; Venn diagrams; Relations among sets of numbers; Distinction between ~ and E; Definitions of equality, union, intersection, difference, and complements of sets; Cartesian products; Formal language; Algorithm for checking a subset relation 5.2 Properties of Sets 244 List of basic set properties; How to prove set properties using element arguments (via procedural versions of definitions); Disproving proposed set properties; Deriving additional set properties from those on a basic list The Empty Set, Partitions, Power Sets, and Boolean Algebras 258 How to prove a set is empty; Set properties that involve the empty set; Partitions of sets; Power sets; Boolean algebras 5.4 Russell's Paradox and the Halting Problem 268 The barber puzzle; Russell's paradox; The halting problem Chapter 6 Counting 6.1 Counting and Probability 274 Concept of sample space; Probability in the equally likely outcomes case; Tossing coins, rolling dice; Counting the elements of lists, sublists, and one-dimensional arrays 6.2 Possibility Trees and the Multiplication Rule 281 Possibility trees; The multiplication rule; Counting possibilities with and without repetition; Permutations; Permutations of selected elements: r-permutations; Proving properties of P(n, r) Counting Elements of Disjoint Sets: The Addition Rule 295 The addition rule; The difference rule; The inclusion/exclusion rule; The number of elements in a general union, the number of elements in an intersection 6.4 Counting Subsets of a Set: Combinations 306 R-combinations; Ordered and unordered selections; Relation between permutations and combinations; Permutations of a set with repeated elements; A common mistake: double counting

5 Contents ix 6.5 R-Combinations with Repetition Allowed 322 R -combinations with repetition allowed; Multisets; How to count these; Applications 6.6 The Algebra of Combinations 330 Combinatorial formulas; New formulas from old by substitution; Pascal's triangle; Algebraic and combinatorial proofs of Pascal's formula 6.7 The Binomial Theorem 336 The binomial theorem; Algebraic and combinatorial proofs; Applications Chapter 7 Functions Functions Defined on General Sets 344 Definition of a function as a relationship between the elements of two sets; Arrow diagram of a function; Function machines; Equality of functions; Examples of functions such as the identity function, sequences, functions defined on a power set, functions defined on a language, logarithmic functions, encoding and decoding functions, and Hamming distance function; Boolean functions; Determining whether a function is well-defined 7.2 Application: Finite-State Automata 357 Definitions and examples of finite-state automata; How to construct a finite-state automaton to do a certain job; The language accepted by a finite-state automaton; The eventual-state function; Simulating a finite-state automaton using software 7.3 One-to-One and Onto, Inverse Functions 369 Definition and examples of one-to-one and onto functions; Application to hash functions; Properties of logarithmic and exponential functions; One-to-one correspondences; Inverse functions 7.4 Application: The Pigeonhole Principle 387 Statement and discussion of principle; Applications; Generalized pigeonhole principle and applications; Proof of the pigeonhole principle 7.5 Composition of Functions 401 Definition and examples; Theorems on composition of one-to-one and onto functions

6 x Contents 7.6 Cardinality with Applications to Computability 411 Definition of cardinality and countability; Countability of the set of all integers, the set of all even integers, and the set of all rational numbers; Uncountability of the real numbers; Countability of the set of all computer programs in a given computer language; Impossibility of computing certain functions Chapter 8 Recursion Recursively Defined Sequences 424 Definition of recurrence relation; Computing terms of recursively defined sequences; The towers of Hanoi; The Fibonacci numbers; Compound interest; Number of bit strings with a certain property; Number of partitions of a set into r subsets; Stirling numbers of the second kind 8.2 Solving Recurrence Relations by Iteration 441 Finding explicit formulas for recursively defined sequences by iteration; Arithmetic and geometric sequences; Using mathematical induction to check whether a recursively defined sequence satisfies a given explicit formula 8.3 Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients 453 Technique for solving these special relations; Formula for the Fibonacci sequence; Gambler's ruin 8.4 General Recursive Definitions 466 Recursive definition of Boolean expressions, parentheses structures, arithmetic expressions, L, n, and finite unions and intersections; Recursively defined functions Chapter 9 0 -Notation and the Efficiency of Algorithms Real-Valued Functions of a Real Variable and Their Graphs 476 Graph of a function; Graphs of integral and fractional power functions; Graph of the floor function; Graphs of functions defined on sets of integers; The graph of a multiple of a function; Increasing and decreasing junctions 9.20-Notation 485 Definition of order; Graphical interpretation; Computing orders of functions from the definition; Orders of polynomial functions; Orders of rational functions; "Best" big-oh approximation

7 Contents xi 9.3 Application: Efficiency of Algorithms I 495 Use of the order notation to discuss algorithm efficiency; Computing orders of simple algorithms; Calculating the efficiency of the sequential search, insertion sort, and selection sort algorithms; Comparing algorithms for polynomial evaluation 9.4 Exponential and Logarithmic Functions: Graphs and Orders 505 Graphs of logarithmic and exponential functions; Consequences of the fact that the logarithmic function with base b > 1 is increasing; The number of bits needed to represent an integer in binary notation; Using logarithms to solve a recurrence relation; Exponential and logarithmic orders 9.5 Application: Efficiency of Algorithms II 519 Divide-and-conquer algorithms; Calculating the efficiency of the binary search and merge sort algorithms Chapter I0 Relations Relations on Sets 533 Definition and notation for relations; Examples of relations; Inverse of a relation; Arrow diagram of a relation; Functions and relations; Directed graph of a relation; n-ary relations and relational databases 10.2 Reflexivity, Symmetry, and Transitivity 546 Reflexive, symmetric, and transitive properties; Transitive closure of a relation; Properties of relations on infinite sets 10.3 Equivalence Relations 555 The relation induced by a partition; Equivalence relations; Examples such as congruence classes modulo n and equivalence of digital logic circuits; Equivalence classes; Lemma that two elements are equivalent if, and only if, they are in the same class; Theorem on the partition of a set by an equivalence relation; Examples of equivalence classes 10.4 Application: Simplifying Finite-State Automata 572 An equivalence relation on the set of states of a finite-state automaton; The quotient automaton; Equivalent automata

8 xii Contents 10.5 Partial Order Relations 585 Definition and examples; Lexicographic order; Hasse diagrams; Partially and totally ordered sets; Topological sorting; PERT and CPM Chapter II Graphs and Trees Graphs: An Introduction 602 Basic terminology and examples of graphs and directed graphs (communication network, representation of a knowledge system, state graph); Special graphs (simple graphs, complete graphs, bipartite graphs); Subgraphs; The concept of degree; Relation of total degree to number of edges; Applications 11.2 Paths and Circuits 619 The puzzle of the Konigsberg bridges; Basic definitions of walks, paths, and circuits; Connectedness; Euler circuits; Euler's theorem; Algorithm for constructing an Euler circuit; Hamiltonian circuits; The traveling salesperson problem 11.3 Matrix Representations of Graphs 640 Matrix notation; Adjacency matrices of directed and undirected graphs; Matrices and connected components; Matrix multiplication; Using matrix entries to find the number of walks of length n in a graph 11.4 Isomorphisms of Graphs 656 Definition of graph isomorphism; Examples; Finding all non isomorphic graphs with certain properties; isomorphic invariants; Using isomorphic invariants to show that graphs are not isomorphic; Graph isomorphism for simple graphs 11.5 Trees 664 Definition and examples of trees (decision tree, derivation tree, structure of hydrocarbon molecules); Equivalent characterizations of trees; Determining number of trees with certain properties; Rooted trees; Binary trees 11.6 Spanning Trees 683 Definition of a spanning tree; Proof of existence; Kruskal's and Prim's algorithms for finding the minimal spanning tree of a weighted graph Appendix A Properties of the Real Numbers 695 Appendix B Solutions and Hints to Selected Exercises 698 Photo Credits 800 Index 1-1

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