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Discrete Mathematics With Applications



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Brooks Cole
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Susanna Epp's Discrete Mathematics with Applications, Second Edition provides a clear introduction to discrete mathematics. Epp has always been recognized for her lucid, accessible prose that explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. The text is suitable for many course structures, including one-semester or full-year classes. Its emphasis on reasoning provides strong preparation for computer science or more advanced mathematics courses.

Table of Contents

Chapter 1 The Logic of Compound Statements
1.1 Logical Form and Logical Equivalence 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 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 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 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 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
2.1 Predicates and Quantified Statements I Predicates; Set notation; Universal and existential statements; Translating between formal and informal language; Universal conditional statements; Equivalent forms of universal and existential statements; Implicit quantification; Negations of universal and existential statements; Negations of universal conditional statements; Vacuous truth of universal statements
2.2 Predicates and Quantified Statements II Alternate forms for universal conditional statements; Statements containing both "for all" and "there exists;" Relation among (XXX) and V; The use of predicates in Prolog
2.3 Arguments with Quantified Statements 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
3.1 Direct Proof and Counterexample I: Introduction 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
3.3 Direct Proof and Counterexample III: Divisibility 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 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 of n/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
3.7 Two Classical Theorems The irrationality of (Square root of 2); The infinitude of the primes
3.8 Application: Algorithms Notation for algorithms; Trace tables; The division algorithm; The Euclidean algorithm
Chapter 4 Sequences and Mathematical Induction
4.1 Sequences 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 Principle of mathematical induction; Sum of the first n integers; Sum of a geometric sequence
4.3 Mathematical Induction II Comparison of mathematical induction and inductive reasoning; Proving divisibility properties; Proving inequalities
4.4 Strong Mathematical Induction and the Well-Ordering Principle 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 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
Chapter 5 Set Theory
5.1 Basic Definitions of Set Theory Definition of subset; Venn diagrams; Relations among sets of numbers; Distinction between (XXX) and (XXX); 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 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
5.3 The Empty Set, Partitions, Power Sets, and Boolean Algebras 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 The barber puzzle; Russell's paradox; The halting problem
Chapter 6 Counting
6.1 Counting and Probability 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 Possibility trees; The multiplication rule; Counting possibilities with and without repetition; Permutations; Permutations of selected elements: r-permutations; Proving properties of P(n, r).
6.3 Counting Elements of Disjoint Sets: The Addition Rule 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 R-combinations; Ordered and unordered selections; Relation between permutations and combinations; Permutations of a set with repeated elements; A common mistake: double counting
6.5 R-Combinations with Repetition Allowed R-combinations with repetition allowed; Multisets; How to count these; Applications
6.6 The Algebra of Combinations Combinatorial formulas; New formulas from old by substitution; Pascal's triangle; Algebraic and combinatorial proofs of Pascal's formula
6.7 The Binomial Theorem The binomial theorem; Algebraic and combinatorial proofs; Applications
Chapter 7 Functions
7.1 Functions Defined on General Sets 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 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 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 Statement and discussion of principle; Applications; Generalized pigeonhole principle and applications; Proof of the pigeonhole principle
7.5 Composition of Functions Definition and examples; Theorems on composition of one-to-one and onto functions
7.6 Cardinality with Applications to Computability 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
8.1 Recursively Defined Sequences 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 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 Technique for solving these special relations; Formula for the Fibonacci sequence; Gambler's ruin
8.4 General Recursive Definitions Recursive definition of Boolean expressions, parentheses structures, arithmetic expressions, (XXX), Pi, and finite unions and intersections; Recursively defined functions
Chapter 9 O-Notation and the Efficiency of Algorithms
9.1 Real-Valued Functions of a Real Variable and Their Graphs 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 functions
9.2 O-Notation Definition of order; Graphical interpretation; Computing orders of functions from the definition; Orders of polynomial functions; Orders of rational functions; "Best" big-oh approximation
9.3 Application: Efficiency of Algorithms I 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 Graphs of logarithmic and exponential functions; Consequences of the fact that the logarithmic function with base b Greater than 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 Divide-and-conquer algorithms; Calculating the efficiency of the binary search and merge sort algorithms
Chapter 10 Relations
10.1 Relations on Sets 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 Reflexive, symmetric, and transitive properties; Transitive closure of a relation; Properties of relations on infinite sets
10.3 Equivalence Relations 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 An equivalence relation on the set of states of a finite-state automaton; The quotient automaton; Equivalent automata
10.5 Partial Order Relations Definition and examples; Lexicographic order; Hasse diagrams; Partially and totally ordered sets; Topological sorting; PERT and CPM
Chapter 11 Graphs and Trees
11.1 Graphs: An Introduction 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 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 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 Definition of graph isomorphism; Examples; Finding all nonisomorphic graphs with certain properties; isomorphic invariants; Using isomorphic invariants to show that graphs are not isomorphic; Graph isomorphism for simple graphs
11.5 Trees 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 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(3)
Appendix B Solutions and Hints to Selected Exercises 698(102)
Photo Credits 800
Index I-1

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