Discrete Mathematics with Applications / ���������� ���������� � �� ���������� ��� �������: 2020 �����: Susanna S. Epp / ���� ������� ���� ��� ��������: Discrete Mathematics ������������: Cengage Learning, Inc ISBN: 978-1337694193 ����: ���������� ������: PDF ��������: ������������ ����� ��� ����� (eBook) ������������� ����������: ��� ���������� �������: 1057 ��������: ���������� ���������� � ������������, 5-� �������, ��������� �������, ����������� ������� � �������� � ��������� � ������������ ������� ������ ��� ������������ ���� � ������ ���������� �������� ������ ������������� ����. ����� ������� ��� ������������ �� ������ �������� ���� ���������� ����������, �� � �����������, ������� � ������ �������������� �����. �������� ��������� ����������� ���������� �������, ������ ���� ������ � ��������������. ������ ����� �������, ��� ���������� ����� � �������� �����������, ������ ����������, ����������� ��������, ������������, ��������, ������������ � �������������, �������� ������������, ��� ���� ���������� ���������� ����� � ������ ����������� ����� � ������� � �������� �� ���������. -------------- DISCRETE MATHEMATICS WITH APPLICATIONS, 5th Edition, explains complex, abstract concepts with clarity and precision and provides a strong foundation for computer science and upper-level mathematics courses of the computer age. Author Susanna Epp presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to today's science and technology.
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Speaking Mathematically 1 Variables 1 Using Variables in Mathematical Discourse; Introduction to Universal, Existential, and Conditional Statements The Language of Sets 6 The Set-Roster and Set-Builder Notations; Subsets; Cartesian Products; Strings The Language of Relations and Functions 15 Definition of a Relation from One Set to Another; Arrow Diagram of a Relation; Definition of Function; Function Machines; Equality of Functions The Language of Graphs 24 Definition and Representation of Graphs and Directed Graphs; Degree of a Vertex; Examples of Graphs Including a Graph Coloring Application The Logic of Compound Statements 37 Logical Form and Logical Equivalence 37 Statements; Compound Statements; Truth Values; Evaluating the Truth of More General Compound Statements; Logical Equivalence; Tautologies and Contradictions; Summary of Logical Equivalences Conditional Statements 53 Logical Equivalences Involving S; Representation of If-Then As Or; The Negation of a Conditional Statement; The Contrapositive of a Conditional Statement; The Converse and Inverse of a Conditional Statement; Only If and the Biconditional; Necessary and Sufficient Conditions; Remarks Valid and Invalid Arguments 66 Modus Ponens and Modus Tollens; Additional Valid Argument Forms: Rules of Inference; Fallacies; Contradictions and Valid Arguments; Summary of Rules of Inference Application: Digital Logic Circuits 79 Black Boxes and Gates; The Input/Output Table for a Circuit; The Boolean Expression Corresponding to a Circuit; The Circuit Corresponding to a Boolean Expression; Finding a Circuit That Corresponds to a Given Input/Output Table; Simplifying Combinational Circuits; NAND and NOR Gates Application: Number Systems and Circuits for Addition 93 Binary Representation of Numbers; Binary Addition and Subtraction; Circuits for Computer Addition; Two�s Complements and the Computer Representation of Negative Integers; 8-Bit Representation of a Number; Computer Addition with Negative Integers; Hexadecimal Notation the Logic of Quantified statements 108 Predicates and Quantified Statements I I08 The Universal Quantifier: 5; The Existential Quantifier: E; Formal versus Informal Language; Universal Conditional Statements; Equivalent Forms of Universal and Existential Statements; Bound Variables and Scope; Implicit Quantification; Tarski�s World Predicates and Quantified Statements II 122 Negations of Quantified Statements; Negations of Universal Conditional Statements; The Relation among 5, E, ` , and ~ ; Vacuous Truth of Universal Statements; Variants of Universal Conditional Statements; Necessary and Sufficient Conditions, Only If Statements with Multiple Quantifiers 131 Translating from Informal to Formal Language; Ambiguous Language; Negations of Multiply-Quantified Statements; Order of Quantifiers; Formal Logical Notation; Prolog Arguments with Quantified Statements 146 Universal Modus Ponens; Use of Universal Modus Ponens in a Proof; Universal Modus Tollens; Proving Validity of Arguments with Quantified Statements; Using Diagrams to Test for Validity; Creating Additional Forms of Argument; Remark on the Converse and Inverse Errors elementary Number theory and methods of Proof 160 Direct Proof and Counterexample I: Introduction 161 Definitions; Proving Existential Statements; Disproving Universal Statements by Counterexample; Proving Universal Statements; Generalizing from the Generic Particular; Method of Direct Proof; Existential Instantiation; Getting Proofs Started; Examples Direct Proof and Counterexample II: Writing Advice 173 Writing Proofs of Universal Statements; Common Mistakes; Examples; Showing That an Existential Statement Is False; Conjecture, Proof, and Disproof Direct Proof and Counterexample III: Rational Numbers 183 More on Generalizing from the Generic Particular; Proving Properties of Rational Numbers; Deriving New Mathematics from Old Direct Proof and Counterexample IV: Divisibility 190 Proving Properties of Divisibility; Counterexamples and Divisibility; The Unique Factorization of Integers Theorem Direct Proof and Counterexample V: Division into Cases and the Quotient-Remainder Theorem 200 Discussion of the Quotient-Remainder Theorem and Examples; div and mod; Alternative Representations of Integers and Applications to Number Theory; Absolute Value and the Triangle Inequality Direct Proof and Counterexample VI: Floor and Ceiling 211 Definition and Basic Properties; The Floor of ny2 Indirect Argument: Contradiction and Contraposition 218 Proof by Contradiction; Argument by Contraposition; Relation between Proof by Contradiction and Proof by Contraposition; Proof as a Problem-Solving Tool Indirect Argument: Two Famous Theorems 228 The Irrationality of Ï 2; Are There Infinitely Many Prime Numbers?; When to Use Indirect Proof; Open Questions in Number Theory Application: The handshake Theorem 235 The Total Degree of a Graph; The Handshake Theorem and Consequences; Applications; Simple Graphs; Complete Graphs; Bipartite Graphs Application: Algorithms 244 An Algorithmic Language; A Notation for Algorithms; Trace Tables; The Division Algorithm; The Euclidean Algorithm sequences, mathematical induction, and recursion 258 Sequences 258 Explicit Formulas for Sequences; Summation Notation; Product Notation; Properties of Summations and Products; Change of Variable; Factorial and n Choose r Notation; Sequences in Computer Programming; Application: Algorithm to Convert from Base 10 to Base 2 Using Repeated Division by 2 Mathematical Induction I: Proving Formulas 275 Principle of Mathematical Induction; Sum of the First n Integers; Proving an Equality; Deducing Additional Formulas; Sum of a Geometric Sequence Mathematical Induction II: Applications 289 Comparison of Mathematical Induction and Inductive Reasoning; Proving Divisibility Properties; Proving Inequalities; Trominoes and Other Applications Strong Mathematical Induction and the Well-Ordering Principle for the Integers 301 Strong Mathematical Induction; The Well-Ordering Principle for the Integers; Binary Representation of Integers and Other Applications Application: Correctness of Algorithms 314 Assertions; Loop Invariants; Correctness of the Division Algorithm; Correctness of the Euclidean Theorem Defining Sequences Recursively 325 Examples of Recursively Defined Sequences; Recursive Definitions of Sum and Product Solving Recurrence Relations by Iteration 340 The Method of Iteration; Using Formulas to Simplify Solutions Obtained by Iteration; Checking the Correctness of a Formula by Mathematical Induction; Discovering That an Explicit Formula Is Incorrect Second-Order Linear homogeneous Recurrence Relations with Constant Coefficients 352 Derivation of a Technique for Solving These Relations; The Distinct-Roots Case; The Single-Root Case General Recursive Definitions and Structural Induction 364 Recursively Defined Sets; Recursive Definitions for Boolean Expressions, Strings, and Parenthesis Structures; Using Structural Induction to Prove Properties about Recursively Defined Sets; Recursive Functions set theory 377 Set Theory: Definitions and the Element Method of Proof 377 Subsets: Introduction to Proof and Disproof for Sets; Set Equality; Venn Diagrams; Operations on Sets; The Empty Set; Partitions of Sets; Power Sets; An Algorithm to Check Whether One Set Is a Subset of Another (Optional) Properties of Sets 391 Set Identities; Proving Subset Relations and Set Equality; Proving That a Set Is the Empty Set Disproofs and Algebraic Proofs 407 Disproving an Alleged Set Property; Problem-Solving Strategy; The Number of Subsets of a Set; �Algebraic� Proofs of Set Identities Boolean Algebras, Russell�s Paradox, and the halting Problem 414 Boolean Algebras: Definition and Properties; Russell�s Paradox; The Halting Problem Properties of Functions 425 Functions Defined on General Sets 425 Dynamic Function Terminology; Equality of Functions; Additional Examples of Functions; Boolean Functions; Checking Whether a Function Is Well Defined; Functions Acting on Sets One-to-One, Onto, and Inverse Functions 439 One-to-One Functions; One-to-One Functions on Infinite Sets; Application: Hash Functions and Cryptographic Hash Functions; Onto Functions; Onto Functions on Infinite Sets; Relations between Exponential and Logarithmic Functions; One-to-One Correspondences; Inverse Functions Composition of Functions 461 Definition and Examples; Composition of One-to-One Functions; Composition of Onto Functions Cardinality with Applications to Computability 473 Definition of Cardinal Equivalence; Countable Sets; The Search for Larger Infinities: The Cantor Diagonalization Process; Application: Cardinality and Computability Properties of relations 487 Relations on Sets 487 Additional Examples of Relations; The Inverse of a Relation; Directed Graph of a Relation; N-ary Relations and Relational Databases Reflexivity, Symmetry, and Transitivity 495 Reflexive, Symmetric, and Transitive Properties; Properties of Relations on Infinite Sets; The Transitive Closure of a Relation Equivalence Relations 505 The Relation Induced by a Partition; Definition of an Equivalence Relation; Equivalence Classes of an Equivalence Relation Modular Arithmetic with Applications to Cryptography 524 Properties of Congruence Modulo n; Modular Arithmetic; Extending the Euclidean Algorithm; Finding an Inverse Modulo n; RSA Cryptography; Euclid�s Lemma; Fermat�s Little Theorem; Why Does the RSA Cipher Work?; Message Authentication; Additional Remarks on Number Theory and Cryptography Partial Order Relations 546 Antisymmetry; Partial Order Relations; Lexicographic Order; Hasse Diagrams; Partially and Totally Ordered Sets; Topological Sorting; An Application; PERT and CPM counting and Probability 564 Introduction to Probability 564 Definition of Sample Space and Event; Probability in the Equally Likely Case; Counting the Elements of Lists, Sublists, and One-Dimensional Arrays Possibility Trees and the Multiplication Rule 573 Possibility Trees; The Multiplication Rule; When the Multiplication Rule Is Difficult or Impossible to Apply; Permutations; Permutations of Selected Elements Counting Elements of Disjoint Sets: The Addition Rule 589 The Addition Rule; The Difference Rule; The Inclusion/Exclusion Rule The Pigeonhole Principle 604 Statement and Discussion of the Principle; Applications; Decimal Expansions of Fractions; Generalized Pigeonhole Principle; Proof of the Pigeonhole Principle Counting Subsets of a Set: Combinations 617 r-Combinations; Ordered and Unordered Selections; Relation between Permutations and Combinations; Permutation of a Set with Repeated Elements; Some Advice about Counting; The Number of Partitions of a Set into r Subsets r-Combinations with Repetition Allowed 634 Multisets and How to Count Them; Which Formula to Use? Pascal�s Formula and the Binomial Theorem 642 Combinatorial Formulas; Pascal�s Triangle; Algebraic and Combinatorial Proofs of Pascal�s Formula; The Binomial Theorem and Algebraic and Combinatorial Proofs for It; Applications Probability Axioms and Expected Value 655 Probability Axioms; Deriving Additional Probability Formulas; Expected Value Conditional Probability, Bayes� Formula, and Independent Events 662 Conditional Probability; Bayes� Theorem; Independent Events theory of Graphs and trees 677 Trails, Paths, and Circuits 677 Definitions; Connectedness; Euler Circuits; Hamiltonian Circuits Matrix Representations of Graphs 698 Matrices; Matrices and Directed Graphs; Matrices and Undirected Graphs; Matrices and Connected Components; Matrix Multiplication; Counting Walks of Length N Isomorphisms of Graphs 713 Definition of Graph Isomorphism and Examples; Isomorphic Invariants; Graph Isomorphism for Simple Graphs Trees: Examples and Basic Properties 720 Definition and Examples of Trees; Characterizing Trees Rooted Trees 732 Definition and Examples of Rooted Trees; Binary Trees and Their Properties; Binary Search Trees Spanning Trees and a Shortest Path Algorithm 742 Definition of a Spanning Tree; Minimum Spanning Trees; Kruskal�s Algorithm; Prim�s Algorithm; Dijkstra�s Shortest Path Algorithm analysis of algorithm efficiency 760 Real-Valued Functions of a Real Variable and Their Graphs 760 Graph of a Function; Power Functions; The Floor Function; Graphing Functions Defined on Sets of Integers; Graph of a Multiple of a Function; Increasing and Decreasing Functions Big-O, Big-Omega, and Big-Theta Notations 769 Definition and General Properties of O-, V-, and Q-Notations; Orders of Power Functions; Orders of Polynomial Functions; A Caution about O-Notation; Theorems about Order Notation Application: Analysis of Algorithm Efficiency I 787 Measuring the Efficiency of an Algorithm; Computing Orders of Simple Algorithms; The Sequential Search Algorithm; The Insertion Sort Algorithm; Time Efficiency of an Algorithm Exponential and Logarithmic Functions: Graphs and Orders 800 Graphs of Exponential and Logarithmic Functions; Application: Number of Bits Needed to Represent an Integer in Binary Notation; Application: Using Logarithms to Solve Recurrence Relations; Exponential and Logarithmic Orders Application: Analysis of Algorithm Efficiency II 813 Binary Search; Divide-and-Conquer Algorithms; The Efficiency of the Binary Search Algorithm; Merge Sort; Tractable and Intractable Problems; A Final Remark on Algorithm Efficiency regular expressions and Finite-state automata 828 Formal Languages and Regular Expressions 829 Definitions and Examples of Formal Languages and Regular Expressions; The Language Defined by a Regular Expression; Practical Uses of Regular Expressions Finite-State Automata 841 Definition of a Finite-State Automaton; The Language Accepted by an Automaton; The Eventual-State Function; Designing a Finite-State Automaton; Simulating a Finite-State Automaton Using Software; Finite-State Automata and Regular Expressions; Regular Languages Simplifying Finite-State Automata 858 * -Equivalence of States; k-Equivalence of States; Finding the * -Equivalence Classes; The Quotient Automaton; Constructing the Quotient Automaton; Equivalent Automata Properties of the real Numbers a-1 solutions and hints to selected exercises a-4 Index I-1
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