Discrete Mathematics with Applications / ���������� ���������� � �� ����������
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���� ��� ��������: Discrete Mathematics
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ISBN: 978-1337694193
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DISCRETE MATHEMATICS WITH APPLICATIONS, 5th Edition, explains complex, abstract concepts with clarity and precision and provides a strong foundation for informatics and upper-level mathematics courses of the computer age. Author Susanna Epp presents not simply the major themes of detached mathematics, but also the reasoning that underlies mathematical idea. Students develop the power to think abstractly as they written report the ideas of logic and proof. While learning almost such concepts every bit logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography and combinatorics, students discover that the ideas of detached mathematics underlie and are essential to today's science and technology.
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Speaking Mathematically one
Variables ane
Using Variables in Mathematical Discourse; Introduction to Universal, Existential,
and Conditional Statements
The Language of Sets vi
The Set-Roster and Ready-Builder Notations; Subsets; Cartesian Products; Strings
The Language of Relations and Functions 15
Definition of a Relation from One Fix to Another; Arrow Diagram of a Relation;
Definition of Function; Office Machines; Equality of Functions
The Linguistic communication 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-And then Equally Or; The Negation of
a Conditional Statement; The Contrapositive of a Provisional Statement; The Antipodal
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
Awarding: Digital Logic Circuits 79
Black Boxes and Gates; The Input/Output Table for a Excursion; The Boolean Expression
Respective to a Excursion; The Circuit Corresponding to a Boolean Expression; Finding
a Excursion That Corresponds to a Given Input/Output Tabular array; Simplifying Combinational
Circuits; NAND and NOR Gates
Application: Number Systems and Circuits for Improver 93
Binary Representation of Numbers; Binary Addition and Subtraction; Circuits for
Reckoner Addition; Two�s Complements and the Computer Representation of Negative
Integers; eight-Chip Representation of a Number; Figurer Addition with Negative Integers;
Hexadecimal Notation
the Logic of Quantified statements 108
Predicates and Quantified Statements I I08
The Universal Quantifier: v; The Existential Quantifier: E; Formal versus Informal
Language; Universal Conditional Statements; Equivalent Forms of Universal and
Existential Statements; Leap Variables and Scope; Implicit Quantification; Tarski�due south
Globe
Predicates and Quantified Statements 2 122
Negations of Quantified Statements; Negations of Universal Conditional Statements;
The Relation among 5, Due east, ` , and ~ ; Vacuous Truth of Universal Statements; Variants of
Universal Conditional Statements; Necessary and Sufficient Conditions, Only If
Statements with Multiple Quantifiers 131
Translating from Breezy to Formal Language; Ambiguous Linguistic communication; Negations of
Multiply-Quantified Statements; Order of Quantifiers; Formal Logical Note; 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 Statement; 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 Communication 173
Writing Proofs of Universal Statements; Common Mistakes; Examples; Showing That an
Existential Statement Is False; Conjecture, Proof, and Disproof
Direct Proof and Counterexample Three: Rational Numbers 183
More on Generalizing from the Generic Particular; Proving Properties of Rational
Numbers; Deriving New Mathematics from Old
Direct Proof and Counterexample 4: Divisibility 190
Proving Properties of Divisibility; Counterexamples and Divisibility; The Unique
Factorization of Integers Theorem
Direct Proof and Counterexample Five: Partition into Cases and the
Quotient-Residuum Theorem 200
Discussion of the Quotient-Residuum Theorem and Examples; div and mod; Alternative
Representations of Integers and Applications to Number Theory; Accented Value and the
Triangle Inequality
Straight Proof and Counterexample VI: Floor and Ceiling 211
Definition and Basic Properties; The Flooring of ny2
Indirect Argument: Contradiction and Contraposition 218
Proof by Contradiction; Argument by Contraposition; Relation between Proof past
Contradiction and Proof by Contraposition; Proof as a Trouble-Solving Tool
Indirect Argument: 2 Famous Theorems 228
The Irrationality of Ï 2; Are In that location Infinitely Many Prime Numbers?; When to Use
Indirect Proof; Open Questions in Number Theory
Awarding: The handshake Theorem 235
The Total Degree of a Graph; The Handshake Theorem and Consequences; Applications;
Unproblematic Graphs; Complete Graphs; Bipartite Graphs
Awarding: Algorithms 244
An Algorithmic Language; A Note 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 Boosted Formulas; Sum of a Geometric Sequence
Mathematical Induction II: Applications 289
Comparison of Mathematical Induction and Anterior Reasoning; Proving Divisibility
Properties; Proving Inequalities; Trominoes and Other Applications
Strong Mathematical Consecration and the Well-Ordering
Principle for the Integers 301
Strong Mathematical Consecration; 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 Segmentation Algorithm; Definiteness of the
Euclidean Theorem
Defining Sequences Recursively 325
Examples of Recursively Divers 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 Wrong
Second-Order Linear homogeneous Recurrence Relations
with Constant Coefficients 352
Derivation of a Technique for Solving These Relations; The Singled-out-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 Evidence Properties about Recursively
Defined Sets; Recursive Functions
set theory 377
Set Theory: Definitions and the Chemical 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; Ability Sets; An Algorithm to
Check Whether Ane Set Is a Subset of Another (Optional)
Properties of Sets 391
Ready Identities; Proving Subset Relations and Set Equality; Proving That a Gear up Is the
Empty Set
Disproofs and Algebraic Proofs 407
Disproving an Alleged Set Belongings; Problem-Solving Strategy; The Number of Subsets
of a Set up; �Algebraic� Proofs of Set Identities
Boolean Algebras, Russell�s Paradox, and the halting Problem 414
Boolean Algebras: Definition and Backdrop; Russell�s Paradox; The Halting Problem
Properties of Functions 425
Functions Divers on General Sets 425
Dynamic Office 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
I-to-1 Functions; One-to-1 Functions on Infinite Sets; Application: Hash
Functions and Cryptographic Hash Functions; Onto Functions; Onto Functions on
Infinite Sets; Relations between Exponential and Logarithmic Functions; Ane-to-One
Correspondences; Inverse Functions
Composition of Functions 461
Definition and Examples; Limerick of Ane-to-One Functions; Limerick of Onto
Functions
Cardinality with Applications to Computability 473
Definition of Key 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; Northward-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 Changed Modulo n; RSA Cryptography; Euclid�due south Lemma; Fermat�s
Little Theorem; Why Does the RSA Nil Work?; Bulletin Authentication; Additional
Remarks on Number Theory and Cryptography
Fractional Order Relations 546
Antisymmetry; Fractional Society 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 Upshot; Probability in the Equally Likely Example; Counting
the Elements of Lists, Sublists, and One-Dimensional Arrays
Possibility Copse and the Multiplication Dominion 573
Possibility Trees; The Multiplication Dominion; When the Multiplication Rule Is Hard or
Impossible to Use; 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 Give-and-take 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 up 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�due south 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 Information technology;
Applications
Probability Axioms and Expected Value 655
Probability Axioms; Deriving Additional Probability Formulas; Expected Value
Provisional 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 Unproblematic 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
Existent-Valued Functions of a Real Variable and Their Graphs 760
Graph of a Function; Power Functions; The Flooring Function; Graphing Functions Defined
on Sets of Integers; Graph of a Multiple of a Function; Increasing and Decreasing
Functions
Big-O, Large-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 nigh O-Notation; Theorems
about Lodge Note
Awarding: Analysis of Algorithm Efficiency I 787
Measuring the Efficiency of an Algorithm; Calculating Orders of Uncomplicated 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 Note; Application: Using Logarithms to Solve
Recurrence Relations; Exponential and Logarithmic Orders
Application: Assay 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 past a Regular Expression; Practical Uses of Regular Expressions
Finite-State Automata 841
Definition of a Finite-State Automaton; The Linguistic communication Accustomed by an Automaton; The
Eventual-Country Function; Designing a Finite-Country 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
Alphabetize I-ane
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