Discrete Mathematics and Its Applications ISE

Höfundur: Kenneth Rosen (Útgáfa: 9)
Discrete Mathematics and Its Applications ISE

Kaup valmöguleikar

Discrete Mathematics and Its Applications eftir Rosen setur stærðfræðileg hugtök fram með nákvæmum, heildstæðum hætti sem á erindi við lesandann. Þessi heimsþekkta metsölubók var skrifuð til að mæta þörfum nemenda í ólíkum greinum og deildum, þar á meðal í stærðfræði, tölvunarfræði og verkfræði. Sem leiðandi bók á sínu sviði er hún sveigjanleg, yfirgripsmikil og reynst hefur vel sem kennslutæki.

Nánar um bókina

Útgefandi
McGraw-Hill Higher Education (International)
ISBN
9781266450662
Print ISBN
9781266191541
Format
ePub
Útgáfa
9
Höfundar
Kenneth Rosen
Tungumál
English
Útgefið
2025-07-01
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • Table of Content and Preface
  • Cover Page
  • Title Page
  • Copyright Information
  • Contents
  • About the Author
  • Preface
  • Goals of a Discrete Mathematics Course
  • Changes in the 2025 Release
  • Features of the Book
  • How to Use This Book
  • Acknowledgments
  • Online Resources
  • 0.1 The Online Learning Center
  • To the Student
  • Why Study Discrete Mathematics?
  • The Exercises
  • Web Resources
  • The Value of This Book
  • Chapter 1: The Foundations: Logic and Proofs
  • Chapter 1 Introduction
  • 1.1 Propositional Logic
  • 1.1.1 Introduction
  • 1.1.2 Propositions
  • 1.1.3 Conditional Statements
  • 1.1.4 Truth Tables of Compound Propositions
  • 1.1.5 Precedence of Logical Operators
  • 1.1.6 Logic and Bit Operations
  • 1.1 Propositional Logic Exercises
  • 1.2 Applications of Propositional Logic
  • 1.2.1 Introduction
  • 1.2.2 Translating English Sentences
  • 1.2.3 System Specifications
  • 1.2.4 Boolean Searches
  • 1.2.5 Logic Puzzles
  • 1.2.6 Logic Circuits
  • 1.2 Applications of Propositional Logic Exercises
  • 1.3 Propositional Equivalences
  • 1.3.1 Introduction
  • 1.3.2 Logical Equivalences
  • 1.3.3 Using De Morgan’s Laws
  • 1.3.4 Constructing New Logical Equivalences
  • 1.3.5 Satisfiability
  • 1.3.6 Applications of Satisfiability
  • 1.3.7 Solving Satisfiability Problems
  • 1.3 Propositional Equivalences Exercises
  • 1.4 Predicates and Quantifiers
  • 1.4.1 Introduction
  • 1.4.2 Predicates
  • 1.4.3 Quantifiers
  • 1.4.4 Quantifiers Over Finite Domains
  • 1.4.5 Quantifiers with Restricted Domains
  • 1.4.6 Precedence of Quantifiers
  • 1.4.7 Binding Variables
  • 1.4.8 Logical Equivalences Involving Quantifiers
  • 1.4.9 Negating Quantified Expressions
  • 1.4.10 Translating from English into Logical Expressions
  • 1.4.11 Using Quantifiers in System Specifications
  • 1.4.12 Examples from Lewis Carroll
  • 1.4.13 Logic Programming
  • 1.4 Predicates and Quantifiers Exercises
  • 1.5 Nested Quantifiers
  • 1.5.1 Introduction
  • 1.5.2 Understanding Statements Involving Nested Quantifiers
  • 1.5.3 The Order of Quantifiers
  • 1.5.4 Translating Mathematical Statements into Statements Involving Nested Quantifiers
  • 1.5.5 Translating from Nested Quantifiers into English
  • 1.5.6 Translating English Sentences into Logical Expressions
  • 1.5.7 Negating Nested Quantifiers
  • 1.5 Nested Quantifiers Exercises
  • 1.6 Rules of Inference
  • 1.6.1 Introduction
  • 1.6.2 Valid Arguments in Propositional Logic
  • 1.6.3 Rules of Inference for Propositional Logic
  • 1.6.4 Using Rules of Inference to Build Arguments
  • 1.6.5 Resolution
  • 1.6.6 Fallacies
  • 1.6.7 Rules of Inference for Quantified Statements
  • 1.6.8 Combining Rules of Inference for Propositions and Quantified Statements
  • 1.6 Rules of Inference Exercises
  • 1.7 Introduction to Proofs
  • 1.7.1 Introduction
  • 1.7.2 Some Terminology
  • 1.7.3 Understanding How Theorems Are Stated
  • 1.7.4 Methods of Proving Theorems
  • 1.7.5 Direct Proofs
  • 1.7.6 Proof by Contraposition
  • 1.7.7 Proofs by Contradiction
  • 1.7.8 Mistakes in Proofs
  • 1.7.9 Just a Beginning
  • 1.7 Introduction to Proofs Exercises
  • 1.8 Proof Methods and Strategy
  • 1.8.1 Introduction
  • 1.8.2 Exhaustive Proof and Proof by Cases
  • 1.8.3 Existence Proofs
  • 1.8.4 Uniqueness Proofs
  • 1.8.5 Proof Strategies
  • 1.8.6 Looking for Counterexamples
  • 1.8.7 Proof Strategy in Action
  • 1.8.8 Tilings
  • 1.8.9 The Role of Open Problems
  • 1.8.10 Additional Proof Methods
  • 1.8 Proof Methods and Strategy Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with the specified input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
  • Chapter 2 Introduction
  • 2.1 Sets
  • 2.1.1 Introduction
  • 2.1.2 Venn Diagrams
  • 2.1.3 Subsets
  • 2.1.4 The Size of a Set
  • 2.1.5 Power Sets
  • 2.1.6 Cartesian Products
  • 2.1.7 Using Set Notation with Quantifiers
  • 2.1.8 Truth Sets and Quantifiers
  • 2.1 Sets Exercises
  • 2.2 Set Operations
  • 2.2.1 Introduction
  • 2.2.2 Set Identities
  • 2.2.3 Generalized Unions and Intersections
  • 2.2.4 Computer Representation of Sets
  • 2.2.5 Multisets
  • 2.2 Set Operations Exercises
  • 2.3 Functions
  • 2.3.1 Introduction
  • 2.3.2 One-to-One and Onto Functions
  • 2.3.3 Inverse Functions and Compositions of Functions
  • 2.3.4 The Graphs of Functions
  • 2.3.5 Some Important Functions
  • 2.3.6 Partial Functions
  • 2.3 Functions Exercises
  • 2.4 Sequences and Summations
  • 2.4.1 Introduction
  • 2.4.2 Sequences
  • 2.4.3 Recurrence Relations
  • 2.4.4 Special Integer Sequences
  • 2.4.5 Summations
  • 2.4 Sequences and Summations Exercises
  • 2.5 Cardinality of Sets
  • 2.5.1 Introduction
  • 2.5.2 Countable Sets
  • 2.5.3 An Uncountable Set
  • 2.5.4 Algebraic and Transcendental Numbers
  • 2.5.5 Results About Cardinality
  • 2.5 Cardinality of Sets Exercises
  • 2.6 Matrices
  • 2.6.1 Introduction
  • 2.6.2 Matrix Arithmetic
  • 2.6.3 Transposes and Powers of Matrices
  • 2.6.4 Zero–One Matrices
  • 2.6 Matrices Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with the specified input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 3: Algorithms
  • Chapter 3 Introduction
  • 3.1 Algorithms
  • 3.1.1 Introduction
  • 3.1.2 Searching Algorithms
  • 3.1.3 Sorting
  • 3.1.4 String Matching
  • 3.1.5 Greedy Algorithms
  • 3.1.6 The Halting Problem
  • 3.1 Algorithms Exercises
  • 3.2 The Growth of Functions
  • 3.2.1 Introduction
  • 3.2.2 Big-O Notation
  • 3.2.3 Big-O Estimates for Some Important Functions
  • 3.2.4 The Growth of Combinations of Functions
  • 3.2.5 Big-Omega and Big-Theta Notation
  • 3.2 The Growth of Functions Exercises
  • 3.3 Complexity of Algorithms
  • 3.3.1 Introduction
  • 3.3.2 Time Complexity
  • 3.3.3 Complexity of Matrix Multiplication
  • 3.3.4 Algorithmic Paradigms
  • 3.3.5 Understanding the Complexity of Algorithms
  • 3.3 Complexity of Algorithms Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these inputs and outputs.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 4: Number Theory and Cryptography
  • Chapter 4 Introduction
  • 4.1 Divisibility and Modular Arithmetic
  • 4.1.1 Introduction
  • 4.1.2 Division
  • 4.1.3 The Division Algorithm
  • 4.1.4 Modular Arithmetic
  • 4.1.5 Arithmetic Modulo m
  • 4.1 Divisibility and Modular Arithmetic Exercises
  • 4.2 Integer Representations and Algorithms
  • 4.2.1 Introduction
  • 4.2.2 Representations of Integers
  • 4.2.3 Algorithms for Integer Operations
  • 4.2.4 Modular Exponentiation
  • 4.2 Integer Representations and Algorithms Exercises
  • 4.3 Primes and Greatest Common Divisors
  • 4.3.1 Introduction
  • 4.3.2 Primes
  • 4.3.3 Trial Division
  • 4.3.4 The Sieve of Eratosthenes
  • 4.3.5 Conjectures and Open Problems About Primes
  • 4.3.6 Greatest Common Divisors and Least Common Multiples
  • 4.3.7 The Euclidean Algorithm
  • 4.3.8 gcds as Linear Combinations
  • 4.3 Primes and Greatest Common Divisors Exercises
  • 4.4 Solving Congruences
  • 4.4.1 Introduction
  • 4.4.2 Linear Congruences
  • 4.4.3 The Chinese Remainder Theorem
  • 4.4.4 Computer Arithmetic with Large Integers
  • 4.4.5 Fermat’s Little Theorem
  • 4.4.6 Pseudoprimes
  • 4.4.7 Primitive Roots and Discrete Logarithms
  • 4.4 Solving Congruences Exercises
  • 4.5 Applications of Congruences
  • 4.5.1 Hashing Functions
  • 4.5.2 Pseudorandom Numbers
  • 4.5.3 Check Digits
  • 4.5 Applications of Congruences Exercises
  • 4.6 Cryptography
  • 4.6.1 Introduction
  • 4.6.2 Classical Cryptography
  • 4.6.3 Public Key Cryptography
  • 4.6.4 The RSA Cryptosystem
  • 4.6.5 RSA Encryption
  • 4.6.6 RSA Decryption
  • 4.6.7 RSA as a Public Key System
  • 4.6.8 Cryptographic Protocols
  • 4.6.9 Homomorphic Encryption
  • 4.6 Cryptography Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these inputs and outputs.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 5: Induction and Recursion
  • Chapter 5 Introduction
  • 5.1 Mathematical Induction
  • 5.1.1 Introduction
  • 5.1.2 Mathematical Induction
  • 5.1.3 Why Mathematical Induction is Valid
  • 5.1.4 Choosing the Correct Basis Step
  • 5.1.5 Guidelines for Proofs by Mathematical Induction
  • 5.1.6 The Good and the Bad of Mathematical Induction
  • 5.1.7 Examples of Proofs by Mathematical Induction
  • 5.1.8 Mistaken Proofs By Mathematical Induction
  • 5.1 Mathematical Induction Exercises
  • 5.2 Strong Induction and Well-Ordering
  • 5.2.1 Introduction
  • 5.2.2 Strong Induction
  • 5.2.3 Examples of Proofs Using Strong Induction
  • 5.2.4 Using Strong Induction in Computational Geometry
  • 5.2.5 Proofs Using the Well-Ordering Property
  • 5.2 Strong Induction and Well-Ordering Exercises
  • 5.3 Recursive Definitions and Structural Induction
  • 5.3.1 Introduction
  • 5.3.2 Recursively Defined Functions
  • 5.3.3 Recursively Defined Sets and Structures
  • 5.3.4 Structural Induction
  • 5.3.5 Generalized Induction
  • 5.3 Recursive Definitions and Structural Induction Exercises
  • 5.4 Recursive Algorithms
  • 5.4.1 Introduction
  • 5.4.2 Proving Recursive Algorithms Correct
  • 5.4.3 Recursion and Iteration
  • 5.4.4 The Merge Sort
  • 5.4 Recursive Algorithms Exercises
  • 5.5 Program Correctness
  • 5.5.1 Introduction
  • 5.5.2 Program Verification
  • 5.5.3 Rules of Inference
  • 5.5.4 Conditional Statements
  • 5.5.5 Loop Invariants
  • 5.5 Program Correctness Exercises
  • Key Terms and Results
  • TERMS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 6: Counting
  • Chapter 6 Introduction
  • 6.1 The Basics of Counting
  • 6.1.1 Introduction
  • 6.1.2 Basic Counting Principles
  • 6.1.3 More Complex Counting Problems
  • 6.1.4 The Subtraction Rule (Inclusion–Exclusion for Two Sets)
  • 6.1.5 The Division Rule
  • 6.1.6 Tree Diagrams
  • 6.1 The Basics of Counting Exercises
  • 6.2 The Pigeonhole Principle
  • 6.2.1 Introduction
  • 6.2.2 The Generalized Pigeonhole Principle
  • 6.2.3 Some Elegant Applications of the Pigeonhole Principle
  • 6.2 The Pigeonhole Principle Exercises
  • 6.3 Permutations and Combinations
  • 6.3.1 Introduction
  • 6.3.2 Permutations
  • 6.3.3 Combinations
  • 6.3 Permutations and Combinations Exercises
  • 6.4 Binomial Coefficients and Identities
  • 6.4.1 The Binomial Theorem
  • 6.4.2 Pascal’s Identity and Triangle
  • 6.4.3 Other Identities Involving Binomial Coefficients
  • 6.4 Binomial Coefficients and Identities Exercises
  • 6.5 Generalized Permutations and Combinations
  • 6.5.1 Introduction
  • 6.5.2 Permutations with Repetition
  • 6.5.3 Combinations with Repetition
  • 6.5.4 Permutations with Indistinguishable Objects
  • 6.5.5 Distributing Objects into Boxes
  • 6.5 Generalized Permutations and Combinations Exercises
  • 6.6 Generating Permutations and Combinations
  • 6.6.1 Introduction
  • 6.6.2 Generating Permutations
  • 6.6.3 Generating Combinations
  • 6.6 Generating Permutations and Combinations Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 7: Discrete Probability
  • Chapter 7 Introduction
  • 7.1 An Introduction to Discrete Probability
  • 7.1.1 Introduction
  • 7.1.2 Finite Probability
  • 7.1.3 Probabilities of Complements and Unions of Events
  • 7.1.4 Probabilistic Reasoning
  • 7.1 An Introduction to Discrete Probability Exercises
  • 7.2 Probability Theory
  • 7.2.1 Introduction
  • 7.2.2 Assigning Probabilities
  • 7.2.3 Probabilities of Complements and Unions of Events
  • 7.2.4 Conditional Probability
  • 7.2.5 Independence
  • 7.2.6 Bernoulli Trials and the Binomial Distribution
  • 7.2.7 Random Variables
  • 7.2.8 The Birthday Problem
  • 7.2.9 Monte Carlo Algorithms
  • 7.2.10 The Probabilistic Method
  • 7.2 Probability Theory Exercises
  • 7.3 Bayes’ Theorem
  • 7.3.1 Introduction
  • 7.3.2 Bayes’ Theorem
  • 7.3.3 Bayesian Spam Filters
  • 7.3 Bayes’ Theorem Exercises
  • 7.4 Expected Value and Variance
  • 7.4.1 Introduction
  • 7.4.2 Expected Values
  • 7.4.3 Linearity of Expectations
  • 7.4.4 Average-Case Computational Complexity
  • 7.4.5 The Geometric Distribution
  • 7.4.6 Independent Random Variables
  • 7.4.7 Variance
  • 7.4.8 Chebyshev’s Inequality
  • 7.4 Expected Value and Variance Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 8: Advanced Counting Techniques
  • Chapter 8 Introduction
  • 8.1 Applications of Recurrence Relations
  • 8.1.1 Introduction
  • 8.1.2 Modeling With Recurrence Relations
  • 8.1.3 Algorithms and Recurrence Relations
  • 8.1 Applications of Recurrence Relations Exercises
  • 8.2 Solving Linear Recurrence Relations
  • 8.2.1 Introduction
  • 8.2.2 Solving Linear Homogeneous Recurrence Relations with Constant Coefficients
  • 8.2.3 Linear Nonhomogeneous Recurrence Relations with Constant Coefficients
  • 8.2 Solving Linear Recurrence Relations Exercises
  • 8.3 Divide-and-Conquer Algorithms and Recurrence Relations
  • 8.3.1 Introduction
  • 8.3.2 Divide-and-Conquer Recurrence Relations
  • 8.3 Divide-and-Conquer Algorithms and Recurrence Relations Exercises
  • 8.4 Generating Functions
  • 8.4.1 Introduction
  • 8.4.2 Useful Facts About Power Series
  • 8.4.3 Counting Problems and Generating Functions
  • 8.4.4 Using Generating Functions to Solve Recurrence Relations
  • 8.4.5 Proving Identities via Generating Functions
  • 8.4 Generating Functions Exercises
  • 8.5 Inclusion–Exclusion
  • 8.5.1 Introduction
  • 8.5.2 The Principle of Inclusion–Exclusion
  • 8.5 Inclusion–Exclusion Exercises
  • 8.6 Applications of Inclusion–Exclusion
  • 8.6.1 Introduction
  • 8.6.2 An Alternative Form of Inclusion–Exclusion
  • 8.6.3 The Sieve of Eratosthenes
  • 8.6.4 The Number of Onto Functions
  • 8.6.5 Derangements
  • 8.6 Applications of Inclusion–Exclusion Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 9: Relations
  • Chapter 9 Introduction
  • 9.1 Relations and Their Properties
  • 9.1.1 Introduction
  • 9.1.2 Functions as Relations
  • 9.1.3 Relations on a Set
  • 9.1.4 Properties of Relations
  • 9.1.5 Combining Relations
  • 9.1 Relations and Their Properties Exercises
  • 9.2 n-ary Relations and Their Applications
  • 9.2.1 Introduction
  • 9.2.2 n-ary Relations
  • 9.2.3 Databases and Relations
  • 9.2.4 Operations on n-ary Relations
  • 9.2.5 SQL
  • 9.2.6 Association Rules from Data Mining
  • 9.2 n-ary Relations and Their Applications Exercises
  • 9.3 Representing Relations
  • 9.3.1 Introduction
  • 9.3.2 Representing Relations Using Matrices
  • 9.3.3 Representing Relations Using Digraphs
  • 9.3 Representing Relations Exercises
  • 9.4 Closures of Relations
  • 9.4.1 Introduction
  • 9.4.2 Different Types of Closures
  • 9.4.3 Paths in Directed Graphs
  • 9.4.4 Transitive Closures
  • 9.4.5 Warshall’s Algorithm
  • 9.4 Closures of Relations Exercises
  • 9.5 Equivalence Relations
  • 9.5.1 Introduction
  • 9.5.2 Equivalence Relations
  • 9.5.3 Equivalence Classes
  • 9.5.4 Equivalence Classes and Partitions
  • 9.5 Equivalence Relations Exercises
  • 9.6 Partial Orderings
  • 9.6.1 Introduction
  • 9.6.2 Lexicographic Order
  • 9.6.3 Hasse Diagrams
  • 9.6.4 Maximal and Minimal Elements
  • 9.6.5 Lattices
  • 9.6.6 Topological Sorting
  • 9.6 Partial Orderings Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 10: Graphs
  • Chapter 10 Introduction
  • 10.1 Graphs and Graph Models
  • 10.1.1 Graph Models
  • 10.1 Graphs and Graph Models Exercises
  • 10.2 Graph Terminology and Special Types of Graphs
  • 10.2.1 Introduction
  • 10.2.2 Basic Terminology
  • 10.2.3 Some Special Simple Graphs
  • 10.2.4 Bipartite Graphs
  • 10.2.5 Bipartite Graphs and Matchings
  • 10.2.6 Some Applications of Special Types of Graphs
  • 10.2.7 New Graphs from Old
  • 10.2 Graph Terminology and Special Types of Graphs Exercises
  • 10.3 Representing Graphs and Graph Isomorphism
  • 10.3.1 Introduction
  • 10.3.2 Representing Graphs
  • 10.3.3 Adjacency Matrices
  • 10.3.4 Incidence Matrices
  • 10.3.5 Isomorphism of Graphs
  • 10.3.6 Determining whether Two Simple Graphs are Isomorphic
  • 10.3 Representing Graphs and Graph Isomorphism Exercises
  • 10.4 Connectivity
  • 10.4.1 Introduction
  • 10.4.2 Paths
  • 10.4.3 Connectedness in Undirected Graphs
  • 10.4.4 How Connected is a Graph?
  • 10.4.5 Connectedness in Directed Graphs
  • 10.4.6 Paths and Isomorphism
  • 10.4.7 Counting Paths Between Vertices
  • 10.4 Connectivity Exercises
  • 10.5 Euler and Hamilton Paths
  • 10.5.1 Introduction
  • 10.5.2 Euler Paths and Circuits
  • 10.5.3 Hamilton Paths and Circuits
  • 10.5.4 Applications of Hamilton Circuits
  • 10.5 Euler and Hamilton Paths Exercises
  • 10.6 Shortest-Path Problems
  • 10.6.1 Introduction
  • 10.6.2 A Shortest-Path Algorithm
  • 10.6.3 The Traveling Salesperson Problem
  • 10.6 Shortest-Path Problems Exercises
  • 10.7 Planar Graphs
  • 10.7.1 Introduction
  • 10.7.2 Euler’s Formula
  • 10.7.3 Kuratowski’s Theorem
  • 10.7 Planar Graphs Exercises
  • 10.8 Graph Coloring
  • 10.8.1 Introduction
  • 10.8.2 Applications of Graph Colorings
  • 10.8 Graph Coloring Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 11: Trees
  • Chapter 11 Introduction
  • 11.1 Introduction to Trees
  • 11.1.1 Rooted Trees
  • 11.1.2 Trees as Models
  • 11.1.3 Properties of Trees
  • 11.1 Introduction to Trees Exercises
  • 11.2 Applications of Trees
  • 11.2.1 Introduction
  • 11.2.2 Binary Search Trees
  • 11.2.3 Decision Trees
  • 11.2.4 Prefix Codes
  • 11.2.5 Game Trees
  • 11.2.6 The Tree of Fractions
  • 11.2 Applications of Trees Exercises
  • 11.3 Tree Traversal
  • 11.3.1 Introduction
  • 11.3.2 Universal Address Systems
  • 11.3.3 Traversal Algorithms
  • 11.3.4 Infix, Prefix, and Postfix Notation
  • 11.3 Tree Traversal Exercises
  • 11.4 Spanning Trees
  • 11.4.1 Introduction
  • 11.4.2 Depth-First Search
  • 11.4.3 Breadth-First Search
  • 11.4.4 Backtracking Applications
  • 11.4.5 Depth-First Search in Directed Graphs
  • 11.4 Spanning Trees Exercises
  • 11.5 Minimum Spanning Trees
  • 11.5.1 Introduction
  • 11.5.2 Algorithms for Minimum Spanning Trees
  • 11.5 Minimum Spanning Trees Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 12: Boolean Algebra
  • Chapter 12 Introduction
  • 12.1 Boolean Functions
  • 12.1.1 Introduction
  • 12.1.2 Boolean Expressions and Boolean Functions
  • 12.1.3 Identities of Boolean Algebra
  • 12.1.4 Duality
  • 12.1.5 The Abstract Definition of a Boolean Algebra
  • 12.1 Boolean Functions Exercises
  • 12.2 Representing Boolean Functions
  • 12.2.1 Sum-of-Products Expansions
  • 12.2.2 Functional Completeness
  • 12.2 Representing Boolean Functions Exercises
  • 12.3 Logic Gates
  • 12.3.1 Introduction
  • 12.3.2 Combinations of Gates
  • 12.3.3 Examples of Circuits
  • 12.3.4 Adders
  • 12.3 Logic Gates Exercises
  • 12.4 Minimization of Circuits
  • 12.4.1 Introduction
  • 12.4.2 Karnaugh Maps
  • 12.4.3 Don’t Care Conditions
  • 12.4.4 The Quine–McCluskey Method
  • 12.4 Minimization of Circuits Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Chapter 13: Modeling Computation
  • Chapter 13 Introduction
  • 13.1 Languages and Grammars
  • 13.1.1 Introduction
  • 13.1.2 Phrase-Structure Grammars
  • 13.1.3 Types of Phrase-Structure Grammars
  • 13.1.4 Derivation Trees
  • 13.1.5 Backus–Naur Form
  • 13.1 Languages and Grammars Exercises
  • 13.2 Finite-State Machines with Output
  • 13.2.1 Introduction
  • 13.2.2 Finite-State Machines with Outputs
  • 13.2 Finite-State Machines with Output Exercises
  • 13.3 Finite-State Machines with No Output
  • 13.3.1 Introduction
  • 13.3.2 Set of Strings
  • 13.3.3 Finite-State Automata
  • 13.3.4 Language Recognition by Finite-State Machines
  • 13.3.5 Nondeterministic Finite-State Automata
  • 13.3 Finite-State Machines with No Output Exercises
  • 13.4 Language Recognition
  • 13.4.1 Introduction
  • 13.4.2 Kleene’s Theorem
  • 13.4.3 Regular Sets and Regular Grammars
  • 13.4.4 A Set Not Recognized by a Finite-State Automaton
  • 13.4.5 More Powerful Types of Machines
  • 13.4 Language Recognition Exercises
  • 13.5 Turing Machines
  • 13.5.1 Introduction
  • 13.5.2 Definition of Turing Machines
  • 13.5.3 Using Turing Machines to Recognize Sets
  • 13.5.4 Computing Functions with Turing Machines
  • 13.5.5 Different Types of Turing Machines
  • 13.5.6 The Church–Turing Thesis
  • 13.5.7 Computational Complexity, Computability, and Decidability
  • 13.5 Turing Machines Exercises
  • Key Terms and Results
  • TERMS
  • RESULTS
  • Review Questions
  • Supplementary Exercises
  • Computer Projects
  • Write programs with these input and output.
  • Computations and Explorations
  • Use a computational program or programs you have written to do these exercises.
  • Writing Projects
  • Respond to these with essays using outside sources.
  • Appendix 1: Axioms for the Real Numbers and the Positive Integers
  • Appendix 1 Axioms for the Real Numbers and the Positive Integers
  • A1.1 Axioms for Real Numbers
  • A1.2 Using Axioms to Prove Basic Facts
  • A1.3 Axioms for the Set of Positive Integers
  • Appendix 1 Axioms for the Real Numbers and the Positive Integers Exercises
  • Appendix 2: Exponential and Logarithmic Functions
  • Appendix 2 Exponential and Logarithmic Functions
  • A2.1 Exponential Functions
  • A2.2 Logarithmic Functions
  • Appendix 2 Exponential and Logarithmic Functions Exercises
  • Appendix 3: Pseudocode
  • Appendix 3 Pseudocode
  • A3.1 Procedure Statements
  • A3.2 Assignments and Other Types of Statements
  • A3.3 Comments
  • A3.4 Conditional Constructions
  • A3.5 Loop Constructions
  • A3.6 Loops within Loops
  • A3.7 Using Procedures in Other Procedures
  • A3.8 Return Statements
  • Appendix 3 Pseudocode Exercises
  • Suggested Readings
  • Suggested Reading
  • Chapter 1
  • Chapter 2
  • Chapter 3
  • Chapter 4
  • Chapter 5
  • Chapter 6
  • Chapter 7
  • Chapter 8
  • Chapter 9
  • Chapter 10
  • Chapter 11
  • Chapter 12
  • Chapter 13
  • Appendixes
  • References
  • Answers to Odd-Numbered Exercises
  • Chapter 1
  • CHAPTER 1
  • Chapter 2
  • Section 2.1
  • Section 2.2
  • Section 2.3
  • Section 2.4
  • Section 2.5
  • Section 2.6
  • Supplementary Exercises
  • Chapter 3
  • Section 3.1
  • Section 3.2
  • Section 3.3
  • Supplementary Exercises
  • Chapter 4
  • Section 4.1
  • Section 4.2
  • Section 4.3
  • Section 4.4
  • Section 4.5
  • Section 4.6
  • Supplementary Exercises
  • Chapter 5
  • Section 5.1
  • Section 5.2
  • Section 5.3
  • Section 5.4
  • Section 5.5
  • Supplementary Exercises
  • Chapter 6
  • Section 6.1
  • Section 6.2
  • Section 6.3
  • Section 6.4
  • Section 6.5
  • Section 6.6
  • Supplementary Exercises
  • Chapter 7
  • Section 7.1
  • Section 7.2
  • Section 7.3
  • Section 7.4
  • Supplementary Exercises
  • Chapter 8
  • Section 8.1
  • Section 8.2
  • Section 8.3
  • Section 8.4
  • Section 8.5
  • Section 8.6
  • Supplementary Exercises
  • Chapter 9
  • Section 9.1
  • Section 9.2
  • Section 9.3
  • Section 9.4
  • Section 9.5
  • Section 9.6
  • Supplementary Exercises
  • Chapter 10
  • Section 10.1
  • Section 10.2
  • Section 10.3
  • Section 10.4
  • Section 10.5
  • Section 10.6
  • Section 10.7
  • Section 10.8
  • Supplementary Exercises
  • Chapter 11
  • Section 11.1
  • Section 11.2
  • Section 11.3
  • Section 11.4
  • Section 11.5
  • Supplementary Exercises
  • Chapter 12
  • Section 12.1
  • Section 12.2
  • Section 12.3
  • Section 12.4
  • Supplementary Exercises
  • Chapter 13
  • Section 13.1
  • Section 13.2
  • Section 13.3
  • Section 13.4
  • Section 13.5
  • Supplementary Exercises
  • Appendixes
  • Appendix 1
  • Appendix 2
  • Appendix 3
  • Index of Biographies
  • Index of Biographies
  • Index
  • Index
  • List of Symbols
  • List of Symbols
  • Accessibility Content: Extended Descriptions for Images
  • Figure 1 Extended Description (Chapter 1)
  • Figure 2 Extended Description (Chapter 1)
  • Figure 3 Extended Description (Chapter 1)
  • The Circuit Consists Extended Description (Chapter 1)
  • Circuit Consists of 2 Inverters Extended Description (Chapter 1)
  • The Circuit Consists of 2 Inverters Extended Description (Chapter 1)
  • A Logical Circuit with 4 Inverters Extended Description (Chapter 1)
  • Figure 1 Extended Description (Chapter 1)
  • Figure 2 Extended Description (Chapter 1)
  • Figure 1 Extended Description (Chapter 1)
  • Figure 7 Extended Description (Chapter 1)
  • Standard Checkerboard with Barriers Extended Description (Chapter 1)
  • Four 8 by 8 Squares Filled with Tetrominoes Extended Description (Chapter 1)
  • Figure 1 Extended Description (Chapter 2)
  • A Venn Diagram with Intersecting Extended Description (Chapter 2)
  • A Venn Diagram with Intersecting Circle Extended Description (Chapter 2)
  • Figure 1 Extended Description (Chapter 2)
  • Figure 2 Extended Description (Chapter 2)
  • Figure 3 Extended Description (Chapter 2)
  • Figure 4 Extended Description (Chapter 2)
  • FIGURE 5 Extended Description (Chapter 2)
  • FIGURE 6 Extended Description (Chapter 2)
  • FIGURE 7 Extended Description (Chapter 2)
  • FIGURE 8 Extended Description (Chapter 2)
  • FIGURE 9 Extended Description (Chapter 2)
  • FIGURE 10 Extended Description (Chapter 2)
  • A Graph with 7 Horizontal Segments Extended Description (Chapter 2)
  • A Graph with 5 Horizontal Segments Extended Description (Chapter 2)
  • A Graph Depicts 7 Horizontal Segments Extended Description (Chapter 2)
  • FIGURE 1 Extended Description (Chapter 2)
  • FIGURE 2 Extended Description (Chapter 2)
  • FIGURE 3 Extended Description (Chapter 2)
  • FIGURE 2 Extended Description (Chapter 2)
  • FIGURE 1 Extended Description (Chapter 3)
  • FIGURE 2 Extended Description (Chapter 3)
  • FIGURE 3 Extended Description (Chapter 3)
  • FIGURE 1 Extended Description (Chapter 3)
  • FIGURE 2 Extended Description (Chapter 3)
  • FIGURE 3 Extended Description (Chapter 3)
  • Graph of 3 Functions Extended Description (Chapter 3)
  • The Graphs of the Functions x Squared Extended Description (Chapter 3)
  • FIGURE 1 Extended Description (Chapter 5)
  • FIGURE 3 Extended Description (Chapter 5)
  • FIGURE 5 Extended Description (Chapter 5)
  • FIGURE 7 Extended Description (Chapter 5)
  • Tiling of an 8 by 8 Checkerboard Extended Description (Chapter 5)
  • Tiling of an 8 by 8 Extended Description (Chapter 5)
  • FIGURE 1 Extended Description (Chapter 5)
  • FIGURE 2 Extended Description (Chapter 5)
  • FIGURE 3 Extended Description (Chapter 5)
  • Two Non Convex Polygons Extended Description (Chapter 5)
  • FIGURE 2 Extended Description (Chapter 5)
  • FIGURE 3 Extended Description (Chapter 5)
  • FIGURE 4 Extended Description (Chapter 5)
  • FIGURE 1 Extended Description (Chapter 5)
  • FIGURE 2 Extended Description (Chapter 5)
  • A Merge Sort Extended Description (Chapter 5)
  • FIGURE 1 Extended Description (Chapter 6)
  • FIGURE 2 Extended Description (Chapter 6)
  • FIGURE 3 Extended Description (Chapter 6)
  • FIGURE 4 Extended Description (Chapter 6)
  • FIGURE 5 Extended Description (Chapter 6)
  • FIGURE 6 Extended Description (Chapter 6)
  • FIGURE 1 Extended Description (Chapter 6)
  • FIGURE 1 Extended Description (Chapter 6)
  • Two Rectangular Grids Extended Description (Chapter 6)
  • FIGURE 2 Extended Description (Chapter 6)
  • FIGURE 1 Extended Description (Chapter 8)
  • FIGURE 4 Extended Description (Chapter 8)
  • FIGURE 5 Extended Description (Chapter 8)
  • FIGURE 1 Extended Description (Chapter 8)
  • FIGURE 2 Extended Description (Chapter 8)
  • FIGURE 1 Extended Description (Chapter 8)
  • FIGURE 2 Extended Description (Chapter 8)
  • FIGURE 3 Extended Description (Chapter 8)
  • FIGURE 4 Extended Description (Chapter 8)
  • FIGURE 1 Extended Description (Chapter 9)
  • FIGURE 2 Extended Description (Chapter 9)
  • FIGURE 3 Extended Description (Chapter 9)
  • FIGURE 2 Extended Description (Chapter 9)
  • FIGURE 3 Extended Description (Chapter 9)
  • FIGURE 4 Extended Description (Chapter 9)
  • FIGURE 5 Extended Description (Chapter 9)
  • FIGURE 6 Extended Description (Chapter 9)
  • A Directed Graph Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices and 4 Edges Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices and 4 Edges and 3 Loops Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices and 4 Edges and 4 Loops Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices and 8 Edges for Exercise 21 c Extended Description (Chapter 9)
  • A Graph with Vertices A, B and C Extended Description (Chapter 9)
  • A Graph with Vertices Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D with Loops Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D with Loops at Vertices Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D with Loops at All Vertices Extended Description (Chapter 9)
  • FIGURE 1 Extended Description (Chapter 9)
  • FIGURE 2 Extended Description (Chapter 9)
  • FIGURE 3 Extended Description (Chapter 9)
  • FIGURE 4 Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D with a Loop at Vertex A Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C and D Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 7 Edges Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 8 Edges Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 6 Edges Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 6 Edges for Exercise Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 8 Edges for Exercise Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 6 Edges for Exercise Extended Description (Chapter 9)
  • A Directed Graph with 4 Vertices, 8 Edges and 4 Loops Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C, D and E Extended Description (Chapter 9)
  • A Graph with Vertices A, B, C, D Extended Description (Chapter 9)
  • A Graph with Vertices A, B Extended Description (Chapter 9)
  • A Graph Which Has Vertices A, B, C Extended Description (Chapter 9)
  • FIGURE 1 Extended Description (Chapter 9)
  • FIGURE 2 Extended Description (Chapter 9)
  • FIGURE 3 Extended Description (Chapter 9)
  • FIGURE 4 Extended Description (Chapter 9)
  • FIGURE 5 Extended Description (Chapter 9)
  • FIGURE 6 Extended Description (Chapter 9)
  • FIGURE 7 Extended Description (Chapter 9)
  • FIGURE 8 Extended Description (Chapter 9)
  • FIGURE 9 Extended Description (Chapter 9)
  • FIGURE 10 Extended Description (Chapter 9)
  • FIGURE 11 Extended Description (Chapter 9)
  • A Graph Which Has Vertices A, B, C and D Extended Description (Chapter 9)
  • A Graph Which Has Vertices Extended Description (Chapter 9)
  • A Graph with Vertices A and B Extended Description (Chapter 9)
  • An Undirected Graph Extended Description (Chapter 9)
  • An Undirected Graph with 7 Vertices Extended Description (Chapter 9)
  • An Undirected Graph with 8 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram with 4 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram with 5 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram with 7 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram with 13 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram Shows 7 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram Shows 8 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram Shows 9 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram Shows 14 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram shows 12 Vertices Extended Description (Chapter 9)
  • A Hasse Diagram shows 14 Vertices Extended Description (Chapter 9)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • FIGURE 9 Extended Description (Chapter 10)
  • FIGURE 10 Extended Description (Chapter 10)
  • FIGURE 11 Extended Description (Chapter 10)
  • FIGURE 12 Extended Description (Chapter 10)
  • FIGURE 13 Extended Description (Chapter 10)
  • FIGURE 14 Extended Description (Chapter 10)
  • FIGURE 15 Extended Description (Chapter 10)
  • A graph with 4 Vertices Extended Description (Chapter 10)
  • A graph with 4 Vertices, 7 Undirected Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 7 Undirected Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 11 Undirected Extended Description (Chapter 10)
  • A Graph with 6 Vertices Extended Description (Chapter 10)
  • An Undirected Graph with 5 Vertices Extended Description (Chapter 10)
  • An Undirected Graph with 5 Vertices and 18 Edges Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices and 20 Edges Extended Description (Chapter 10)
  • An Undirected Graph with 5 Vertices and 8 Edges Extended Description (Chapter 10)
  • An Undirected Graph with 6 Vertices and 13 Edges Extended Description (Chapter 10)
  • A Niche Overlap Graph with 6 Vertices Extended Description (Chapter 10)
  • A Graph with 18 Vertices Extended Description (Chapter 10)
  • An Influence Graph with 5 Vertices Extended Description (Chapter 10)
  • A Semantic Network of Nouns Extended Description (Chapter 10)
  • A Directed Graph with 4 Vertices Extended Description (Chapter 10)
  • A Precedence Graph with 7 Vertices Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • FIGURE 9 Extended Description (Chapter 10)
  • Figure 10 Extended Description (Chapter 10)
  • FIGURE 12 Extended Description (Chapter 10)
  • FIGURE 13 Extended Description (Chapter 10)
  • FIGURE 14 Extended Description (Chapter 10)
  • FIGURE 16 Extended Description (Chapter 10)
  • FIGURE 17 Extended Description (Chapter 10)
  • A Graph Shows 6 Vertices and 6 Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices, 10 Edges and 2 Loops Extended Description (Chapter 10)
  • A Graph with 9 Vertices and 12 Undirected Extended Description (Chapter 10)
  • A Graph with 4 Vertices, 6 Directed Extended Description (Chapter 10)
  • Figure for Exercise 8. A Graph with 4 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 9. A Graph with 5 Vertices Extended Description (Chapter 10)
  • A Graph with 6 Vertices, 8 Undirected Extended Description (Chapter 10)
  • Figure for Exercise 21. A Graph with 5 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 22. A Graph with 5 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 23. A Graph with 6 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 24. A Graph with 6 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 25. A Graph with 6 Vertices Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 7 Undirected Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 7 Undirected Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 4 Undirected Edges Extended Description (Chapter 10)
  • Figure for Exercise 53 Extended Description (Chapter 10)
  • Figure for Exercise 58 Extended Description (Chapter 10)
  • Figure for Exercise 59 Extended Description (Chapter 10)
  • A Graph with 7 Vertices and 10 Undirected Extended Description (Chapter 10)
  • Figure for Exercise 60 Extended Description (Chapter 10)
  • A Graph with 5 Vertices, 7 Directed Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices, 1 11 Directed Edges Extended Description (Chapter 10)
  • A Graph with 6 Vertices, 9 Directed Edges Extended Description (Chapter 10)
  • A Graph with 9 Vertices Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • FIGURE 9 Extended Description (Chapter 10)
  • FIGURE 10 Extended Description (Chapter 10)
  • FIGURE 11 Extended Description (Chapter 10)
  • FIGURE 12 Extended Description (Chapter 10)
  • Figure for Exercise 1 Extended Description (Chapter 10)
  • Figure for Exercise 2 Extended Description (Chapter 10)
  • Figure for Exercise 3 Extended Description (Chapter 10)
  • Figure for Exercise 4 Extended Description (Chapter 10)
  • A Graph with 4 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 13 Extended Description (Chapter 10)
  • Figure for Exercise 14 Extended Description (Chapter 10)
  • Figure for Exercise 15 Extended Description (Chapter 10)
  • An Undirected Graph with 4 Vertices Extended Description (Chapter 10)
  • Figure for Exercise 19 Extended Description (Chapter 10)
  • Figure for Exercise 20 Extended Description (Chapter 10)
  • Figure for Exercise 21 Extended Description (Chapter 10)
  • A Directed Graph with 3 Vertices Extended Description (Chapter 10)
  • Two Graphs with 5 Vertices Each Extended Description (Chapter 10)
  • Two Undirected Graphs with 5 Vertices Extended Description (Chapter 10)
  • Two Undirected Graphs with 5 Vertices Each Extended Description (Chapter 10)
  • Figure for Exercise 41 Extended Description (Chapter 10)
  • Figure for Exercise 42 Extended Description (Chapter 10)
  • Figure for Exercise 43 Extended Description (Chapter 10)
  • 2 Graphs with 6 Vertices Each Extended Description (Chapter 10)
  • Figure for Exercise 45 Extended Description (Chapter 10)
  • 2 Graphs with 10 Vertices Each Extended Description (Chapter 10)
  • Figure for Exercise 46 Extended Description (Chapter 10)
  • Figure for Exercise 46 has 2 Graphs Extended Description (Chapter 10)
  • Figure for Exercise 67 Extended Description (Chapter 10)
  • Figure for Exercise 68 Extended Description (Chapter 10)
  • Figure for Exercise 69 Extended Description (Chapter 10)
  • A Bipartite Graph., A Directed Graph Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • Figure for exercise 1 Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices Extended Description (Chapter 10)
  • A Graph with 7 Vertices Extended Description (Chapter 10)
  • A Graph with 11 Vertices Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices and 6 Edges Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices and 8 Edges Extended Description (Chapter 10)
  • A Directed Graph with 7 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Directed Graph with 6 Vertices and 5 Edges Extended Description (Chapter 10)
  • A Directed Graph with 6 Vertices and 9 Edges Extended Description (Chapter 10)
  • A Directed Graph with 7 Vertices and 6 Edges Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Directed Graph with 6 Vertices and 8 Edges Extended Description (Chapter 10)
  • A Directed Graph with 9 Vertices and 12 Edges Extended Description (Chapter 10)
  • A Directed Graph with 6 Vertices and 11 Edges Extended Description (Chapter 10)
  • A Directed Graph with 8 Vertices and 12 Edges Extended Description (Chapter 10)
  • A Directed Graph with 9 Vertices and 19 Edges Extended Description (Chapter 10)
  • Two Graphs G and H with 8 Vertices Each Extended Description (Chapter 10)
  • Two Graphs G and H Extended Description (Chapter 10)
  • Two Graphs G and H with 8 Vertices Extended Description (Chapter 10)
  • Two Graphs Labeled G and H Show 8 Vertices Each Extended Description (Chapter 10)
  • Graph with 6 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Graph Shows with 6 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Graph with 9 Vertices and 9 Edges Extended Description (Chapter 10)
  • A Graph With 7 Vertices and 8 Edges Extended Description (Chapter 10)
  • A Graph with 14 Vertices and 16 Edges Extended Description (Chapter 10)
  • A Graph with 8 Vertices and 15 Edges Extended Description (Chapter 10)
  • A graph with 12 vertices and 18 Edges Extended Description (Chapter 10)
  • Graph with 6 Vertices and 12 Edges Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • FIGURE 9 Extended Description (Chapter 10)
  • FIGURE 10 Extended Description (Chapter 10)
  • FIGURE 11 Extended Description (Chapter 10)
  • FIGURE 12 Extended Description (Chapter 10)
  • FIGURE 13 Extended Description (Chapter 10)
  • FIGURE 14 Extended Description (Chapter 10)
  • A Graph with 5 Vertices Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 10 Edges Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 11 Edges Extended Description (Chapter 10)
  • Graph with 9 Vertices and 15 Edges Extended Description (Chapter 10)
  • Graph with 9 Vertices and 19 Edges Extended Description (Chapter 10)
  • A Graph with 15 Vertices and 26 Edges Extended Description (Chapter 10)
  • A Map Shows Bridges Extended Description (Chapter 10)
  • A Rectangle with Two Diagonals Extended Description (Chapter 10)
  • A Directed Graph with 4 Vertices and 8 Edges Extended Description (Chapter 10)
  • A Directed Graph with 4 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Directed Graph with 5 Vertices and 9 Edges Extended Description (Chapter 10)
  • A directed graph with 5 vertices and 12 Edges Extended Description (Chapter 10)
  • A Directed Graph with 6 Vertices and 16 Edges Extended Description (Chapter 10)
  • A Directed Graph with 12 Vertices and 17 Edges Extended Description (Chapter 10)
  • Figure for Exercise 30 Extended Description (Chapter 10)
  • A Graph with 17 Vertices and 24 Edges Extended Description (Chapter 10)
  • A Pentagram Inside a Pentagon Extended Description (Chapter 10)
  • Graph with 6 Vertices and 9 Edges Extended Description (Chapter 10)
  • An Undirected Graph with 12 Vertices Extended Description (Chapter 10)
  • An Undirected Graph with 12 Vertices and 18 Edges Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • An Undirected Graph with 12 Vertices and 18 Edges Extended Description (Chapter 10)
  • A Weighted Graph with 8 Vertices Extended Description (Chapter 10)
  • A Weighted Graph with 21 Vertices Extended Description (Chapter 10)
  • A Weighted Graph Shows The Distances Between 7 Cities Extended Description (Chapter 10)
  • A Complete Weighted Graph with 4 Vertices Extended Description (Chapter 10)
  • A Complete Weighted Graph with 4 Extended Description (Chapter 10)
  • A Complete Weighted Graph Showing the Airfare Extended Description (Chapter 10)
  • A Complete Weighted Graph Showing the Airfare in Dollars Extended Description (Chapter 10)
  • Square Graph with Vertices Extended Description (Chapter 10)
  • An Undirected Weighted Graph with 3 Vertices Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 10 Extended Description (Chapter 10)
  • FIGURE 11 Extended Description (Chapter 10)
  • FIGURE 12 Extended Description (Chapter 10)
  • FIGURE 13 Extended Description (Chapter 10)
  • FIGURE 14 Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 6 Edges Extended Description (Chapter 10)
  • A Bipartite Graph with 5 Vertices and 6 Edges Extended Description (Chapter 10)
  • An Undirected Weighted Graph with 5 Vertices Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 9 Edges Extended Description (Chapter 10)
  • A Graph with 6 Vertices Extended Description (Chapter 10)
  • A Bipartite Graph with 6 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 11 Edges Extended Description (Chapter 10)
  • An Undirected Graph with 6 Vertices Extended Description (Chapter 10)
  • A Graph Has a Shape of an Octagon Extended Description (Chapter 10)
  • A Graph with 9 Vertices and 15 Edges Extended Description (Chapter 10)
  • Graph with 8 Vertices and 10 Edges Extended Description (Chapter 10)
  • A Graph with 8 Vertices and 12 Edges Extended Description (Chapter 10)
  • A Graph with 12 Vertices and 15 Edges Extended Description (Chapter 10)
  • A Bipartite Graph with 8 Vertices and 12 Edges Extended Description (Chapter 10)
  • A Graph with 9 Vertices and 19 Edges Extended Description (Chapter 10)
  • A Graph with 7 Vertices and 18 Edges Extended Description (Chapter 10)
  • Five Vertical Black Dots are Connected Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 10)
  • Figure 2 Extended Description (Chapter 10)
  • FIGURE 3 Extended Description (Chapter 10)
  • FIGURE 4 Extended Description (Chapter 10)
  • FIGURE 5 Extended Description (Chapter 10)
  • FIGURE 6 Extended Description (Chapter 10)
  • FIGURE 7 Extended Description (Chapter 10)
  • FIGURE 8 Extended Description (Chapter 10)
  • FIGURE 9 Extended Description (Chapter 10)
  • A Map Divided into 5 Regions Labeled A, B, C, D, and E Extended Description (Chapter 10)
  • An Undirected Graph with 5 Vertices and 9 Edges Extended Description (Chapter 10)
  • A Map Divided into 4 Regions Extended Description (Chapter 10)
  • Map Divided into 6 Regions Labeled A, B Extended Description (Chapter 10)
  • An Undirected Graph with 6 Vertices and 8 Edges Extended Description (Chapter 10)
  • A Map Divided into 6 Regions Labeled Extended Description (Chapter 10)
  • A Graph with 7 Vertices and 11 Edges Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 12 Edges Extended Description (Chapter 10)
  • A Graph with 9 Vertices and 20 Edges Extended Description (Chapter 10)
  • A Graph with 15 Vertices Extended Description (Chapter 10)
  • A Graph with 10 Vertices Extended Description (Chapter 10)
  • A Graph with 4 Vertices Extended Description (Chapter 10)
  • A pentagon A Extended Description (Chapter 10)
  • A Graph with 15 Vertices and 15 Edges Extended Description (Chapter 10)
  • Two Graphs Show 8 Vertices Each Extended Description (Chapter 10)
  • A Graph with 8 Vertices Extended Description (Chapter 10)
  • A Graph with 8 Vertices Extended Description (Chapter 10)
  • A Graph with 7 Vertices and 13 Edges Extended Description (Chapter 10)
  • A Graph with 9 Vertices Extended Description (Chapter 10)
  • A Graph with 11 Vertices Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 8 Edges Extended Description (Chapter 10)
  • Graph with 2 Triangles and an Edge Connecting Them Extended Description (Chapter 10)
  • A Graph with 12 Vertices Extended Description (Chapter 10)
  • A Queen on a Chessboard 4 by 4 Extended Description (Chapter 10)
  • An Undirected Graph with 9 Vertices Extended Description (Chapter 10)
  • An Undirected Graph with 16 Vertices and 78 Edges Extended Description (Chapter 10)
  • A Directed Graph with Vertices A, B, C, D Extended Description (Chapter 10)
  • A Directed Graph with Vertices Extended Description (Chapter 10)
  • Graph with Square a-b-d-e, Diagonal Extended Description (Chapter 10)
  • A Graph with 5 Vertices and 6 Edges Extended Description (Chapter 10)
  • A Graph with 6 Vertices and 7 Edges Extended Description (Chapter 10)
  • A Graph with 8 Vertices and 10 Edges Extended Description (Chapter 10)
  • FIGURE 1 Extended Description (Chapter 11)
  • FIGURE 2 Extended Description (Chapter 11)
  • FIGURE 3 Extended Description (Chapter 11)
  • FIGURE 4 Extended Description (Chapter 11)
  • FIGURE 5 Extended Description (Chapter 11)
  • FIGURE 6 Extended Description (Chapter 11)
  • FIGURE 7 Extended Description (Chapter 11)
  • FIGURE 8 Extended Description (Chapter 11)
  • FIGURE 9 Extended Description (Chapter 11)
  • FIGURE 10 Extended Description (Chapter 11)
  • FIGURE 11 Extended Description (Chapter 11)
  • FIGURE 12 Extended Description (Chapter 11)
  • FIGURE 13 Extended Description (Chapter 11)
  • FIGURE 14 Extended Description (Chapter 11)
  • FIGURE 15 Extended Description (Chapter 11)
  • A Graph with 6 Vertices and 8 Edges Extended Description (Chapter 11)
  • A Graph Has 7 Vertices and 6 Edges Extended Description (Chapter 11)
  • A Tree with 21 Vertices Labeled from A Through U Extended Description (Chapter 11)
  • A Tree with 19 Vertices Labeled from A Through S Extended Description (Chapter 11)
  • A Complete Binary Tree of Height 3 Extended Description (Chapter 11)
  • A Tree with 12 Vertices Labeled from A Through L Extended Description (Chapter 11)
  • A Graph with 11 Vertices Labeled from A Through K Extended Description (Chapter 11)
  • A Tree with 14 Vertices Labeled from A Through N Extended Description (Chapter 11)
  • The First Seven Rooted Fibonacci Trees Extended Description (Chapter 11)
  • FIGURE 1 Extended Description (Chapter 11)
  • FIGURE 2 Extended Description (Chapter 11)
  • FIGURE 3 Extended Description (Chapter 11)
  • FIGURE 4 Extended Description (Chapter 11)
  • FIGURE 5 Extended Description (Chapter 11)
  • FIGURE 6 Extended Description (Chapter 11)
  • FIGURE 7 Extended Description (Chapter 11)
  • FIGURE 8 Extended Description (Chapter 11)
  • FIGURE 9 Extended Description (Chapter 11)
  • FIGURE 10 Extended Description (Chapter 11)
  • A Binary Search Tree of Height 4 Extended Description (Chapter 11)
  • A Binary Search Tree of Height 6 with 8 Vertices Extended Description (Chapter 11)
  • Figure for Exercise 12 Extended Description (Chapter 11)
  • A Binary Tree with Root 17 Extended Description (Chapter 11)
  • A Binary Tree with Root 14 Extended Description (Chapter 11)
  • A Binary Tree with Root 11 Extended Description (Chapter 11)
  • A Binary Tree with Root 9 Extended Description (Chapter 11)
  • A Binary Tree with Root 3 Extended Description (Chapter 11)
  • Figure for Exercise 21 A Extended Description (Chapter 11)
  • A Game Tree for a Game of Nim Extended Description (Chapter 11)
  • Figure for Exercise 37 A Extended Description (Chapter 11)
  • Figure for Exercise 37 B Extended Description (Chapter 11)
  • Figure for Exercise 37 C Extended Description (Chapter 11)
  • Figure for Exercise 37 D Extended Description (Chapter 11)
  • A Game Tree for Tic-Tac-Toe Extended Description (Chapter 11)
  • A Game Tree for Tic-Tac-Toe with 3 Levels Extended Description (Chapter 11)
  • A Game Tree for Tic-Tac-Toe with 4 Levels Extended Description (Chapter 11)
  • Figure for Exercise 38 A Extended Description (Chapter 11)
  • Figure for Exercise 38 B Extended Description (Chapter 11)
  • Figure for Exercise 38 C Extended Description (Chapter 11)
  • Figure for Exercise 38 D Extended Description (Chapter 11)
  • Three Game Trees for Tic-Tac-Toe Extended Description (Chapter 11)
  • FIGURE 1 Extended Description (Chapter 11)
  • FIGURE 2 Extended Description (Chapter 11)
  • FIGURE 3 Extended Description (Chapter 11)
  • FIGURE 4 Extended Description (Chapter 11)
  • FIGURE 5 Extended Description (Chapter 11)
  • FIGURE 6 Extended Description (Chapter 11)
  • FIGURE 7 Extended Description (Chapter 11)
  • FIGURE 8 Extended Description (Chapter 11)
  • FIGURE 9 Extended Description (Chapter 11)
  • FIGURE 10 Extended Description (Chapter 11)
  • FIGURE 11 Extended Description (Chapter 11)
  • FIGURE 12 Extended Description (Chapter 11)
  • FIGURE 13 Extended Description (Chapter 11)
  • FIGURE 14 Extended Description (Chapter 11)
  • An Ordered Rooted Tree with 3 Extended Description (Chapter 11)
  • Figure for Exercise 3 Extended Description (Chapter 11)
  • An Ordered Rooted Tree with 6 Extended Description (Chapter 11)
  • Figure for Exercise 8 Extended Description (Chapter 11)
  • Figure for Exercise 9 Extended Description (Chapter 11)
  • Two Binary Trees Extended Description (Chapter 11)
  • An Ordered Rooted Tree with 3 Levels Extended Description (Chapter 11)
  • An Ordered Rooted Tree with 4 Levels Extended Description (Chapter 11)
  • Figure for Exercise 28 Extended Description (Chapter 11)
  • Figure for Exercise 29 Extended Description (Chapter 11)
  • FIGURE 1 Extended Description (Chapter 11)
  • FIGURE 2 Extended Description (Chapter 11)
  • FIGURE 3 Extended Description (Chapter 11)
  • FIGURE 4 Extended Description (Chapter 11)
  • FIGURE 5 Extended Description (Chapter 11)
  • FIGURE 6 Extended Description (Chapter 11)
  • FIGURE 7 Extended Description (Chapter 11)
  • FIGURE 8 Extended Description (Chapter 11)
  • FIGURE 9 Extended Description (Chapter 11)
  • FIGURE 10 Extended Description (Chapter 11)
  • FIGURE 11 Extended Description (Chapter 11)
  • FIGURE 12 Extended Description (Chapter 11)
  • FIGURE 13 Extended Description (Chapter 11)
  • FIGURE 14 Extended Description (Chapter 11)
  • A Graph with 5 Vertices and 6 Edges Extended Description (Chapter 11)
  • Tree with 7 Vertices Extended Description (Chapter 11)
  • Figure for Exercise 4 Extended Description (Chapter 11)
  • Figure for Exercise 5 Extended Description (Chapter 11)
  • A Tree with 12 Vertices Extended Description (Chapter 11)
  • Figure for Exercise 6 Extended Description (Chapter 11)
  • Tree with 8 Vertices Extended Description (Chapter 11)
  • A Tree with 8 Vertices Extended Description (Chapter 11)
  • Tree with 6 Vertices Extended Description (Chapter 11)
  • 16 Trees with 8 Vertices Each Extended Description (Chapter 11)
  • A Tree with 10 Vertices Extended Description (Chapter 11)
  • Figure for Exercise 14 Extended Description (Chapter 11)
  • A Tree with 20 Vertices Extended Description (Chapter 11)
  • Figure for Exercise 23 Extended Description (Chapter 11)
  • A 5 by 5 Chessboard Extended Description (Chapter 11)
  • A 6 by 6 Chessboard Extended Description (Chapter 11)
  • Figure for Exercise 30 Extended Description (Chapter 11)
  • A Graph with Vertices A to L Extended Description (Chapter 11)
  • FIGURE 1 Extended Description (Chapter 11)
  • FIGURE 2 Extended Description (Chapter 11)
  • FIGURE 3 Extended Description (Chapter 11)
  • FIGURE 4 Extended Description (Chapter 11)
  • FIGURE 5 Extended Description (Chapter 11)
  • Figure for Exercise 1 Extended Description (Chapter 11)
  • Figure for Exercise 2 Extended Description (Chapter 11)
  • Figure for Exercise 3 Extended Description (Chapter 11)
  • Figure for Exercise 4 Extended Description (Chapter 11)
  • Tree with 5 Vertices Consequently Extended Description (Chapter 11)
  • A Tree Shows 5 Vertices Consequently Extended Description (Chapter 11)
  • A Tree with 16 Vertices Extended Description (Chapter 11)
  • Tree with 12 Vertices Extended Description (Chapter 11)
  • A Tree Shows 5 Vertices Consequently Connected Extended Description (Chapter 11)
  • A Tree with 12 Vertices Arranged Extended Description (Chapter 11)
  • Five Routed Trees Labeled from B 0 To B 4 Extended Description (Chapter 11)
  • Five Routed Trees Labeled from S 0 Extended Description (Chapter 11)
  • Figure for Supplementary Exercise 27 B Extended Description (Chapter 11)
  • Figure for Supplementary Exercise 27 C Extended Description (Chapter 11)
  • A Tree with 6 Vertices Arranged Extended Description (Chapter 11)
  • Figure for Supplementary Exercise 33 Extended Description (Chapter 11)
  • A Tree with 9 Vertices Consequently Connected Extended Description (Chapter 11)
  • A Tree with 5 Vertices Extended Description (Chapter 11)
  • A Tree with 6 Vertices Extended Description (Chapter 11)
  • A Tree with 7 Vertices Extended Description (Chapter 11)
  • Figure for Supplementary Exercise 44 A Extended Description (Chapter 11)
  • Figure for Supplementary Exercise 44 B Extended Description (Chapter 11)
  • FIGURE 1 Extended Description (Chapter 12)
  • A Cube with Highlighted Vertices 010 Extended Description (Chapter 12)
  • A Cube with Highlighted Vertices 110 Extended Description (Chapter 12)
  • A Cube with all Highlighted Vertices Except 111 Extended Description (Chapter 12)
  • A Cube with Highlighted Vertices 111 and 100 Extended Description (Chapter 12)
  • FIGURE 2 Extended Description (Chapter 12)
  • FIGURE 3 Extended Description (Chapter 12)
  • FIGURE 4 Extended Description (Chapter-12)
  • FIGURE 5 Extended Description (Chapter 12)
  • FIGURE 6 Extended Description (Chapter 12)
  • FIGURE 7 Extended Description (Chapter 12)
  • FIGURE 8 Extended Description (Chapter 12)
  • FIGURE 9 Extended Description (Chapter 12)
  • FIGURE 10 Extended Description (Chapter 12)
  • A Logical Circuit with 1 OR Gate Extended Description (Chapter 12)
  • A Logical Circuit with 3 Inverters Extended Description (Chapter 12)
  • A Logical Circuit with 2 Inverters Extended Description (Chapter 12)
  • A Logical Circuit with 4 Inverters Extended Description (Chapter 12)
  • A Logical Circuit with 4 Inverters, and 4 OR Gates Extended Description (Chapter 12)
  • A Logical Circuit with 10 AND Gates Extended Description (Chapter 12)
  • A Circuit with 4 Full Adders Extended Description (Chapter 12)
  • Logical Circuit with 6 Inverters Extended Description (Chapter-12)
  • A Circuit with 4 Inverters Extended Description (Chapter 12)
  • Two Logic Gates Extended Description (Chapter 12)
  • A Circuit with 3 NAND Gates Extended Description (Chapter 12)
  • Circuit with 3 NAND Gates Extended Description (Chapter 12)
  • A Circuit with 5 NAND Gates Extended Description (Chapter 12)
  • Two Circuits with Inputs X and Y Extended Description (Chapter 12)
  • A Logical Circuit with 4 Inverters Extended Description (Chapter 12)
  • FIGURE 1 Extended Description (Chapter 12)
  • FIGURE 2 Extended Description (Chapter 12)
  • FIGURE 3 Extended Description (Chapter 12)
  • FIGURE 4 Extended Description (Chapter 12)
  • FIGURE 5 Extended Description (Chapter 12)
  • FIGURE 6 Extended Description (Chapter 12)
  • FIGURE 7 Extended Description (Chapter 12)
  • FIGURE 8 Extended Description (Chapter 12)
  • FIGURE 9 Extended Description (Chapter 12)
  • FIGURE 10 Extended Description (Chapter 12)
  • FIGURE 11 Extended Description (Chapter 12)
  • K-Map with 2 Rows and 2 Columns Has Entry Extended Description (Chapter 12)
  • K-map with 2 Rows and 2 Columns Has Entries in the Following Cells Extended Description (Chapter 12)
  • K-map with 2 Rows Extended Description (Chapter 12)
  • K-map with 2 Rows and 2 Columns Has Entries in All the Cells Extended Description (Chapter 12)
  • K-Map with 2 Rows and 2 Columns Has Entry in First Row Extended Description (Chapter 12)
  • K-Map with 2 Rows Extended Description (Chapter 12)
  • K-Map with 2 Rows and 2 Columns Has Entry in All the Cells Extended Description (Chapter 12)
  • K-Map with 2 Rows and 4 Columns Shows 1 Entry Extended Description (Chapter 12)
  • A Logical Circuit with 2 AND Gates Extended Description (Chapter 12)
  • A Logical Circuit with 4 AND Gates Extended Description (Chapter 12)
  • A Logical Circuit Extended Description (Chapter 12)
  • A 4-Cube for Exercise 11 in Section 12.4 Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 4 Columns Has Entry in 1 Cell Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 8 Columns Has Entries in 2 Cells Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 8 Columns Has Entries in 4 Cells Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 8 Columns Has Entries in 8 Cells Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 8 Columns Depicts Entries in 8 Cells Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 8 Columns with Entries in 8 Cells Extended Description (Chapter 12)
  • K-Map with 4 Rows and 8 Columns Depicts Entries in 8 Cells Extended Description (Chapter 12)
  • A Circuit with 3 AND Gates and 1 OR Gate Extended Description (Chapter 12)
  • A Logical Circuit with 6 Inverters Extended Description (Chapter 12)
  • A K-Map with 2 Rows and 4 Columns Shows 1 Entry Extended Description (Chapter 12)
  • A K-Map with 2 Rows and 4 Columns Shows Entries in 2 Cells Extended Description (Chapter 12)
  • A K-Map with 2 Rows and 4 Columns Shows Entries in 2 Cells Extended Description (Chapter 12)
  • A K-Map with 2 Rows and 4 Columns Shows Entries in 4 cells Extended Description (Chapter-12)
  • A K-Map with 4 Rows and 4 Columns Extended Description (Chapter 12)
  • A K-Map with 4 Rows and 4 Columns Shows Entries in 8 Cells Extended Description (Chapter 12)
  • K-Map with 4 Rows and 4 Columns Extended Description (Chapter 12)
  • A Logical Circuit with 2 XOR Gates and 1 Inverter Extended Description (Chapter 12)
  • A Logical Circuit with 5 XOR Gates and 3 Inverters Extended Description (Chapter 12)
  • A Circuit with 1 AND Gate and 1 XOR Gate Extended Description (Chapter 12)
  • A Circle Divided into 2 Parts with N Inputs and an Output Extended Description (Chapter 12)
  • A Circle Divided into 2 Parts with 3 Inputs and an Output Extended Description (Chapter 12)
  • FIGURE 1 Extended Description (Chapter 13)
  • A Derivation Tree with Root Sentence Extended Description (Chapter 13)
  • A Derivation Tree for Exercise 23 Extended Description (Chapter 13)
  • A Derivation Tree with Root Labeled Signed Integer Extended Description (Chapter 13)
  • A Derivation Tree with Root Labeled Signed Decimal Number Extended Description (Chapter 13)
  • FIGURE 1 Extended Description (Chapter 13)
  • FIGURE 2 Extended Description (Chapter 13)
  • FIGURE 3 Extended Description (Chapter 13)
  • FIGURE 4 Extended Description (Chapter 13)
  • FIGURE 5 Extended Description (Chapter 13)
  • FIGURE 6 Extended Description (Chapter 13)
  • A State Diagram with 3 States Extended Description (Chapter 13)
  • A State Diagram with 4 States Extended Description (Chapter 13)
  • A State Diagram with 5 States Extended Description (Chapter 13)
  • A State Diagram with 8 States Extended Description (Chapter 13)
  • A State Diagram with 3 States for Exercise 9 Extended Description (Chapter 13)
  • A State Diagram with 4 States for Exercise 11 Extended Description (Chapter 13)
  • A State Diagram with 5 States for Exercise 13 Extended Description (Chapter 13)
  • A State Diagram with 7 States for Exercise 17 Extended Description (Chapter 13)
  • A State Diagram with 8 States for Exercise 19 Extended Description (Chapter 13)
  • A State Diagram with 4 States from S0 Through S3 Extended Description (Chapter 13)
  • A State Diagram with 3 States from S0 Through S2 Extended Description (Chapter 13)
  • State Diagram with 4 States from S0 Through S3 Extended Description (Chapter 13)
  • State Diagram with 3 States from S0 Through S2 Extended Description (Chapter 13)
  • A State Diagram with 4 States for Exercise 25 Extended Description (Chapter 13)
  • FIGURE 1 Extended Description (Chapter 13)
  • FIGURE 2 Extended Description (Chapter 13)
  • FIGURE 3 Extended Description (Chapter 13)
  • FIGURE 4 Extended Description (Chapter 13)
  • FIGURE 5 Extended Description (Chapter 13)
  • FIGURE 6 Extended Description (Chapter 13)
  • FIGURE 7 Extended Description (Chapter 13)
  • FIGURE 8 Extended Description (Chapter 13)
  • State Diagram with 3 States from S0 (Start State) Through S2 Extended Description (Chapter 13)
  • A State Diagram with 3 States Extended Description (Chapter 13)
  • A State Diagram with 3 States from S0 Extended Description (Chapter 13)
  • State Diagram with 3 States from S0 Extended Description (Chapter 13)
  • A State Diagram with 4 States Extended Description (Chapter 13)
  • A State Diagram with 6 States from S0 (Start State) Extended Description (Chapter 13)
  • A State Diagram with 6 States Extended Description (Chapter 13)
  • A State Diagram with 7 States for Exercise 31 Extended Description (Chapter 13)
  • A State Diagram with 4 States for Exercise 35 Extended Description (Chapter 13)
  • A state diagram with 3 states from S0 Extended Description (Chapter 13)
  • State Diagram with 3 States from S0 Extended Description (Chapter 13)
  • A State Diagram with 3 States Extended Description (Chapter 13)
  • A State Diagram with 3 States from S0 (Start State) Extended Description (Chapter 13)
  • A State Diagram with 4 States from S0 (Start State) Extended Description (Chapter 13)
  • A State Diagram with 6 States from S0 (Start State) Extended Description (Chapter 13)
  • A State Diagram with 6 States Extended Description (Chapter 13)
  • A State Diagram with 6 States for Exercise 51 Extended Description (Chapter 13)
  • A State Diagram with 6 States for Exercise 51 Extended Description (Chapter 13)
  • A State Diagram with 6 States from S0 Extended Description (Chapter 13)
  • Figure 2 Extended Description (Chapter 13)
  • Figure 3 Extended Description (Chapter 13)
  • FIGURE 4 Extended Description (Chapter 13)
  • FIGURE 5 Extended Description (Chapter 13)
  • FIGURE 6 Extended Description (Chapter 13)
  • A State Diagram with 6 States for Exercise 13 Extended Description (Chapter 13)
  • A State Diagram with 3 States from S0 Extended Description (Chapter 13)
  • A State Diagram with 3 States from S0 (Start State) Extended Description (Chapter 13)
  • A State Diagram with 4 States from S0 (Start State) Through S3 Extended Description (Chapter 13)
  • FIGURE 1 Extended Description (Chapter 13)
  • FIGURE 2 Extended Description (Chapter 13)
  • Three Derivation Trees for Supplementary Exercise 3 Extended Description (Chapter 13)
  • Two Derivation Trees for Supplementary Exercise 5 Extended Description (Chapter 13)
  • A State Diagram with 5 States from S0 Extended Description (Chapter 13)
  • A State Diagram with 5 States from S0 (Start State) Through S4 Extended Description (Chapter 13)
  • State Diagram with 5 States from S0 (Start State) Through S4 Extended Description (Chapter 13)
  • A State Diagram with 5 States from S0 Extended Description (Chapter 13)
  • A State Diagram with 16 Sets of States for Supplementary Exercise 19 Extended Description (Chapter 13)
  • A State Diagram with 3 States for Supplementary Exercise 21 Extended Description (Chapter 13)
  • A State Diagram with 7 States for Supplementary Exercise 25 Extended Description (Chapter 13)
  • A State Diagram with 4 States Extended Description (Chapter 13)
  • A State Diagram with 7 States from S0 Through S6 Extended Description (Chapter 13)
  • A State Diagram with 4 States from S0 Through S3 Extended Description (Chapter 13)
  • FIGURE 1 Extended Description (Appendix-2)
  • FIGURE 2 Extended Description (Appendix-2)
  • A Concave up Increasing Curve Extended Description (Appendix-2)
  • A Tree Diagram Showing the Alternative Chapters Extended Description (FM)
  • A Logical Circuit with 4 Inverters Extended Description (Answers)
  • Four 8 By 8 Squares Filled with Tetrominoes for Exercise 51 B Extended Description (Answers)
  • A Venn Diagram with Intersecting Circles Extended Description (Answers)
  • A Venn Diagram with Intersecting Circles Labeled A, B, and C Extended Description (Answers)
  • A Graph with 7 Horizontal Segments Extended Description (Answers)
  • A Graph with 5 Horizontal Segments Extended Description (Answers)
  • A Graph Depicts 7 Horizontal Extended Description (Answers)
  • Graph Depicts 7 Horizontal Extended Description (Answers)
  • Graph with 7 Horizontal Segments Extended Description (Answers)
  • Graph with 8 Horizontal Segments Extended Description (Answers)
  • A Graph with 4 Segments and 5 Points Extended Description (Answers)
  • A Graph with 5 Segments and 2 Points Extended Description (Answers)
  • Graph of 3 Functions Extended Description (Answers)
  • The Graphs of the Functions X Squared Extended Description (Answers)
  • Tiling of an 8 by 8 Checkerboard Using Right Extended Description (Answers)
  • Tiling of an 8 by 8 Checkerboard Extended Description (Answers)
  • A Merge Sort of b, d, a Extended Description (Answers)
  • A Directed Graph with 4 Vertices and 6 Edges for Exercise 19 Extended Description (Answers)
  • A Directed Graph with 4 Vertices and 12 Edges for Exercise 19 C Extended Description (Answers)
  • A Directed Graph with 4 Vertices and 4 Edges for Exercise 19 D Extended Description (Answers)
  • A Directed Graph with 4 Vertices Extended Description (Answers)
  • A Directed Graph with 4 Vertices, 6 Edges and 4 Loops Extended Description (Answers)
  • A Directed Graph with 4 Vertices, 7 Edges and 4 Loops Extended Description (Answers)
  • A Directed Graph with 4 Vertices, 8 Edges Extended Description (Answers)
  • A Directed Graph with 4 Vertices, 8 Edges Extended Description (Answers)
  • An Undirected Graph with 8 Vertices and 8 Edges Extended Description (Answers)
  • An Undirected Graph with 5 Vertices and 5 Edges Extended Description (Answers)
  • An Undirected Graph with 5 Vertices and 18 Edges Extended Description (Answers)
  • A Directed Graph with 5 Vertices and 9 Edges Extended Description (Answers)
  • A Directed Graph with 5 Vertices and 20 Edges Extended Description (Answers)
  • An Undirected Graph with 5 Vertices Extended Description (Answers)
  • An Undirected Graph with 6 Vertices and 13 EdgesExtended Description (Answers)
  • A Niche Overlap Graph with 6 Vertices Extended Description (Answers)
  • A Graph with 18 Vertices and 24 Undirected Edge Extended Description (Answers)
  • An Influence Graph with 5 Vertices Extended Description (Answers)
  • A Semantic Network of Nouns with Similar Extended Description (Answers)
  • A Directed Graph with 4 Vertices and 6 Edges Extended Description (Answers)
  • A Precedence Graph with 7 Vertices Extended Description (Answers)
  • A Graph with 6 Vertices Extended Description (Answers)
  • A Graph with 6 Vertices and 7 Undirected Extended Description (Answers)
  • A Graph with 5 Vertices and 7 Undirected Extended Description (Answers)
  • A Graph with 5 Vertices and 4 Undirected Edges Extended Description (Answers)
  • 34 Graphs for Exercise 53 in Section 10.2 Extended Description (Answers)
  • A Graph with 7 vertices and 10 Undirected Edges Extended Description (Answers)
  • A Graph with 5 Vertices, 7 Directed Edges Extended Description (Answers)
  • A Graph with 5 Vertices Extended Description (Answers)
  • A Graph with 6 Vertices, 9 Directed Edges Extended Description (Answers)
  • A Graph with 9 Vertices, Labeled from P(0, 0) to P(2, 2) Extended Description (Answers)
  • A Graph with 4 Vertices and 9 Directed Edges Extended Description (Answers)
  • An Undirected Graph with 4 Vertices, 7 Edges Extended Description (Answers)
  • A Directed Graph with 3 Vertices, 7 Edges and 3 Loops Extended Description (Answers)
  • An Undirected Graph with 12 Vertices and 14 Edges Extended Description (Answers)
  • An Undirected Weighted Graph with 3 Vertices and 3 Edges Extended Description (Answers)
  • An Undirected Weighted Graph with 5 Vertices and 6 Edges Extended Description (Answers)
  • An Undirected Graph with 6 Vertices and 11 Edges Extended Description (Answers)
  • A Two Dimensional Representation of an Undirected Graph Extended Description (Answers)
  • An Undirected Graph with 5 Vertices and 9 Edges Extended Description (Answers)
  • An Undirected Graph with 6 Vertices and 8 Edges Extended Description (Answers)
  • A pentagon A Extended Description (Answers)
  • An Undirected Graph with 9 Vertices and 27 Edges Extended Description (Answers)
  • An Undirected Graph with 16 Vertices and 78 Edges Extended Description (Answers)
  • A Complete Binary Tree of Height 3 for Exercise 27 Extended Description (Answers)
  • The First Seven Rooted Fibonacci Trees for Exercise Extended Description (Answers)
  • A Binary Search Tree of Height 4 with 7 Vertices Extended Description (Answers)
  • A Binary Search Tree of Height 6 with 8 Vertices Extended Description (Answers)
  • An Ordered Binary Tree with Root 17 Extended Description (Answers)
  • A Game Tree for a Game of Nim for Exercise 33 Extended Description (Answers)
  • A Game Tree for Tic-Tac-Toe with 3 Levels Extended Description (Answers)
  • A Game Tree Shows Tic-Tac-Toe with 3 Levels Extended Description (Answers)
  • A Game Tree Shows Tic-Tac-Toe with 4 Levels Extended Description (Answers)
  • Three Game Trees for Tic-Tac-Toe Extended Description (Answers)
  • An Ordered Rooted Tree with 3 Levels and 6 Vertices Extended Description (Answers)
  • An Ordered Rooted Tree with 6 Levels and 16 Vertices Extended Description (Answers)
  • Two Binary Trees Extended Description (Answers)
  • An Ordered Rooted Tree with 3 Levels and 9 Vertices Extended Description (Answers)
  • An Ordered Rooted Tree with 4 Levels and 12 Vertices Extended Description (Answers)
  • Tree with 7 Vertices Extended Description (Answers)
  • A Tree with 12 Vertices Extended Description (Answers)
  • Tree with 8 Vertices Extended Description (Answers)
  • A Tree with 8 Vertices Extended Description (Answers)
  • Tree with 6 Vertices Extended Description (Answers)
  • 16 Trees with 8 Vertices each Extended Description (Answers)
  • A Tree with 10 Vertices Extended Description (Answers)
  • A Tree with 20 Vertices Extended Description (Answers)
  • A 5 by 5 Chessboard with 5 Queens on it Extended Description (Answers)
  • A 6 by 6 Chessboard with 6 Queens on it Extended Description (Answers)
  • A Graph with Vertices a to l Extended Description (Answers)
  • Tree with 5 Vertices Consequently Connected to one Another Extended Description (Answers)
  • A Tree Shows 5 Vertices Consequently Connected Extended Description (Answers)
  • A Tree with 16 Vertices Arranged in a 4 by 4 Square Matrix Extended Description (Answers)
  • Tree with 12 Vertices Extended Description (Answers)
  • A Tree with 5 Vertices Consequently Connected to one Another Extended Description (Answers)
  • A Tree with 12 Vertices Arranged in a 3 by 4 Matrix Extended Description (Answers)
  • Five Routed Trees Labeled from B 0 to B 4 Extended Description (Answers)
  • Five Routed Trees Labeled from S 0 to S 4 Extended Description (Answers)
  • A Tree with 6 Vertices Arranged in a 2 by 3 Matrix Extended Description (Answers)
  • A Tree with 9 Vertices Consequently Connected Extended Description (Answers)
  • A Tree with 5 Vertices Extended Description (Answers)
  • A Tree with 6 Vertices Extended Description (Answers)
  • A Tree with 7 Vertices Extended Description (Answers)
  • A Cube with Highlighted Vertices Extended Description (Answers)
  • A Cube with Highlighted Vertices 110 Extended Description (Answers)
  • A Cube with all Highlighted Vertices Except 111 Extended Description (Answers)
  • A Cube with Highlighted Vertices 111 and 100 Extended Description (Answers)
  • A Logical Circuit with 10 AND Gates, and 1 OR Gate Extended Description (Answers)
  • A Circuit with 4 Full Adders and 1 Half Adder Extended Description (Answers)
  • Logical Circuit with 6 Inverters Extended Description (Answers)
  • A Circuit with 4 Inverters Extended Description (Answers)
  • A Circuit with 3 NAND Gates Extended Description (Answers)
  • Circuit with 3 NAND Gates Extended Description (Answers)
  • A Circuit with 5 NAND Gates Extended Description (Answers)
  • Two Circuits with Inputs x and y Extended Description (Answers)
  • A Logical Circuit with 4 Inverters Extended Description (Answers)
  • K-map with 2 Rows and 2 Columns has Entry Extended Description (Answers)
  • K-map with 2 Rows and 2 Columns has Entry in Second Row Extended Description (Answers)
  • K-map with 2 Rows and 2 Columns Extended Description (Answers)
  • K-map with 2 Rows and 2 Columns Has Entries Extended Description (Answers)
  • K-map With 2 Rows and 4 Columns Shows 1 Entry Extended Description (Answers)
  • A K-map With 2 Rows and 4 Columns Shows 1 Entry Extended Description (Answers)
  • A K-map With 2 Rows and 4 Columns Shows 1 Entries Extended Description (Answers)
  • A K-map with 2 Rows and 4 Columns Shows Entries in 4 Cells Extended Description (Answers)
  • A 4-Cube for Exercise 11 in Section 12.4 Extended Description (Answers)
  • A K-map with 4 Rows and 4 Columns has Entry in 1 Cell Extended Description (Answers)
  • A K-map with 4 Rows and 8 Columns Has Entries in 2 Cells Extended Description (Answers)
  • A K-map with 4 Rows and 8 Columns Has Entries in 4 Cells Extended Description (Answers)
  • A K-map with 4 Rows and 8 Columns Has Entries in 8 Cells Extended Description (Answers)
  • A K-map with 4 Rows and 8 Columns Depicts Entries in 8 Cells Extended Description (Answers)
  • A K-map with 4 Rows and 8 Columns with Entries in 8 Cells Extended Description (Answers)
  • K-map with 4 Rows and 8 Columns Depicts Entries in 8 Cells Extended Description (Answers)
  • A Circuit with 3 AND Gates and 1 OR Gate Extended Description (Answers)
  • A Logical Circuit with 6 Inverters Extended Description (Answers)
  • A Circuit with 1 AND Gate and 1 XOR Gate Extended Description (Answers)
  • A Derivation Tree for Exercise 23 A in Section 13.1 Extended Description (Answers)
  • A Derivation Tree for Exercise 23 B in Section 13.1 Extended Description (Answers)
  • A Derivation Tree for Exercise 23 C in Section 13.1 Extended Description (Answers)
  • A Derivation Tree with Root Labeled Sentence Extended Description (Answers)
  • A Derivation Tree with Root Labeled Signed Integer Extended Description (Answers)
  • A Derivation Tree with Root Labeled Signed Decimal Number Extended Description (Answers)
  • A State Diagram with 3 States for Exercise 42 A Extended Description (Answers)
  • A State Diagram with 4 States for Exercise 1 B Extended Description (Answers)
  • A State Diagram with 5 States for Exercise 1 C in Section 13.2 Extended Description (Answers)
  • A State Diagram with 8 States for Exercise 7 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 5 States for Exercise 9 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 4 States for Exercise 11 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 5 States for Exercise 13 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 7 States for Exercise 17 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 8 States for Exercise 19 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 4 States for Exercise 25 in Section 13.2 Extended Description (Answers)
  • A State Diagram with 7 States for Exercise 31 in Section 13.3 Extended Description (Answers)
  • A State Diagram with 4 States for Exercise 35 in Section 13.3 Extended Description (Answers)
  • A State Diagram with 6 States for Exercise 51 in Section 13.3 Extended Description (Answers)
  • A State Diagram with 3 States for Exercise 55 A in section 13 Extended Description (Answers)
  • A State Diagram with 5 States for Exercise 55 B in Section 13.3 Extended Description (Answers)
  • A State Diagram with 4 States for Exercise 55 C in Section 13.3 Extended Description (Answers)
  • A State Diagram with 6 States for Exercise 13 A in Section 13.4 Extended Description (Answers)
  • A State Diagram with 8 States for Exercise 13 B in Section 13.4 Extended Description (Answers)
  • A State Diagram with 13 States for Exercise 13 C in Section 13.4 Extended Description (Answers)
  • Three Derivation Trees for Supplementary Exercise 3 Extended Description (Answers)
  • Two Derivation Trees for Supplementary Exercise 5 Extended Description (Answers)
  • A State Siagram with 5 States from S0 (Start State) Through S4 Extended Description (Answers)
  • A State Diagram with 5 States from S0 (Start State) to S4 (Final State) Extended Description (Answers)
  • A State Diagram with 16 Sets of States for Supplementary Exercise 19 Extended Description (Answers)
  • A State Diagram with 3 States for Supplementary Exercise 21 A Extended Description (Answers)
  • A State Diagram with 6 States for Supplementary Exercise 21 B Extended Description (Answers)
  • A State Diagram with 3 States and 4 Edges for Supplementary Exercise 21 C Extended Description (Answers)
  • A State Diagram with 7 States for Supplementary Exercise 25 A Extended Description (Answers)
  • A State Diagram with 8 States and 12 Edges for Supplementary Exercise 25 B Extended Description (Answers)
  • A State Diagram with 21 States for Supplementary Exercise 25 C Extended Description (Answers)
  • A Concave up Increasing Curve Extended Description (Answers)