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Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

ISBN: 9780716743583 | 0716743582
Edition: 5th
Format: Hardcover
Publisher: W. H. Freeman
Pub. Date: 7/19/2002

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SummaryTable of Contents
New edition of the classic discrete mathematics text for computer science majors.
http://www.netread.com/jcusers2/bk1388/583/9780716743583/image/lgcover.9780716743583.jpg
Prefacexvii
Note to the Studentxx
Formal Logic
1(82)
Statements, Symbolic Representation, and Tautologies
... MORE2(18)
Connectives and Truth Values
2(6)
Tautologies
8(2)
Logical Connectives in the Real World
10(1)
An Algorithm
11(3)
Exercises 1.1
14(6)
Propositional Logic
20(13)
Valid Arguments
20(2)
Derivation Rules for Propositional Logic
22(4)
Deduction Method and Other Rules
26(2)
Verbal Arguments
28(2)
Exercises 1.2
30(3)
Quantifiers, Predicates, and Validity
33(12)
Quantifiers and Predicates
33(3)
Translation
36(3)
Validity
39(2)
Exercises 1.3
41(4)
Predicate Logic
45(15)
Derivation Rules for Predicate Logic
46(1)
Universal Instantiation
47(1)
Existential Instantiation
48(1)
Universal Generalization
49(1)
Existential Generalization
50(1)
More Work with Rules
50(5)
Verbal Arguments
55(1)
Conclusion
55(2)
Exercises 1.4
57(3)
Logic Programming
60(10)
Prolog
60(2)
Horn Clauses and Resolution
62(4)
Recursion
66(2)
Expert Systems
68(1)
Exercises 1.56
69(1)
Proof of Correctness
70(13)
Assertions
71(1)
Assignment Rule
72(3)
Conditional Rule
75(3)
Exercises 1.6
78(2)
Chapter 1 Review
80(2)
On the Computer
82(1)
Proofs, Recursion, and Analysis of Algorithms
83(79)
Proof Techniques
84(10)
Theorems and Informal Proofs
84(1)
To Prove or Not to Prove
85(1)
Exhaustive Proof
86(1)
Direct Proof
87(1)
Contraposition
88(1)
Contradiction
89(2)
Serendipity
91(1)
Exercises 2.1
92(2)
Induction
94(15)
First Principle of Induction
94(2)
Proofs by Mathematical Induction
96(5)
Second Principle of Induction
101(4)
Exercises 2.2
105(4)
More on Proof of Correctness
109(11)
Loop Rule
110(3)
Euclidean Algorithm
113(3)
Exercises 2.3
116(4)
Recursion and Recurrence Relations
120(28)
Recursive Definitions
120(1)
Recursively Defined Sequences
121(2)
Recursively Defined Sets
123(2)
Recursively Defined Operations
125(1)
Recursively Defined Algorithms
126(5)
Solving Recurrence Relations
131(8)
Exercises 2.4
139(9)
Analysis of Algorithms
148(14)
Analysis Using Recurrence Relations (Binary Search)
149(4)
Upper Bound (Euclidean Algorithm)
153(1)
Exercises 2.5
154(3)
Chapter 2 Review
157(2)
On the Computer
159(3)
Sets, Combinatorics, and Probability
162(84)
Sets
163(25)
Notation
163(2)
Relationships Between Sets
165(3)
Sets of Sets
168(1)
Binary and Unary Operations
168(2)
Operations on Sets
170(3)
Set Identities
173(2)
Countable and Uncountable Sets
175(3)
Exercises 3.1
178(10)
Counting
188(13)
Multiplication Principle
188(4)
Addition Principle
192(1)
Using the Principles Together
193(2)
Decision Trees
195(1)
Exercises 3.2
196(5)
Principle of Inclusion and Exclusion; Pigeonhole Principle
201(8)
Principle of Inclusion and Exclusion
201(5)
Pigeonhole Principle
206(1)
Exercises 3.3
207(2)
Permutations and Combinations
209(14)
Permutations
209(2)
Combinations
211(3)
Eliminating Duplicates
214(1)
Permutations and Combinations with Repetitions
215(2)
Exercises 3.4
217(6)
Probability
223(13)
Introduction to Finite Probability
223(2)
Probability Distributions
225(2)
Conditional Probability
227(1)
Expected Value
228(2)
Average Case Analysis of Algorithms
230(2)
Exercises 3.5
232(4)
Binomial Theorem
236(10)
Pascal's Triangle
236(2)
Binomial Theorem and Its Proof
238(1)
Applying the Binomial Theorem
239(1)
Exercises 3.6
240(2)
Chapter 3 Review
242(2)
On the Computer
244(2)
Relations, Functions, and Matrices
246(93)
Relations
247(24)
Binary Relations
247(3)
Properties of Relations
250(2)
Closures of Relations
252(2)
Partial Orderings
254(2)
Equivalence Relations
256(6)
Exercises 4.1
262(9)
Topological Sorting
271(7)
Exercises 4.2
276(2)
Relations and Databases
278(11)
Entity-Relationship Model
278(1)
Relational Model
279(3)
Operations on Relations
282(3)
Null Values and Three-Valued Logic
285(1)
Database Integrity
286(1)
Exercises 4.3
287(2)
Functions
289(32)
Definition
289(6)
Properties of Functions
295(1)
Onto Functions
295(1)
One-to-One Functions
296(1)
Bijections
297(1)
Composition of Functions
297(2)
Inverse Functions
299(1)
Permutation Functions
300(3)
How Many Functions
303(3)
Equivalent Sets
306(1)
Order of Magnitude of Functions
307(5)
Exercises 4.4
312(9)
Matrices
321(18)
Terminology
321(2)
Matrix Operations
323(4)
Boolean Matrices
327(2)
Exercises 4.5
329(5)
Chapter 4 Review
334(2)
On the Computer
336(3)
Graphs and Trees
339(71)
Graphs and Their Representations
340(31)
Definitions of a Graph
340(3)
Applications of Graphs
343(2)
Graph Terminology
345(4)
Isomorphic Graphs
349(3)
Planar Graphs
352(5)
Computer Representation of Graphs
357(1)
Adjacency Matrix
357(2)
Adjacency List
359(2)
Exercises 5.1
361(10)
Trees and Their Representations
371(16)
Tree Terminology
371(1)
Applications of Trees
372(2)
Binary Tree Representation
374(1)
Tree Traversal Algorithms
375(5)
Results About Trees
380(1)
Exercises 5.2
381(6)
Decision Trees
387(10)
Searching
388(2)
Lower Bounds on Searching
390(1)
Binary Tree Search
391(1)
Sorting
392(2)
Exercises 5.3
394(3)
Huffman Codes
397(13)
Problem and Trial Solution
397(2)
Huffman Encoding Algorithm
399(2)
Justification
401(2)
Applications of Huffman Codes
403(1)
Exercises 5.4
404(3)
Chapter 5 Review
407(1)
On the Computer
408(2)
Graph Algorithms
410(51)
Directed Graphs and Binary Relations; Warshall's Algorithm
411(14)
Directed Graphs and Binary Relations
412(2)
Reachability
414(4)
Warshall's Algorithm
418(4)
Exercises 6.1
422(3)
Euler Path and Hamiltonian Circuit
425(8)
Euler Path Problem
425(4)
Hamiltonian Circuit Problem
429(1)
Exercises 6.2
430(3)
Shortest Path and Minimal Spanning Tree
433(14)
Shortest-Path Problem
433(6)
Minimal Spanning Tree Problem
439(2)
Exercises 6.3
441(6)
Traversal Algorithms
447(14)
Depth-First Search
447(2)
Breadth-First Search
449(3)
Analysis
452(1)
Applications
453(3)
Exercises 6.4
456(2)
Chapter 6 Review
458(1)
On the Computer
459(2)
Boolean Algebra and Computer Logic
461(57)
Boolean Algebra Structure
462(17)
Models or Abstractions
463(1)
Definition and Properties
464(4)
Isomorphic Boolean Algebras
468(1)
What Is Isomorphism?
468(3)
Isomorphism as Applied to Boolean Algebra
471(3)
Exercises 7.1
474(5)
Logic Networks
479(21)
Combinational Networks
479(1)
Basic Logic Elements
479(1)
Boolean Expressions
480(1)
Truth Functions
481(1)
Networks and Expressions
481(2)
Canonical Form
483(2)
Minimization
485(2)
Programmable Logic Arrays
487(2)
A Useful Network
489(2)
Other Logic Elements
491(1)
Constructing Truth Functions
492(1)
Exercises 7.2
493(7)
Minimization
500(18)
Minimization Process
500(1)
Karnaugh Map
501(1)
Maps for Three and Four Variables
502(2)
Using the Karnaugh Map
504(4)
Quine-McCluskey Procedure
508(4)
Exercises 7.3
512(3)
Chapter 7 Review
515(1)
On the Computer
516(2)
Modeling Arithmetic, Computation, and Languages
518(94)
Algebraic Structures
519(25)
Definitions and Examples
519(8)
Basic Results about Groups
527(3)
Subgroups
530(3)
Isomorphic Groups
533(6)
Exercises 8.1
539(5)
Finite-State Machines
544(28)
Definition
544(1)
Examples of Finite-State Machines
545(3)
Recognition
548(2)
Regular Sets and Kleene's Theorem
550(2)
Machine Minimization
552(1)
Unreachable States
552(1)
Minimization Procedure
553(5)
Sequential Networks and Finite-State Machines
558(4)
Exercises 8.2
562(10)
Turing Machines
572(21)
Definition
573(4)
Turing Machines as Set Recognizers
577(3)
Turing Machines as Function Computers
580(2)
Church-Turing Thesis
582(2)
Decision Problems and Uncomputability
584(1)
Examples of Decision Problems
584(2)
Halting Problem
586(1)
Computational Complexity
587(3)
Exercises 8.3
590(3)
Formal Languages
593(19)
Classes of Grammars
599(3)
Formal Languages and Computational Devices
602(1)
Context-Free Grammars
603(3)
Exercises 8.4
606(2)
Chapter 8 Review
608(2)
On the Computer
610(2)
Appendix A Summation Notation612(3)
Appendix B The Logarithm Function615(4)
Answers to Practice Problems619(34)
Answers to Selected Exercises653(58)
Answers to Self-Tests711(8)
Index719

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