Discrete Structures, Logic, and Computability

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Edition: 2nd
Format: Hardcover
Pub. Date: 2002-08-01
Publisher(s): Jones & Bartlett
List Price: $101.95

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Summary

Winner at the 46th Annual New England Book Show (2003) in the College Covers & Jackets category This introduction to discrete mathematics prepares future computer scientists, engineers, and mathematicians for success by providing extensive and concentrated coverage of logic, functions, algorithmic analysis, and algebraic structures.  Discrete Mathematics, Second Edition illustrates the relationships between key concepts through its thematic organization and provides a seamless transition between subjects.  Distinct for the depth with which it covers logic, this text emphasizes problem solving and the application of theory as it carefully guides the reader from basic to more complex topics.  Discrete Mathematics is an ideal resource for discovering the fundamentals of discrete math.     Discrete Mathematics, Second Edition is designed for an introductory course in discrete mathematics for the prospective computer scientist, applied mathematician, or engineer who wants to learn how the ideas apply to computer sciences. The choice of topics-and the breadth of coverage-reflects the desire to provide students with the foundations needed to successfully complete courses at the upper division level in undergraduate computer science courses.This book differs in several ways from current books about discrete mathematics. It presents an elementary and unified introduction to a collection of topics that has not been available in a single source. A major feature of the book is the unification of the material so that it does not fragment into a collection of seemingly unrelated ideas.

Table of Contents

Elementary Notions and Notations
1(73)
A Proof Primer
2(11)
Logical Statements
2(3)
Something to Talk About
5(1)
Proof Techniques
6(6)
Exercises
12(1)
Sets
13(22)
Definition of a Set
13(5)
Operations on Sets
18(8)
Counting Finite Sets
26(3)
Bags (Multisets)
29(1)
Sets Should Not Be Too Complicated
30(1)
Exercises
31(4)
Ordered Structures
35(20)
Tuples
35(4)
Lists
39(2)
Strings and Languages
41(5)
Relations
46(3)
Counting Tuples
49(3)
Exercises
52(3)
Graphs and Trees
55(17)
Definition of a Graph
55(4)
Paths and Graphs
59(2)
Graph Traversals
61(2)
Trees
63(5)
Spanning Trees
68(2)
Exercises
70(2)
Chapter Summary
72(1)
Facts about Functions
73(54)
Definitions and Examples
74(17)
Definition of a Function
74(5)
Some Useful Functions
79(8)
Partial Functions
87(1)
Exercises
88(3)
Constructing Functions
91(9)
Composition of Functions
91(5)
The Map Function
96(2)
Exercises
98(2)
Properties of Functions
100(15)
Injections and Surjections
100(2)
Bijections and Inverses
102(3)
The Pigeonhole Principle
105(1)
Simple Ciphers
106(3)
Hash Functions
109(2)
Exercises
111(4)
Countability
115(10)
Comparing the Size of Sets
115(1)
Sets that Are Countable
116(3)
Diagonalization
119(2)
Limits on Computability
121(3)
Exercises
124(1)
Chapter Summary
125(2)
Construction Techniques
127(66)
Inductively Defined Sets
128(17)
Numbers
129(3)
Strings
132(2)
Lists
134(4)
Binary Trees
138(2)
Cartesian Products of Sets
140(2)
Exercises
142(3)
Recursive Functions and Procedures
145(28)
Numbers
146(4)
Strings
150(3)
Lists
153(6)
Binary Trees
159(4)
Two More Problems
163(2)
Infinite Sequences
165(3)
Exercises
168(5)
Grammars
173(18)
Recalling English Grammar
173(1)
Structure of Grammars
174(3)
Derivations
177(4)
Constructing Grammars
181(5)
Meaning and Ambiguity
186(2)
Exercises
188(3)
Chapter Summary
191(2)
Equivalence, Order, and Inductive Proof
193(80)
Properties of Binary Relations
194(19)
Composition of Relations
195(4)
Closures
199(5)
Path Problems
204(5)
Exercises
209(4)
Equivalence Relations
213(19)
Definition and Examples
214(4)
Equivalence Classes
218(1)
Partitions
219(6)
Generating Equivalence Relations
225(4)
Exercises
229(3)
Order Relations
232(21)
Partial Orders
233(6)
Topological Sorting
239(3)
Well-Founded Orders
242(8)
Ordinal Numbers
250(1)
Exercises
251(2)
Inductive Proof
253(19)
Proof by Mathematical Induction
253(6)
Proof by Well-Founded Induction
259(2)
A Variety of Examples
261(6)
Exercises
267(5)
Chapter Summary
272(1)
Analysis Techniques
273(72)
Analyzing Algorithms
274(7)
Worst-Case Running Time
274(3)
Decision Trees
277(4)
Exercises
281(1)
Finding Closed Forms
281(8)
Closed Forms for Sums
282(5)
Exercises
287(2)
Counting and Discrete Probability
289(23)
Permutations (Order Is Important)
289(4)
Combinations (Order Is Not Important)
293(5)
Discrete Probability
298(11)
Exercises
309(3)
Solving Recurrences
312(22)
Solving Simple Recurrences
313(6)
Generating Functions
319(13)
Exercises
332(2)
Comparing Rates of Growth
334(8)
Big Theta
334(4)
Little Oh
338(1)
Big Oh and Big Omega
339(2)
Exercises
341(1)
Chapter Summary
342(3)
Elementary Logic
345(52)
How Do We Reason?
346(2)
What Is a Calculus?
347(1)
How Can We Tell Whether Something Is a Proof?
348(1)
Propositional Calculus
348(21)
Well-Formed Formulas and Semantics
349(4)
Equivalence
353(5)
Truth Functions and Normal Forms
358(7)
Complete Sets of Connectives
365(2)
Exercises
367(2)
Formal Reasoning
369(15)
Inference Rules
370(2)
Formal Proof
372(8)
Proof Notes
380(1)
Exercises
381(3)
Formal Axiom Systems
384(10)
An Example Axiom System
384(7)
Other Axiom Systems
391(1)
Exercises
392(2)
Chapter Summary
394(3)
Predicate Logic
397(60)
First-Order Predicate Calculus
397(19)
Predicates and Quantifiers
398(4)
Well-Formed Formulas
402(2)
Semantics and Interpretations
404(5)
Validity
409(4)
The Validity Problem
413(1)
Exercises
413(3)
Equivalent Formulas
416(16)
Equivalence
416(8)
Normal Forms
424(3)
Formalizing English Sentences
427(2)
Summary
429(1)
Exercises
430(2)
Formal Proofs in Predicate Calculus
432(24)
Universal Instantiation (UI)
433(4)
Existential Generalization (EG)
437(1)
Existential Instantiation (EI)
438(2)
Universal Generalization (UG)
440(3)
Examples of Formal Proofs
443(7)
Summary of Quantifier Proof Rules
450(1)
Exercises
451(5)
Chapter Summary
456(1)
Applied Logic
457(48)
Equality
458(8)
Describing Equality
458(6)
Extending Equals for Equals
464(1)
Exercises
465(1)
Program Correctness
466(25)
Imperative Program Correctness
467(11)
Array Assignment
478(4)
Termination
482(4)
Exercises
486(5)
Higher-Order Logics
491(12)
Classifying Higher-Order Logics
492(4)
Semantics
496(2)
Higher-Order Reasoning
498(3)
Exercises
501(2)
Chapter Summary
503(2)
Computational Logic
505(52)
Automatic Reasoning
505(28)
Clauses and Clausal Forms
506(6)
Resolution for Propositions
512(2)
Substitution and Unification
514(7)
Resolution: The General Case
521(5)
Theorem Proving with Resolution
526(3)
Remarks
529(1)
Exercises
530(3)
Logic Programming
533(22)
Family Trees
534(2)
Definition of a Logic Program
536(1)
Resolution and Logic Programming
537(12)
Logic Programming Techniques
549(4)
Exercises
553(2)
Chapter Summary
555(2)
Algebraic Structures and Techniques
557(76)
What Is an Algebra?
558(14)
Definition of an Algebra
560(2)
Concrete Versus Abstract
562(2)
Working in Algebras
564(6)
Exercises
570(2)
Boolean Algebra
572(13)
Simplifying Boolean Expressions
574(4)
Digital Circuits
578(5)
Exercises
583(2)
Abstract Data Types as Algebras
585(16)
Natural Numbers
585(4)
Lists and Strings
589(3)
Stacks and Queues
592(4)
Binary Trees and Priority Queues
596(3)
Exercises
599(2)
Computational Algebras
601(12)
Relational Algebras
601(6)
Functional Algebras
607(4)
Exercises
611(2)
Other Algebraic Ideas
613(19)
Congruence
613(3)
Cryptology: The RSA Algorithm
616(5)
Subalgebras
621(2)
Morphisms
623(6)
Exercises
629(3)
Chapter Summary
632(1)
Answers to Selected Exercises 633(74)
Bibliography 707(4)
Greek Alphabet 711(2)
Symbol Glossary 713(6)
Index 719

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