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Mathematics : A Discrete Introduction

ISBN: 9780534398989 | 0534398987
Edition: 2nd
Format: Hardcover
Publisher: Cengage Learning
Pub. Date: 6/6/2005

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SummaryTable of Contents
This book has two primary objectives: It teaches students fundamental concepts in discrete mathematics (from counting to basic cryptography to graph theory), and it teaches students proof-writing skills. With a wealth of learning aids and a clear presentation, the book teaches students not only how to write proofs, but how to think clearly and present cases logically beyond this course. Overall, this book is an introduction to mathematics. In particular, it is an introduction to discrete mathematics. All of the material is directly applicable t... MORE
... MORE
To the Studentxv
How to Read a Mathematics Bookxvi
Exercisesxvii
To the Instructorxix
Audience and Prerequisitesxix
Topics Covered and Navigating the Sectionsxix
Sample Course Outlinesxxi
Special Featuresxxi
What's New in This Second Editionxxiii
Acknowledgmentsxxv
This New Editionxxv
From the First Editionxxv
Fundamentals
1(36)
Joy
1(1)
Why?
1(1)
The Agony and the Ecstasy
2(1)
Exercise
2(1)
Definition
2(6)
Recap
5(1)
Exercises
5(3)
Theorem
8(8)
The Nature of Truth
8(1)
If-Then
9(2)
If and Only If
11(1)
And, Or, and Not
12(1)
What Theorems Are Called
13(1)
Vacuous Truth
14(1)
Recap
14(1)
Exercises
15(1)
Proof
16(9)
A More Involved Proof
20(2)
Proving If-and-Only-If Theorems
22(2)
Proving Equations and Inequalities
24(1)
Recap
25(1)
Exercises
25(1)
Counterexample
25(2)
Recap
27(1)
Exercises
27(1)
Boolean Algebra
27(10)
More Operations
31(1)
Recap
32(1)
Exercises
32(2)
Chapter 1 Self Test
34(3)
Collections
37(46)
Lists
37(8)
Counting Two-Element Lists
37(3)
Longer Lists
40(3)
Recap
43(1)
Exercises
43(2)
Factorial
45(4)
Much Ado About 0!
46(1)
Product Notation
47(1)
Recap
48(1)
Exercises
48(1)
Sets I: Introduction, Subsets
49(9)
Equality of Sets
51(2)
Subset
53(2)
Counting Subsets
55(2)
Power Set
57(1)
Recap
57(1)
Exercises
57(1)
Quantifiers
58(6)
There Is
58(1)
For All
59(1)
Negating Quantified Statements
60(1)
Combining Quantifiers
61(1)
Recap
62(1)
Exercises
63(1)
Sets II: Operations
64(12)
Union and Intersection
64(2)
The Size of a Union
66(2)
Difference and Symmetric Difference
68(5)
Cartesian Product
73(1)
Recap
74(1)
Exercises
74(2)
Combinatorial Proof: Two Examples
76(7)
Recap
80(1)
Exercises
80(1)
Chapter 2 Self Test
80(3)
Counting and Relations
83(52)
Relations
83(6)
Properties of Relations
86(1)
Recap
87(1)
Exercises
87(2)
Equivalence Relations
89(9)
Equivalence Classes
92(3)
Recap
95(1)
Exercises
96(2)
Partitions
98(6)
Counting Classes/Parts
100(2)
Recap
102(1)
Exercises
102(2)
Binomial Coefficients
104(13)
Calculating (n/k)
107(2)
Pascal's Triangle
109(2)
A Formula for (n/k)
111(2)
Recap
113(1)
Exercises
113(4)
Counting Multisets
117(6)
Multisets
117(2)
Formulas for (n/k)
119(3)
Recap
122(1)
Exercises
122(1)
Inclusion-Exclusion
123(12)
How to Use Inclusion-Exclusion
126(3)
Derangements
129(3)
A Ghastly Formula
132(1)
Recap
132(1)
Exercises
132(1)
Chapter 3 Self Test
133(2)
More Proof
135(58)
Contradiction
135(7)
Proof by Contrapositive
135(2)
Reductio ad Absurdum
137(4)
A Matter of Style
141(1)
Recap
141(1)
Exercises
141(1)
Smallest Counterexample
142(13)
Well-Ordering
148(5)
Recap
153(1)
Exercises
153(1)
And Finally
154(1)
Induction
155(16)
The Induction Machine
155(2)
Theoretical Underpinnings
157(1)
Proof by Induction
157(3)
Proving Equations and Inequalities
160(2)
Other Examples
162(1)
Strong Induction
163(2)
A More Complicated Example
165(3)
A Matter of Style
168(1)
Recap
168(1)
Exercises
168(3)
Recurrence Relations
171(22)
First-Order Recurrence Relations
172(3)
Second-Order Recurrence Relations
175(3)
The Case of the Repeated Root
178(2)
Sequences Generated by Polynomials
180(7)
Recap
187(1)
Exercises
188
Chapter 4 Self Test
180(13)
Functions
193(52)
Functions
193(12)
Domain and Image
195(1)
Pictures of Functions
196(1)
Counting Functions
197(1)
Inverse Functions
198(4)
Counting Functions, Again
202(1)
Recap
203(1)
Exercises
203(2)
The Pigeonhole Principle
205(6)
Cantor's Theorem
208(2)
Recap
210(1)
Exercises
210(1)
Composition
211(5)
Identity Function
214(1)
Recap
215(1)
Exercises
215(1)
Permutations
216(15)
Cycle Notation
217(3)
Calculations with Permutations
220(1)
Transpositions
221(5)
A Graphical Approach
226(2)
Recap
228(1)
Exercises
228(3)
Symmetry
231(5)
Symmetries of a Square
231(1)
Symmetries as Permutations
232(1)
Combining Symmetries
233(2)
Formal Definition of Symmetry
235(1)
Recap
236(1)
Exercises
236(1)
Assorted Notation
236(9)
Big oh
236(3)
Ω and Θ
239(1)
Little oh
240(1)
Floor and Ceiling
241(1)
Recap
242(1)
Exercises
242(1)
Chapter 5 Self Test
242(3)
Probability
245(48)
Sample Space
245(4)
Recap
248(1)
Exercises
248(1)
Events
249(8)
Combining Events
252(1)
The Birthday Problem
253(1)
Recap
254(1)
Exercises
255(2)
Conditional Probability and Independence
257(9)
Independence
259(2)
Independent Repeated Trials
261(1)
The Monty Hall Problem
262(1)
Recap
263(1)
Exercises
263(3)
Random Variables
266(5)
Random Variables as Events
267(2)
Independent Random Variables
269(1)
Recap
270(1)
Exercises
270(1)
Expectation
271(22)
Linearity of Expectation
276(3)
Product of Random Variables
279(3)
Expected Value as a Measure of Centrality
282(1)
Variance
283(4)
Recap
287(1)
Exercises
287(2)
Chapter 6 Self Test
289(4)
Number Theory
293(44)
Dividing
293(5)
Div and Mod
296(1)
Recap
297(1)
Exercises
297(1)
Greatest Common Divisor
298(11)
Calculating the gcd
299(2)
Correctness
301(1)
How Fast?
302(2)
An Important Theorem
304(3)
Recap
307(1)
Exercises
307(2)
Modular Arithmetic
309(11)
A New Context for Basic Operations
309(1)
Modular Addition and Multiplication
310(1)
Modular Subtraction
311(2)
Modular Division
313(5)
A Note on Notation
318(1)
Recap
318(1)
Exercises
318(2)
The Chinese Remainder Theorem
320(5)
Solving One Equation
320(2)
Solving Two Equations
322(2)
Recap
324(1)
Exercises
324(1)
Factoring
325(12)
Infinitely Many Primes
327(1)
A Formula for Greatest Common Divisor
328(1)
Irrationality of √2
329(2)
Recap
331(1)
Exercises
331(4)
Chapter 7 Self Test
335(2)
Algebra
337(52)
Groups
337(10)
Operations
337(1)
Properties of Operations
338(2)
Groups
340(2)
Examples
342(3)
Recap
345(1)
Exercises
345(2)
Group Isomorphism
347(6)
The Same?
347(2)
Cyclic Groups
349(3)
Recap
352(1)
Exercises
352(1)
Subgroups
353(9)
Lagrange's Theorem
356(3)
Recap
359(1)
Exercises
359(3)
Fermat's Little Theorem
362(8)
First Proof
362(1)
Second Proof
363(3)
Third Proof
366(1)
Euler's Theorem
367(1)
Primality Testing
368(1)
Recap
369(1)
Exercises
369(1)
Public Key Cryptography I: Introduction
370(3)
The Problem: Private Communication in Public
370(1)
Factoring
370(1)
Words to Numbers
371(2)
Cryptography and the Law
373(1)
Recap
373(1)
Exercises
373(1)
Public Key Cryptography II: Rabin's Method
373(7)
Square Roots Modulo n
374(4)
The Encryption and Decryption Procedures
378(1)
Recap
379(1)
Exercises
379(1)
Public Key Cryptography III: RSA
380(9)
The RSA Encryption and Decryption Functions
381(2)
Security
383(1)
Recap
384(1)
Exercises
384(1)
Chapter 8 Self Test
385(4)
Graphs
389(60)
Fundamentals of Graph Theory
389(10)
Map Coloring
389(2)
Three Utilities
391(1)
Seven Bridges
391(1)
What Is a Graph?
392(1)
Adjacency
393(1)
A Matter of Degree
394(2)
Further Notation and Vocabulary
396(1)
Recap
397(1)
Exercises
397(2)
Subgraphs
399(7)
Induced and Spanning Subgraphs
400(2)
Cliques and Independent Sets
402(1)
Complements
403(1)
Recap
404(1)
Exercises
404(2)
Connection
406(7)
Walks
406(1)
Paths
407(3)
Disconnection
410(1)
Recap
411(1)
Exercises
411(2)
Trees
413(8)
Cycles
413(1)
Forests and Trees
413(1)
Properties of Trees
414(2)
Leaves
416(2)
Spanning Trees
418(1)
Recap
419(1)
Exercises
420(1)
Eulerian Graphs
421(6)
Necessary Conditions
422(1)
Main Theorems
423(2)
Unfinished Business
425(1)
Recap
426(1)
Exercises
426(1)
Coloring
427(8)
Core Concepts
427(2)
Bipartite Graphs
429(4)
The Ease of Two-Coloring and the Difficulty of Three-Coloring
433(1)
Recap
434(1)
Exercises
434(1)
Planar Graphs
435(14)
Dangerous Curves
435(1)
Embedding
436(1)
Euler's Formula
437(3)
Nonplanar Graphs
440(2)
Coloring Planar Graphs
442(2)
Recap
444(1)
Exercises
444(2)
Chapter 9 Self Test
446(3)
Partially Ordered Sets
449(38)
Fundamentals of Partially Ordered Sets
449(6)
What Is a Poset?
449(3)
Notation and Language
452(2)
Recap
454(1)
Exercises
454(1)
Max and Min
455(3)
Recap
457(1)
Exercises
457(1)
Linear Orders
458(3)
Recap
460(1)
Exercises
461(1)
Linear Extensions
461(8)
Sorting
465(2)
Linear Extensions of Infinite Posets
467(1)
Recap
468(1)
Exercises
468(1)
Dimension
469(8)
Realizers
469(2)
Dimension
471(2)
Embedding
473(3)
Recap
476(1)
Exercises
476(1)
Lattices
477(10)
Meet and Join
477(2)
Lattices
479(2)
Recap
481(1)
Exercises
482(1)
Chapter 10 Self Test
483(4)
Appendices
487(68)
A Lots of Hints and Comments; Some Answers
487(28)
B Solutions to Self Tests
515(29)
Chapter 1
515(1)
Chapter 2
516(2)
Chapter 3
518(2)
Chapter 4
520(4)
Chapter 5
524(2)
Chapter 6
526(4)
Chapter 7
530(2)
Chapter 8
532(3)
Chapter 9
535(4)
Chapter 10
539(5)
C Glossary
544(8)
D Fundamentals
552(3)
Numbers
552(1)
Operations
552(1)
Ordering
553(1)
Complex Numbers
553(1)
Substitution
553(2)
Index555

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