Mathematical Thinking : Problem Solving and Proofs

by D'Angelo, John P.; West, Douglas Brent

• ISBN13: 9780132633932
• ISBN10: 0132633930
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: Addison Wesley

Summary

This course will differ from other math courses you have taken, because it emphasizes writing and language skills. We do not ask that you memorize formulas, but rather that you learn to express yourself clearly and accurately.

Table of Contents

Preface for the Student	ix
Preface for the Instructor	xiv
PART I Elementary Concepts	1	(74)
Sets and Numbers	2	(15)
The Quadratic Formula	2	(2)
Numbers and Problem-Solving	4	(3)
Sets	7	(3)
Summary of the Axioms	10	(3)
Exercises	13	(4)
Language and Proofs	17	(18)
Two Theorems about Equations	17	(2)
Quantifiers and Logical Statements	19	(4)
Compound Statements and Sets	23	(3)
Elementary Proof Techniques	26	(5)
Exercises	31	(4)
Functions	35	(21)
Definitions and Examples	36	(3)
Graphs of Functions	39	(2)
Bijections	41	(7)
Composition and Functional Digraphs	48	(3)
Operators (optional)	51	(1)
Exercises	52	(4)
Induction	56	(19)
The Principle of Induction	56	(7)
Applications	63	(5)
Strong Induction	68	(2)
Exercises	70	(5)
PART II Properties of Numbers	75	(60)
Counting and Cardinality	76	(17)
Representation of Natural Numbers	77	(3)
Cardinality	80	(5)
Binomial Coefficients and Counting	85	(5)
Exercises	90	(3)
Divisibility	93	(12)
The Euclidean Algorithm	94	(3)
Prime Factorization	97	(1)
The Dart Board Problem	98	(2)
Exercises	100	(5)
Modular Arithmetic	105	(16)
Equivalence Relations	106	(4)
Applications	110	(2)
Fermat's Little Theorem	112	(2)
Congruence and Groups (optional)	114	(2)
Exercises	116	(5)
The Rational Numbers	121	(14)
Constructing the Rationals	122	(3)
Irrational Numbers	125	(2)
Pythagorean Triples	127	(2)
Fractions and Probability	129	(2)
Further Properties of Rationals (optional)	131	(1)
Exercises	132	(3)
PART III Discrete Mathematics	135	(90)
Combinatorial Reasoning	136	(22)
More on the Binomial Coefficients	137	(3)
Analysis of Gaussian Elimination (optional)	140	(1)
Probability and Expectation	141	(6)
Multinomial Coefficients	147	(4)
Generating Functions (optional)	151	(2)
Exercises	153	(5)
Two Principles of Counting	158	(13)
The Pigeonhole Principle	158	(4)
The Inclusion-Exclusion Principle	162	(5)
Exercises	167	(4)
Graph Theory	171	(32)
The Konigsberg Bridge Problem	172	(4)
Isomorphism of Graphs	176	(5)
Connection and Trees	181	(4)
Bipartite Graphs	185	(4)
Coloring Problems	189	(3)
Planar Graphs	192	(6)
Exercises	198	(5)
Recurrence Relations	203	(22)
First-Order Recurrences	204	(5)
Second-Order Recurrences	209	(3)
General Linear Recurrences	212	(3)
Other Classical Recurrences (optional)	215	(4)
Exercises	219	(6)
PART IV Continuous Mathematics	225	(104)
The Real Numbers	226	(9)
The Completeness Axiom	226	(5)
Decimal Expansion and Uncountability	231	(2)
Exercises	233	(2)
Sequences and Series	235	(18)
Cauchy Sequences	235	(9)
Infinite Series	244	(5)
Exercises	249	(4)
Continuity	253	(14)
Limits and Continuity	254	(4)
Applications of Continuity	258	(4)
Continuity and Closed Intervals	262	(3)
Exercises	265	(2)
Differentiation	267	(28)
The Derivative	268	(5)
Applications of the Derivative	273	(4)
Newton's Method	277	(2)
Convexity and Curvature	279	(5)
Series of Functions	284	(6)
Exercises	290	(5)
Integration	295	(23)
Definition of the Integral	296	(6)
The Fundamental Theorem of Calculus	302	(4)
Exponentials and Logarithms	306	(2)
Trigonometric Functions and π	308	(3)
A Return to Infinite Series	311	(3)
Exercises	314	(4)
The Complex Numbers	318	(11)
Properties of the Complex Numbers	318	(4)
Limits and Convergence	322	(2)
The Fundamental Theorem of Algebra	324	(2)
Exercises	326	(3)
Appendix A From N to R	329	(15)
The Natural Numbers	330	(3)
The Integers	333	(2)
The Rational Numbers	335	(1)
The Real Numbers	335	(7)
Exercises	342	(2)
Appendix B Hints to Selected Exercises	344	(11)
General Discussion	344	(2)
Specific Hints	346	(9)
Appendix C Suggestions for Further Reading	355	(2)
Appendix D List of Notation	357	(2)
Index	359