Proofs and Fundamentals

by Bloch, Ethan D.

• ISBN13: 9780817641115
• ISBN10: 0817641114
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: Birkhauser

Summary

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.

Table of Contents

Introduction	ix
To the Student	xiii
To the Instructor	xix
Part I PROOFS	1	(104)
Informal Logic	3	(52)
Introduction	3	(1)
Statements	4	(14)
Relations Between Statements	18	(13)
Valid Arguments	31	(10)
Quantifiers	41	(14)
Strategies for Proofs	55	(50)
Mathematical Proofs --- What They Are and Why We Need Them	55	(8)
Direct Proofs	63	(4)
Proofs by Contrapositive and Contradiction	67	(7)
Cases, and If and Only If	74	(7)
Quantifiers in Theorems	81	(12)
Writing Mathematics	93	(12)
Part II FUNDAMENTALS	105	(144)
Sets	107	(28)
Introduction	107	(2)
Sets --- Basic Definitions	109	(10)
Set Operations	119	(10)
Indexed Families of Sets	129	(6)
Functions	135	(42)
Functions	135	(10)
Image and Inverse Image	145	(7)
Composition and Inverse Functions	152	(9)
Injectivity, Surjectivity and Bijectivity	161	(9)
Sets of Functions	170	(7)
Relations	177	(26)
Relations	177	(7)
Congruence	184	(7)
Equivalence Relations	191	(12)
Infinite and Finite Sets	203	(46)
Cardinality of Sets	203	(15)
Cardinality of the Number Systems	218	(8)
Mathematical Induction	226	(10)
Recursion	236	(13)
Part III EXTRAS	249	(126)
Selected Topics	251	(72)
Binary Operations	251	(7)
Groups	258	(9)
Homomorphisms and Isomorphisms	267	(6)
Partially Ordered Sets	273	(11)
Lattices	284	(10)
Counting: Products and Sums	294	(10)
Counting: Permutations and Combinations	304	(19)
Number Systems	323	(40)
Back to the Beginning	323	(1)
The Natural Numbers	324	(9)
Further Properties of the Natural Numbers	333	(5)
The Integers	338	(10)
The Rational Numbers	348	(4)
The Real Numbers and the Complex Numbers	352	(9)
Appendix: Proof of Theorem 8.2.1	361	(2)
Explorations	363	(12)
Introduction	363	(1)
Greatest Common Divisors	364	(2)
Divisibility Tests	366	(1)
Real-Valued Functions	367	(1)
Iterations of Functions	368	(1)
Fibonacci Numbers and Lucas Numbers	369	(2)
Fuzzy Sets	371	(4)
Appendix: Properties of Numbers	375	(4)
Hints of Selected Exercises	379	(26)
References	405	(8)
Index	413