Chapter Zero Fundamental Notions of Abstract Mathematics

by Schumacher, Carol

• ISBN13: 9780201437249
• ISBN10: 0201437244
• Edition: 2nd
• Format: Paperback
• Copyright: 2000-11-17
• Publisher: Pearson

Summary

This book is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which readers construct their own understanding. However, while readers are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers "proof sketches" and helpful technique tips to help readers as they develop their proof writing skills. This book is most successful in a small, seminar style class. Logic, Sets, Induction, Relations, Functions, Elementary Number Theory, Cardinality, The Real Numbers For all readers interested in abstract mathematics.

Table of Contents

Preface	vii
A Note to the Student: What This Book Expects from You	xiii
Introduction---An Essay	1	(6)
Mathematical Reasoning	1	(1)
Deciding What to Assume	2	(2)
What is Needed to Do Mathematics?	4	(2)
Chapter Zero	6	(1)
Logic	7	(32)
True or False?	7	(2)
Thought Experiment: True or False?	7	(2)
Statements and Predicates	9	(2)
Quantification	11	(3)
Mathematical Statements	14	(1)
Mathematical Implication	15	(3)
Compound Statements and Truth Tables	18	(3)
Learning from Truth Tables	21	(3)
Tautologies	21	(1)
What About the Converse?	21	(1)
Equivalence and Rephrasing	22	(2)
Negating Statements	24	(4)
Existence Theorems	28	(1)
Uniqueness Theorems	29	(1)
Examples and Counterexamples	30	(2)
Direct Proof	32	(1)
Proof by Contrapositive	33	(1)
Proof by Contradiction	34	(2)
Proving Theorems: What Now?	36	(3)
Problems	37	(1)
Questions to Ponder	38	(1)
Sets	39	(18)
Sets and Set Notation	39	(3)
Subsets	42	(1)
Set Operations	43	(4)
The Algebra of Sets	47	(3)
The Power Set	50	(2)
Russell's Paradox	52	(5)
Problems	53	(2)
Questions to Ponder	55	(2)
Induction	57	(8)
Mathematical Induction	57	(4)
Using Induction	61	(1)
Complete Induction	62	(3)
Questions to Ponder	64	(1)
Relations	65	(38)
Relations	65	(4)
Orderings	69	(9)
Equivalence Relations	78	(5)
Graphs	83	(20)
Coloring Maps	95	(1)
Problems	96	(4)
Questions to Ponder	100	(3)
Functions	103	(34)
Basic Ideas	103	(6)
Composition and Inverses	109	(3)
Images and Inverse Images	112	(3)
Order Isomorphisms	115	(3)
Sequences	118	(10)
Sequences with Special Properties	121	(2)
Subsequences	123	(1)
Constructing Subsequences Recursively	124	(4)
Binary Operations	128	(9)
Problems	130	(5)
Questions to Ponder	135	(2)
Elementary Number Theory	137	(20)
Natural Numbers and Integers	137	(2)
Divisibility in the Integers	139	(5)
The Euclidean Algorithm	144	(2)
Relatively Prime Integers	146	(1)
Prime Factorization	147	(1)
Congruence Modulo n	148	(4)
Divisibility Modulo n	152	(5)
Problems	155	(1)
Questions to Ponder	156	(1)
Cardinality	157	(22)
Galileo's Paradox	157	(4)
Infinite Sets	161	(2)
Countable Sets	163	(3)
Beyond Countability	166	(3)
Comparing Cardinalities	169	(5)
The Continuum Hypothesis	174	(5)
Problems	176	(1)
Questions to Ponder	176	(3)
The Real Numbers	179	(18)
Constructing the Axioms	179	(1)
Arithmetic	179	(4)
Order	183	(4)
The Least Upper Bound Axiom	187	(3)
Sequence Convergence in R	190	(7)
Problems	195	(1)
Questions to Ponder	196	(1)
A Axiomatic Set Theory	197	(16)
A.1 Elementary Axioms	197	(6)
A.2 The Axiom of Infinity	203	(6)
A.3 Axioms of Choice and Substitution	209	(4)
B Constructing R	213	(14)
B.1 From N to Z	214	(4)
B.2 From Z to Q	218	(3)
B.3 From Q to R	221	(6)
Index	227