The Nuts and Bolts of Proofs, Fourth Edition: An Introduction to Mathematical Proofs

Front Cover......Page 1
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs......Page 4
Copyright......Page 5
Table of Contents......Page 6
Selected Relations, Definitions, and Properties of Integer, Rational, and Irrational Numbers......Page 8
Some Facts and Properties of Functions......Page 9
Introduction and Basic Terminology......Page 12
General Suggestions......Page 14
What Does “If/Then” Mean?......Page 18
The Negation of a Statement: AND/OR......Page 27
Proof by Contrapositive......Page 32
Proof by Contradiction......Page 36
“If and Only If” or “Equivalence Theorems”......Page 44
Use of Counterexamples......Page 51
Mathematical Induction......Page 53
Existence Theorems......Page 63
Uniqueness Theorems......Page 65
Multiple Hypotheses......Page 69
Multiple Conclusions......Page 72
Equality of Numbers......Page 77
Basic Set Theory and Indexed Families......Page 82
Cartesian Product of Sets......Page 93
Indexed Families of Sets......Page 95
About Functions......Page 98
Composition of Functions......Page 104
A Little More about Functions and Sets......Page 109
Relations......Page 113
Most Common Properties of Relations......Page 116
More about Equivalence Relations......Page 120
A Special Relation and More Facts about Equivalence Classes......Page 122
The Basics of Groups......Page 129
Some Properties of Binary Operations......Page 132
Special Elements......Page 139
When the Properties Fit Together…......Page 142
Sizes and Structures......Page 149
Groups (mod m) and Arithmetic (mod m)......Page 153
Permutations and Symmetric Groups......Page 159
Isomorphism and Subgroups......Page 163
An Important Theorem......Page 167
Limits......Page 170
Getting Closer......Page 173
Functions and Limits......Page 180
Sizes of Infinity......Page 190
5 Review Exercises......Page 200
General Topics......Page 203
Basic Set Theory......Page 206
Relations......Page 207
Limits......Page 208
Collection of “Proofs”......Page 209
Direct Proof......Page 215
The Negation of a Statement: AND/OR......Page 217
Proof by Contrapositive......Page 218
Proof by Contradiction......Page 220
“If and Only If” or Equivalence Theorems......Page 221
Use of Counterexamples......Page 223
Mathematical Induction......Page 225
Existence Theorems......Page 230
Uniqueness Theorems......Page 231
Composite Statements......Page 234
Equality of Numbers......Page 235
Basic Set Theory and Indexed Families......Page 236
About Functions......Page 242
Relations......Page 245
The Basics of Groups......Page 251
Limits......Page 260
Sizes of Infinity......Page 266
Solutions for the Review Exercises......Page 269
Other Books on the Subject of Proofs and Mathematical Writing......Page 289
F......Page 292
P......Page 293
W......Page 294