The Foundations of Mathematics
1st Edition
Published on 21. April 2008
Book
Hardback
408 pages
978-0-470-08501-1 (ISBN)
Description
The Foundations of Mathematics provides a careful introduction to proofs in mathematics, along with basic concepts of logic, set theory and other broadly used areas of mathematics. The concepts are introduced in a pedagogically effective manner without compromising mathematical accuracy and completeness. Thus, in Part I students explore concepts before they use them in proofs. The exercises range from reading comprehension questions and many standard exercises to proving more challenging statements, formulating conjectures and critiquing a variety of false and questionable proofs. The discussion of metamathematics, including Goedel's Theorems, and philosophy of mathematics provides an unusual and valuable addition compared to other similar texts
More details
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Dimensions
Height: 236 mm
Width: 155 mm
Thickness: 28 mm
Weight
680 gr
ISBN-13
978-0-470-08501-1 (9780470085011)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Person
Thomas Q. Sibley is Professor of Mathematics at St. John's University in Collegeville, Minnesota.
Content
PART I Chapter 1: LANGUAGE, LOGIC, AND SETS
1.1 Logic and Language
1.2 Implication
1.3 Quantifiers and Definitions
1.4 Introduction to Sets
1.5 Introduction to Number Theory
1.6 Additional Set Theory
Definitions from Chapter 1
Algebraic and Order Properties of Number Systems
Chapter 2: PROOFS
2.1 Proof Format I: Direct Proofs
2.2 Proof Format II: Contrapositive and Contradition
2.3 Proof Format III: Existence, Uniqueness, Or
2.4 Proof Format IV: Mathematical Induction
The Fundamental Theorem of Arithmetic
2.5 Further Advice and Practice in Proving
Proof Formats
Chapter 3: FUNCTIONS
3.1 Definitions
3.2 Composition, One-to-One, Onto, and Inverses
3.3 Images and Pre-Images of Sets
Definitions from Chapter 3
Chapter 4: RELATIONS
4.1 Relations
4.2 Equivalence Relations
4.3 Partitions and Equivalence Relations
4.4 Partial Orders
Definitions from Chapter 4
PART II
Chapter 5: INFINTE SETS
5.1 The Sizes of Sets
5.2 Countable Sets
5.3 Uncountable Sets
5.4 The Axiom of Choice and Its Equivalents
Definitions from Chapter 5
Chapter 6: INTRODUCTION TO DISCRETE MATHEMATICS
6.1 Graph Theory
6.2 Trees and Algorithms
6.3 Counting Principles I
6.4 Counting Principles II
Definitions from Chapter 6
Chapter 7: INTRODUCTION TO ABSTRACT ALGEBRA
7.1 Operations and Properties
7.2 Groups
Groups in Geometry
7.3 Rings and Fields
7.4 Lattices
7.5 Homomorphisms
Definitions from Chapter 7
Chapter 8: INTRODUCTION TO ANALYSIS
8.1 Real Numbers, Approximations, and Exact Values
Zeno's Paradoxes
8.2 Limits of Functions
8.3 Continuous Functions and Counterexamples
Counterexamples in Rational Analysis
8.4 Sequences and Series
8.5 Discrete Dynamical Systems
The Intermediate Value Theorem
Definitions for Chapter 8
Chapter 9: METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS
9.1 Metamathematics
9.2 The Philosophy of Mathematics
Definitions for Chapter 9
Appendix: THE GREEK ALPHABET
Answers: SELECTED ANSWERS
Index
List of Symbols
1.1 Logic and Language
1.2 Implication
1.3 Quantifiers and Definitions
1.4 Introduction to Sets
1.5 Introduction to Number Theory
1.6 Additional Set Theory
Definitions from Chapter 1
Algebraic and Order Properties of Number Systems
Chapter 2: PROOFS
2.1 Proof Format I: Direct Proofs
2.2 Proof Format II: Contrapositive and Contradition
2.3 Proof Format III: Existence, Uniqueness, Or
2.4 Proof Format IV: Mathematical Induction
The Fundamental Theorem of Arithmetic
2.5 Further Advice and Practice in Proving
Proof Formats
Chapter 3: FUNCTIONS
3.1 Definitions
3.2 Composition, One-to-One, Onto, and Inverses
3.3 Images and Pre-Images of Sets
Definitions from Chapter 3
Chapter 4: RELATIONS
4.1 Relations
4.2 Equivalence Relations
4.3 Partitions and Equivalence Relations
4.4 Partial Orders
Definitions from Chapter 4
PART II
Chapter 5: INFINTE SETS
5.1 The Sizes of Sets
5.2 Countable Sets
5.3 Uncountable Sets
5.4 The Axiom of Choice and Its Equivalents
Definitions from Chapter 5
Chapter 6: INTRODUCTION TO DISCRETE MATHEMATICS
6.1 Graph Theory
6.2 Trees and Algorithms
6.3 Counting Principles I
6.4 Counting Principles II
Definitions from Chapter 6
Chapter 7: INTRODUCTION TO ABSTRACT ALGEBRA
7.1 Operations and Properties
7.2 Groups
Groups in Geometry
7.3 Rings and Fields
7.4 Lattices
7.5 Homomorphisms
Definitions from Chapter 7
Chapter 8: INTRODUCTION TO ANALYSIS
8.1 Real Numbers, Approximations, and Exact Values
Zeno's Paradoxes
8.2 Limits of Functions
8.3 Continuous Functions and Counterexamples
Counterexamples in Rational Analysis
8.4 Sequences and Series
8.5 Discrete Dynamical Systems
The Intermediate Value Theorem
Definitions for Chapter 8
Chapter 9: METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS
9.1 Metamathematics
9.2 The Philosophy of Mathematics
Definitions for Chapter 9
Appendix: THE GREEK ALPHABET
Answers: SELECTED ANSWERS
Index
List of Symbols