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The Foundations of Mathematics
Thomas Q. Sibley (Author)
9780470085011, Wiley
Hardback, published 21 April 2008
408 pages
23.6 x 15.5 x 2.8 cm, 0.68 kg
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 Gödel’s Theorems, and philosophy of mathematics provides an unusual and valuable addition compared to other similar texts
PART I Chapter 1: LANGUAGE, LOGIC, AND SETS Chapter 2: PROOFS Chapter 3: FUNCTIONS Chapter 4: RELATIONS PART II Chapter 5: INFINTE SETS Chapter 6: INTRODUCTION TO DISCRETE MATHEMATICS Chapter 7: INTRODUCTION TO ABSTRACT ALGEBRA Chapter 8: INTRODUCTION TO ANALYSIS Chapter 9: METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS Appendix: THE GREEK ALPHABET Index
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
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
3.1 Definitions
3.2 Composition, One-to-One, Onto, and Inverses
3.3 Images and Pre-Images of Sets
Definitions from Chapter 3
4.1 Relations
4.2 Equivalence Relations
4.3 Partitions and Equivalence Relations
4.4 Partial Orders
Definitions from Chapter 4
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
6.1 Graph Theory
6.2 Trees and Algorithms
6.3 Counting Principles I
6.4 Counting Principles II
Definitions from Chapter 6
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
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
9.1 Metamathematics
9.2 The Philosophy of Mathematics
Definitions for Chapter 9
Answers: SELECTED ANSWERS
List of Symbols
Subject Areas: Mathematics [PB]
