The foundations of mathematics /
Thomas Q. Sibley.
- Hoboken, NJ : John Wiley & Sons, c2009.
- xv, 392 p. : ill. ; 25 cm.
Includes index
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
Finally there's an easy-to-follow book that will help readers succeed in the art of proving theorems. Sibley not only conveys the spirit of mathematics but also uncovers the skills required to succeed. Key definitions are introduced while readers are encouraged to develop an intuition about these concepts and practice using them in problems.