The Foundations of Mathematics

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. With this approach, they′ll gain a strong understanding of the mathematical language as they discover how to apply it in order to find proofs.

Spis treści:
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