Textbook

The Foundations of Mathematics, 1st Edition

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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

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Table of Contents

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

Hallmark Features

Introduces concepts before asking students to use them in proofs so students can separate the learning of concepts from their use in proofs, leading to greater success in both.
Builds on students' understanding of functions in general before introducing relations.
Over 3/4 of the sections have true/false reading comprehension questions to allow students to discover if they missed an important distinction in a concept.
Contains many false and questionable proofs to critique for practice in critically reading proofs and recognizing common errors.
Students are asked to make conjectures when a concept is introduced in order to better understand the concepts and the need for proofs and counter-examples.
Discusses advanced techniques of proof and an optional section carefully presents the axiom of choice and its equivalents.
Discusses the difference between formal and everyday use of terms and the idea of mathematical definitions to make the distinction between how mathematicians use terms and how these terms are used in everyday language.
Introduces metamathematics and the philosophy of mathematics.

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