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ISBN13: 978-0201437249

ISBN10: 0201437244

Edition: 2ND 01

Copyright: 2001

Publisher: Addison-Wesley Longman, Inc.

Published: 2001

International: No

ISBN10: 0201437244

Edition: 2ND 01

Copyright: 2001

Publisher: Addison-Wesley Longman, Inc.

Published: 2001

International: No

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Chapter Zero is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which students construct their own understandings. However, while students are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers ''proof sketches'' and helpful technique tips to help students as they develop their proof writing skills. This book is most successful in a small, seminar style class.

Author Bio

**Schumacher, Carol : Kenyon College**

1. Introduction--an Essay

Mathematical Reasoning

Deciding What to Assume

What Is Needed to Do Mathematics?

Chapter Zero

2. Logic

Statements and Predicates

Mathematical Implication

Direct Proofs

Compound Statements and Truth Tables

Equivalence

Proof by Contrapositive

Negating Statements

Proof by Contradiction

Existence and Uniqueness

Proving Theorems: What Now?

3. Sets

Sets and Set Notation

Set Operations

Russell's Paradox

4. Relations and Ordering

Relations

Orderings

Equivalence Relations

5. Functions

Basic Ideas

Composition and Inverses

Order Isomorphisms

Sequences

Binary Operations

6. Induction

Inductive Reasoning and Mathematical Induction

Using Induction

Complete Induction

7. Elementary Number Theory

Natural Numbers and Integers

Divisibility in the Integers

The Euclidean Algorithm

Relatively Prime Integers

Prime Factorization

Congruence Modulo n

Divisibility Modulo n

8. Cardinality

Galileo's Paradox

Infinite Sets

Countable Sets

Beyond Countability

Comparing Cardinalities

The Continuum Hypothesis

Order Isomorphisms (Revisited)

9. The Real Numbers

Constructing the Axioms

Arithmetic

Order

The Least Upper Bound Axiom

Sequence Convergence in Real Numbers

A. Axiomatic Set Theory

Elementary Axioms

The Axiom of Infinity

Axioms of Choice and Substitution

B. Constructing R

From Natural Numbers to Integers

From Integers to Rational Numbers

From Rational Real Numbers to Real Numbers

Index

ISBN10: 0201437244

Edition: 2ND 01

Copyright: 2001

Publisher: Addison-Wesley Longman, Inc.

Published: 2001

International: No

Chapter Zero is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which students construct their own understandings. However, while students are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers ''proof sketches'' and helpful technique tips to help students as they develop their proof writing skills. This book is most successful in a small, seminar style class.

Author Bio

**Schumacher, Carol : Kenyon College**

Table of Contents

1. Introduction--an Essay

Mathematical Reasoning

Deciding What to Assume

What Is Needed to Do Mathematics?

Chapter Zero

2. Logic

Statements and Predicates

Mathematical Implication

Direct Proofs

Compound Statements and Truth Tables

Equivalence

Proof by Contrapositive

Negating Statements

Proof by Contradiction

Existence and Uniqueness

Proving Theorems: What Now?

3. Sets

Sets and Set Notation

Set Operations

Russell's Paradox

4. Relations and Ordering

Relations

Orderings

Equivalence Relations

5. Functions

Basic Ideas

Composition and Inverses

Order Isomorphisms

Sequences

Binary Operations

6. Induction

Inductive Reasoning and Mathematical Induction

Using Induction

Complete Induction

7. Elementary Number Theory

Natural Numbers and Integers

Divisibility in the Integers

The Euclidean Algorithm

Relatively Prime Integers

Prime Factorization

Congruence Modulo n

Divisibility Modulo n

8. Cardinality

Galileo's Paradox

Infinite Sets

Countable Sets

Beyond Countability

Comparing Cardinalities

The Continuum Hypothesis

Order Isomorphisms (Revisited)

9. The Real Numbers

Constructing the Axioms

Arithmetic

Order

The Least Upper Bound Axiom

Sequence Convergence in Real Numbers

A. Axiomatic Set Theory

Elementary Axioms

The Axiom of Infinity

Axioms of Choice and Substitution

B. Constructing R

From Natural Numbers to Integers

From Integers to Rational Numbers

From Rational Real Numbers to Real Numbers

Index

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