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Cover type: Hardback

Edition: 96

Copyright: 1996

Publisher: Addison-Wesley Longman, Inc.

Published: 1996

International: No

Edition: 96

Copyright: 1996

Publisher: Addison-Wesley Longman, Inc.

Published: 1996

International: No

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This text is a sophomore- or junior-level introduction to the fundamental concepts and techniques used in abstract mathematics. It guides students through the transition from a more computational course to higher level work. It does so by actively engaging students in every step of the development of a central core of ideas common to all branches of mathematics.

**FEATURES:**

- Following the foundation work of Chapters 16, the remaining, self-contained, chapters offer three alternative avenues for continued study. This text uses a unique approach in which students construct their own understandings.
- Logic is used as a tool for analyzing the content of mathematical assertions and for constructing valid mathematical proofs.
- Order structure is used primarily in Chapter 4 and extensively throughout the rest of the text.
- Rigorous axiomatic treatment of set theory is introduced in Appendices A and B (which are written in the same style as the text's chapters).

**Schumacher, Carol : Kenyon College**

**Chapter 1: Introduction--an Essay**

Mathematical Reasoning

Deciding What to Assume

What Is Needed to Do Mathematics?

Chapter Zero

**Chapter 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?

**Chapter 3: Sets**

Sets and Set Notation

Set Operations

Russell's Paradox

**Chapter 4: Relations and Ordering**

Relations

Orderings

Equivalence Relations

**Chapter 5: Functions**

Basic Ideas

Composition and Inverses

Order Isomorphisms

Sequences

Binary Operations

**Chapter 6: Induction**

Inductive Reasoning and Mathematical Induction

Using Induction

Complete Induction

**Chapter 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

**Chapter 8: Cardinality**

Galileo's Paradox

Infinite Sets

Countable Sets

Beyond Countability

Comparing Cardinalities

The Continuum Hypothesis

Order Isomorphisms (Revisited)

**Chapter 9: The Real Numbers**

Constructing the Axioms

Arithmetic

Order

The Least Upper Bound Axiom

Sequence Convergence in R

**Chapter 10: Axiomatic Set Theory**

Elementary Axioms

The Axiom of Infinity

Axioms of Choice and Substitution

**Chapter 11: Constructing R**

From N to Integers

From Integers to Rationals

From Rationals to R

Index

Summary

This text is a sophomore- or junior-level introduction to the fundamental concepts and techniques used in abstract mathematics. It guides students through the transition from a more computational course to higher level work. It does so by actively engaging students in every step of the development of a central core of ideas common to all branches of mathematics.

**FEATURES:**

- Following the foundation work of Chapters 16, the remaining, self-contained, chapters offer three alternative avenues for continued study. This text uses a unique approach in which students construct their own understandings.
- Logic is used as a tool for analyzing the content of mathematical assertions and for constructing valid mathematical proofs.
- Order structure is used primarily in Chapter 4 and extensively throughout the rest of the text.
- Rigorous axiomatic treatment of set theory is introduced in Appendices A and B (which are written in the same style as the text's chapters).

Author Bio

**Schumacher, Carol : Kenyon College**

Table of Contents

**Chapter 1: Introduction--an Essay**

Mathematical Reasoning

Deciding What to Assume

What Is Needed to Do Mathematics?

Chapter Zero

**Chapter 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?

**Chapter 3: Sets**

Sets and Set Notation

Set Operations

Russell's Paradox

**Chapter 4: Relations and Ordering**

Relations

Orderings

Equivalence Relations

**Chapter 5: Functions**

Basic Ideas

Composition and Inverses

Order Isomorphisms

Sequences

Binary Operations

**Chapter 6: Induction**

Inductive Reasoning and Mathematical Induction

Using Induction

Complete Induction

**Chapter 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

**Chapter 8: Cardinality**

Galileo's Paradox

Infinite Sets

Countable Sets

Beyond Countability

Comparing Cardinalities

The Continuum Hypothesis

Order Isomorphisms (Revisited)

**Chapter 9: The Real Numbers**

Constructing the Axioms

Arithmetic

Order

The Least Upper Bound Axiom

Sequence Convergence in R

**Chapter 10: Axiomatic Set Theory**

Elementary Axioms

The Axiom of Infinity

Axioms of Choice and Substitution

**Chapter 11: Constructing R**

From N to Integers

From Integers to Rationals

From Rationals to R

Index

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 1996

International: No

Published: 1996

International: No

This text is a sophomore- or junior-level introduction to the fundamental concepts and techniques used in abstract mathematics. It guides students through the transition from a more computational course to higher level work. It does so by actively engaging students in every step of the development of a central core of ideas common to all branches of mathematics.

**FEATURES:**

- Following the foundation work of Chapters 16, the remaining, self-contained, chapters offer three alternative avenues for continued study. This text uses a unique approach in which students construct their own understandings.
- Logic is used as a tool for analyzing the content of mathematical assertions and for constructing valid mathematical proofs.
- Order structure is used primarily in Chapter 4 and extensively throughout the rest of the text.
- Rigorous axiomatic treatment of set theory is introduced in Appendices A and B (which are written in the same style as the text's chapters).

**Schumacher, Carol : Kenyon College**

**Chapter 1: Introduction--an Essay**

Mathematical Reasoning

Deciding What to Assume

What Is Needed to Do Mathematics?

Chapter Zero

**Chapter 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?

**Chapter 3: Sets**

Sets and Set Notation

Set Operations

Russell's Paradox

**Chapter 4: Relations and Ordering**

Relations

Orderings

Equivalence Relations

**Chapter 5: Functions**

Basic Ideas

Composition and Inverses

Order Isomorphisms

Sequences

Binary Operations

**Chapter 6: Induction**

Inductive Reasoning and Mathematical Induction

Using Induction

Complete Induction

**Chapter 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

**Chapter 8: Cardinality**

Galileo's Paradox

Infinite Sets

Countable Sets

Beyond Countability

Comparing Cardinalities

The Continuum Hypothesis

Order Isomorphisms (Revisited)

**Chapter 9: The Real Numbers**

Constructing the Axioms

Arithmetic

Order

The Least Upper Bound Axiom

Sequence Convergence in R

**Chapter 10: Axiomatic Set Theory**

Elementary Axioms

The Axiom of Infinity

Axioms of Choice and Substitution

**Chapter 11: Constructing R**

From N to Integers

From Integers to Rationals

From Rationals to R

Index