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by Ethan Bloch

Edition: 00Copyright: 2000

Publisher: Birkhauser Boston, Inc.

Published: 2000

International: No

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Proofs and Fundamentals: A First Course in Abstract Mathematics is designed as a 'transition' course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra, and real analysis.

This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material including sets, functions, and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics, and the Peano Postulates. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and writing are never compromised. The material is presented in the way that mathematicians actually use it; good mathematical taste is preferred to overly clever pedagogy. There is a key section devoted to the proper writing of proofs. The text has over 400 exercises, ranging from straightforward examples to very challenging proofs.

The excellent exposition, organization and choice of topics will make this text valuable for classroom use as well as for the general reader who wants to gain a deeper understanding of how modern mathematics is currently practiced by mathematicians.

**Bloch, Ethan : Bard College, Annandale-on-Hudson, NY **

Introduction

To the Student

To the Instructor

**Part I: PROOFS 1. Informal Logic **

1.1 Introduction

1.2 Statements

1.3 Relations Between Statements

1.4 Valid Arguments

1.5 Quantifiers

2.2 Direct Proofs

2.3 Proofs by Contrapositive and Contradiction

2.4 Cases, and If and Only If

2.5 Quantifiers in Theorems

2.6 Writing Mathematics

3.2 Sets-Basic Definitions

3.3 Set Operations

3.4 Indexed Families of Sets

4.2 Image and Inverse Image

4.3 Composition and Inverse Functions

4.4 Injectivity, Surjectivity and Bijectivity

4.5 Set of Functions

5.2 Congruence

5.3 Equivalence Relations

6.2 Cardinality of the Number Systems

6.3 Mathematical Induction

6.4 Recursion

7.2 Groups

7.3 Homomorphisms and Isomorphisms

7.4 Partially Ordered Sets

7.5 Lattices

7.6 Counting: Products and Sums

7.7 Counting: Permutations and Combinations

8.2 The Natural Numbers

8.3 Further Properties of the Natural Numbers

8.4 The Integers

8.5 The Rational Numbers

8.6 The Real Numbers and the Complex Numbers

8.7 Appendix: Proof of Theorem 8.2.1

Appendix: Properties of Numbers

Hints for Selected Exercises

References

Index

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Summary

Proofs and Fundamentals: A First Course in Abstract Mathematics is designed as a 'transition' course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra, and real analysis.

This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material including sets, functions, and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics, and the Peano Postulates. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and writing are never compromised. The material is presented in the way that mathematicians actually use it; good mathematical taste is preferred to overly clever pedagogy. There is a key section devoted to the proper writing of proofs. The text has over 400 exercises, ranging from straightforward examples to very challenging proofs.

The excellent exposition, organization and choice of topics will make this text valuable for classroom use as well as for the general reader who wants to gain a deeper understanding of how modern mathematics is currently practiced by mathematicians.

Author Bio

**Bloch, Ethan : Bard College, Annandale-on-Hudson, NY **

Table of Contents

Introduction

To the Student

To the Instructor

**Part I: PROOFS 1. Informal Logic **

1.1 Introduction

1.2 Statements

1.3 Relations Between Statements

1.4 Valid Arguments

1.5 Quantifiers

2.2 Direct Proofs

2.3 Proofs by Contrapositive and Contradiction

2.4 Cases, and If and Only If

2.5 Quantifiers in Theorems

2.6 Writing Mathematics

3.2 Sets-Basic Definitions

3.3 Set Operations

3.4 Indexed Families of Sets

4.2 Image and Inverse Image

4.3 Composition and Inverse Functions

4.4 Injectivity, Surjectivity and Bijectivity

4.5 Set of Functions

5.2 Congruence

5.3 Equivalence Relations

6.2 Cardinality of the Number Systems

6.3 Mathematical Induction

6.4 Recursion

7.2 Groups

7.3 Homomorphisms and Isomorphisms

7.4 Partially Ordered Sets

7.5 Lattices

7.6 Counting: Products and Sums

7.7 Counting: Permutations and Combinations

8.2 The Natural Numbers

8.3 Further Properties of the Natural Numbers

8.4 The Integers

8.5 The Rational Numbers

8.6 The Real Numbers and the Complex Numbers

8.7 Appendix: Proof of Theorem 8.2.1

Appendix: Properties of Numbers

Hints for Selected Exercises

References

Index

Publisher Info

Publisher: Birkhauser Boston, Inc.

Published: 2000

International: No

Published: 2000

International: No

This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material including sets, functions, and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics, and the Peano Postulates. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and writing are never compromised. The material is presented in the way that mathematicians actually use it; good mathematical taste is preferred to overly clever pedagogy. There is a key section devoted to the proper writing of proofs. The text has over 400 exercises, ranging from straightforward examples to very challenging proofs.

The excellent exposition, organization and choice of topics will make this text valuable for classroom use as well as for the general reader who wants to gain a deeper understanding of how modern mathematics is currently practiced by mathematicians.

**Bloch, Ethan : Bard College, Annandale-on-Hudson, NY **

To the Student

To the Instructor

**Part I: PROOFS 1. Informal Logic **

1.1 Introduction

1.2 Statements

1.3 Relations Between Statements

1.4 Valid Arguments

1.5 Quantifiers

2.2 Direct Proofs

2.3 Proofs by Contrapositive and Contradiction

2.4 Cases, and If and Only If

2.5 Quantifiers in Theorems

2.6 Writing Mathematics

3.2 Sets-Basic Definitions

3.3 Set Operations

3.4 Indexed Families of Sets

4.2 Image and Inverse Image

4.3 Composition and Inverse Functions

4.4 Injectivity, Surjectivity and Bijectivity

4.5 Set of Functions

5.2 Congruence

5.3 Equivalence Relations

6.2 Cardinality of the Number Systems

6.3 Mathematical Induction

6.4 Recursion

7.2 Groups

7.3 Homomorphisms and Isomorphisms

7.4 Partially Ordered Sets

7.5 Lattices

7.6 Counting: Products and Sums

7.7 Counting: Permutations and Combinations

8.2 The Natural Numbers

8.3 Further Properties of the Natural Numbers

8.4 The Integers

8.5 The Rational Numbers

8.6 The Real Numbers and the Complex Numbers

8.7 Appendix: Proof of Theorem 8.2.1

Appendix: Properties of Numbers

Hints for Selected Exercises

References

Index