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Passage to Abstract Mathematics (Subscription), 1st edition

Published by (November 21st 2011) - Copyright ©

1st edition

Passage to Abstract Mathematics (Subscription)

ISBN-13: 9780321830876

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Overview

Table of contents

Preface for the Instructor vii

Preface for the Student xiii

 

1. Logic and Proof

1.1 Proofs, what and why?

1.2 Statements and Non-statements

1.3 Logical Operations and Logical Equivalence

1.4 Conditionals, Tautologies and Contradictions

1.5 Methods of Proof

1.6 Quantifiers

1.7 Further Exercises

 

2. Numbers

2.1 Basic Ideas of Sets

2.2 Sets of Numbers

2.3 Some properties of N and Z

2.4 Prime Numbers

2.5 gcd’s and lcm’s

2.6 Euclid’s Algorithm

2.7 Rational Numbers and Algebraic Numbers

2.8 Further Exercises

 

3. Sets

3.1 Subsets

3.2 Operations with Sets

3.3 The Complement of a Set

3.4 The Cartesian Product

3.5 Families of Sets

3.6 Further Exercises

 

4. Induction

4.1 An Inductive Example

4.2 The Principle of Mathematical Induction

4.3 The Principle of Strong Induction

4.4 The Binomial Theorem

4.5 Further Exercises

 

5. Functions

5.1 Functional Notation

5.2 Operations on Functions

5.3 Induced Set Functions

5.4 Surjections, Injections, and Bijections

5.5 Identity Functions, Cancellation, Inverse Functions, and Restrictions

5.6 Further Exercises

 

6. Binary Relations

6.1 Partitions

6.2 Equivalence Relations

6.3 Order Relations

6.4 Bounds and Extremal Elements

6.5 Applications to Calculus

6.6 Functions Revisited

6.7 Further Exercises

 

7. Infinite Sets and Cardinality

7.1 Counting

7.2 Properties of Countable Sets

7.3 Counting Countable Sets

7.4 Binary Relations on Cardinal Numbers

7.5 Uncountable Sets

7.6 Further Exercises

 

8. Algebraic Systems

8.1 Binary Operations

8.2 Modular Arithmetic

8.3 Numbers Revisited

8.4 Complex Numbers

8.5 Further Exercises

 

Index of Symbols and Notation

Index

 

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