# Passage to Abstract Mathematics, Coursesmart eTextbook

Published Date: Jan 14, 2011

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## Description

Passage to Abstract Mathematics facilitates the transition from introductory mathematics courses to the more abstract work that occurs in advanced courses. The text covers logic, proof, numbers, sets, induction, functions, and more—material which instructors of upper-level courses often presume their students have already mastered but is generally missing from lower-level courses. Students learn how to read and write mathematics—especially proofs—the way that mathematicians do. The text emphasizes the use of complete, correct definitions and mathematical syntax.

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The text assumes students have taken at least two semesters of calculus.¿ Occasionally, the authors also refer to multivariable calculus and linear algebra, but these are not essential to the understanding of the text.

CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.

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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Passage to Abstract Mathematics, Coursesmart eTextbook
Format: Safari Book

\$68.99 | ISBN-13: 978-0-321-73866-0