Mathematical Proofs:A Transition to Advanced Mathematics: International Edition

Description

Features

• Proof Strategies encourage students to plan what is needed to present a proof of the result in question.
• Proof Analysis segments appear after presentations of proofs and discuss key details considered for the creation of each proof.
• Chapter 0, Communicating Mathematics, at the beginning of the text, provides a valuable reference for students as the course progresses. This chapter prepares students to write effective and clear exposition by emphasizing the correct usage of mathematical symbols, mathematical expressions, and key mathematical terminology.
• Early introduction of Sets (Chapter 1) prepares students for the coverage of logic that follows.
• Early introduction of Logic (Chapter 2) presents the needed prerequisites to get into proofs as quickly as possible. Much of the chapter's emphasis is on statements, implications, and an introduction to qualified statements.
• Proof by Contradiction is covered in an entire chapter.
• A wide variety of exercises is provided in the text. Some exercises present students with statements, asking them to decide whether they are true or false. Others propose proofs of statements, asking students if an argument is valid. Another type, proofs without a statement given, asks students to supply a statement of what has been proved.
• Writing better proofs is a major goal of the text. Students learn to construct proofs that are not only mathematically correct but also clearly-written, convincing, and consistent in notation.
• Web site for "Mathematical Proofs" contains three additional chapters: Chapter 14, Proofs in Ring Theory; Chapter 15, Proofs in Linear Algebra; and Chapter 16, Proofs in Topology.  These can be found at www.aw-bc.com/chartrand

New to this Edition

• New organization: each chapter and its exercises have been divided more formally into sections, so that students can refer back to specific content when doing exercises.
• Improved artwork: all figures in this edition have been re-rendered to improve their quality.
• New examples: Chapters 1-11 have been heavily revised with new proofs and exercises, adding support for the material to give students better understanding.
• 50% more exercises: This edition includes exercises that relate to new examples, more proof evaluation exercises, and exercises in which the proof of an unknown result is given with the goal of determining the result being proved. The exercises in each set increase in difficulty.

Table of Contents

0. Communicating Mathematics

Learning Mathematics

What Others Have Said About Writing

Mathematical Writing

Using Symbols

Writing Mathematical Expressions

Common Words and Phrases in Mathematics

Some Closing Comments about Writing

1. Sets

1.1 Describing a Set

1.2 Subsets

1.3 Set Operations

1.4 Indexed Collections of Sets

1.5 Partitions of Sets

1.6 Cartesian Products of Sets

Exercises for Chapter 1

2. Logic

2.1 Statements

2.2 The Negation of a Statement

2.3 The Disjunction and Conjunction of Statements

2.4 The Implication

2.5 More on Implications

2.6 The Biconditional

2.7 Tautologies and Contradictions

2.8 Logical Equivalence

2.9 Some Fundamental Properties of Logical Equivalence

2.10 Quantified Statements

2.11 Characterizations of Statements

Exercises for Chapter 2

3. Direct Proof and Proof by Contrapositive

3.1 Trivial and Vacuous Proofs

3.2 Direct Proofs

3.3 Proof by Contrapositive

3.4 Proof by Cases

3.5 Proof Evaluations

Exercises for Chapter 3

4. More on Direct Proof and Proof by Contrapositive

4.1 Proofs Involving Divisibility of Integers

4.2 Proofs Involving Congruence of Integers

4.3 Proofs Involving Real Numbers

4.4 Proofs Involving Sets

4.5 Fundamental Properties of Set Operations

4.6 Proofs Involving Cartesian Products of Sets

Exercises for Chapter 4

5. Existence and Proof by Contradiction

5.1 Counterexamples

5.2 Proof by Contradiction

5.3 A Review of Three Proof Techniques

5.4 Existence Proofs

5.5 Disproving Existence Statements

Exercises for Chapter 5

6. Mathematical Induction

6.1 The Principle of Mathematical Induction

6.2 A More General Principle of Mathematical Induction

6.3 Proof by Minimum Counterexample

6.4 The Strong Principle of Mathematical Induction

Exercises for Chapter 6

7. Prove or Disprove

7.1 Conjectures in Mathematics

7.2 Revisiting Quantified Statements

7.3 Testing Statements

7.4 A Quiz of "Prove or Disprove" Problems

Exercises for Chapter 7

8. Equivalence Relations

8.1 Relations

8.2 Properties of Relations

8.3 Equivalence Relations

8.4 Properties of Equivalence Classes

8.5 Congruence Modulo n

8.6 The Integers Modulo n

Exercises for Chapter 8

9. Functions

9.1 The Definition of Function

9.2 The Set of All Functions from A to B

9.3 One-to-one and Onto Functions

9.4 Bijective Functions

9.5 Composition of Functions

9.6 Inverse Functions

9.7 Permutations

Exercises for Chapter 9

10. Cardinalities of Sets

10.1 Numerically Equivalent Sets

10.2 Denumerable Sets

10.3 Uncountable Sets

10.4 Comparing Cardinalities of Sets

10.5 The Schröder-Bernstein Theorem

Exercises for Chapter 10

11. Proofs in Number Theory

11.1 Divisibility Properties of Integers

11.2 The Division Algorithm

11.3 Greatest Common Divisors

11.4 The Euclidean Algorithm

11.5 Relatively Prime Integers

11.6 The Fundamental Theorem of Arithmetic

11.7 Concepts Involving Sums of Divisors

Exercises for Chapter 11

12. Proofs in Calculus

12.1 Limits of Sequences

12.2 Infinite Series

12.3 Limits of Functions

12.4 Fundamental Properties of Limits of Functions

12.5 Continuity

12.6 Differentiability

Exercises for Chapter 12

13. Proofs in Group Theory

13.1 Binary Operations

13.2 Groups

13.3 Permutation Groups

13.4 Fundamental Properties of Groups

13.5 Subgroups

13.6 Isomorphic Groups

Exercises for Chapter 13

Answers and Hints to Selected Odd-Numbered Exercises

References

Index of Symbols

Index of Mathematical Terms

Author