Mathematical Proofs: A Transition to Advanced Mathematics

Gary Chartrand / Albert D. Polimeni / Ping Zhang  
Total pages
November 2013
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Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.


  • 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 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 receives an entire chapter, with sections covering counterexamples, existence proofs, and uniqueness.
  • A wide variety of exercises is provided in the text.
    • Difficulty ranges from routine to medium to moderately challenging
    • True/False exercises present statements and ask students to determine whether they are correct, asking for justification as part of the process.
    • Proposed proofs of statements ask students if an argument is valid.
    • Proofs without a statement ask students to supply a statement of what has been proved.
    • Finally, there are exercises that call upon students to ask questions of their own and to provide answers.


New to this Edition

  • New Exercises: More than 250 exercises have been added, including many challenging exercises at the end of exercise sets. New exercises include some dealing with making conjectures to give students practice with this important aspect of advanced mathematics.
  • New and Revised Examples: Examples have been added and heavily revised with new proofs, adding support for the material to give students better understanding and helping them to solve new exercises.
  • Additional sections: Chapter 13 has been expanded to include coverage of cosets and Lagrange’s theorem. The important topic of quantified statements is now introduced in Section 2.10 and reviewed in Section 7.2 in reinforce the student’s understanding.


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

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


14. Proofs in Ring Theory (Online)

14.1 Rings

14.2 Elementary Properties of Rings

14.3 Subrings

14.4 Integral Domains

14.5 Fields

Exercises for Chapter 14


15. Proofs in Linear Algebra (Online)

15.1 Properties of Vectors in 3-Space

15.2 Vector Spaces

15.3 Matrices

15.4 Some Properties of Vector Spaces

15.5 Subspaces

15.6 Spans of Vectors

15.7 Linear Dependence and Independence

15.8 Linear Transformations

15.9 Properties of Linear Transformations

Exercises for Chapter 15


16. Proofs in Topology (Online)

16.1 Metric Spaces

16.2 Open Sets in Metric Spaces

16.3 Continuity in Metric Spaces

16.4 Topological Spaces

16.5 Continuity in Topological Spaces

Exercises for Chapter 16


Answers and Hints to Odd-Numbered Section Exercises


Index of Symbols

Index of Mathematical Terms