Gary Chartrand is Professor Emeritus of Mathematics at Western Michigan University. He received his Ph.D. in mathematics from Michigan State University. His research is in the area of graph theory. Professor Chartrand has authored or co-authored more than 275 research papers and a number of textbooks in discrete mathematics and graph theory as well as the textbook on mathematical proofs. He has given over 100 lectures at regional, national and international conferences and has been a co-director of many conferences. He has supervised 22 doctoral students and numerous undergraduate research projects and has taught a wide range of subjects in undergraduate and graduate mathematics. He is the recipient of the University Distinguished Faculty Scholar Award and the Alumni Association Teaching Award from Western Michigan University and the Distinguished Faculty Award from the State of Michigan. He was the first managing editor of the Journal of Graph Theory. He is a member of the Institute of Combinatorics and Its Applications, the American Mathematical Society, the Mathematical Association of America and the editorial boards of the Journal of Graph Theory and Discrete Mathematics.

Albert D. Polimeni is an Emeritus Professor of Mathematics at the State University of New York at Fredonia. He received his Ph.D. degree in mathematics from Michigan State University. During his tenure at Fredonia he taught a full range of undergraduate courses in mathematics and graduate mathematics. In addition to the textbook on mathematical proofs, he co-authored a textbook in discrete mathematics. His research interests are in the area of finite group theory and graph theory, having published several papers in both areas. He has given addresses in mathematics to regional, national and international conferences. He served as chairperson of the Department of Mathematics for nine years.

Ping Zhang is Professor of Mathematics at Western Michigan University. She received her Ph.D. in mathematics from Michigan State University. Her research is in the area of graph theory and algebraic combinatorics. Professor Zhang has authored or co-authored more than 200 research papers and four textbooks in discrete mathematics and graph theory as well as the textbook on mathematical proofs. She serves as an editor for a series of books on special topics in mathematics. She has supervised 7 doctoral students and has taught a wide variety of undergraduate and graduate mathematics courses including courses on introduction to research. She has given over 60 lectures at regional, national and international conferences. She is a council member of the Institute of Combinatorics and Its Applications and a member of the American Mathematical Society and the Association of Women in 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

3.1. 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

4.1. 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

5.1. 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. Reviewing Proof Techniques

7.1 Reviewing Direct Proof and Proof by Contrapositive

7.2 Reviewing Proof by Contradiction and Existence Proofs

7.3 Reviewing Induction Proofs

7.4 Reviewing Evaluations of Proposed Proofs

Exercises for Chapter 7

8. Prove or Disprove

8.1 Conjectures in Mathematics

8.2 Revisiting Quantified Statements

8.3 Testing Statements

Exercises for Chapter 8

9. Equivalence Relations

9.1 Relations

9.2 Properties of Relations

9.3 Equivalence Relations

9.4 Properties of Equivalence Classes

9.5 Congruence Modulo n

9.6 The Integers Modulo n

Exercises for Chapter 9

10. Functions

10.1 The Definition of Function

10.2 The Set of All Functions From A to B

10.3 One-to-one and Onto Functions

10.4 Bijective Functions

10.5 Composition of Functions

10.6 Inverse Functions

10.7 Permutations

Exercises for Chapter 10

11. Cardinalities of Sets

11.1 Numerically Equivalent Sets

11.2 Denumerable Sets

11.3 Uncountable Sets

11.4 Comparing Cardinalities of Sets

11.5 The Schröder - Bernstein Theorem

Exercises for Chapter 11

12. Proofs in Number Theory

12.1 Divisibility Properties of Integers

12.2 The Division Algorithm

12.3 Greatest Common Divisors

12.4 The Euclidean Algorithm

12.5 Relatively Prime Integers

12.6 The Fundamental Theorem of Arithmetic

12.7 Concepts Involving Sums of Divisors

Exercises for Chapter 12

13. Proofs in Combinatorics

13.1 The Multiplication and Addition Principles

13.2 The Principle of Inclusion-Exclusion

13.3 The Pigeonhole Principle

13.4 Permutations and Combinations

13.5 The Pascal Triangle

13.6 The Binomial Theorem

13.7 Permutations and Combinations with Repetition

Exercises for Chapter 13

14. Proofs in Calculus

14.1 Limits of Sequences

14.2 Infinite Series

14.3 Limits of Functions

14.4 Fundamental Properties of Limits of Functions

14.5 Continuity

14.6 Differentiability

Exercises for Chapter 14

15. Proofs in Group Theory

15.1 Binary Operations

15.2 Groups

15.3 Permutation Groups

15.4 Fundamental Properties of Groups

15.5 Subgroups

15.6 Isomorphic Groups

Exercises for Chapter 15

16. Proofs in Ring Theory (Online)

16.1 Rings

16.2 Elementary Properties of Rings

16.3 Subrings

16.4 Integral Domains

16.5 Fields

Exercises for Chapter 16

17. Proofs in Linear Algebra (Online)

17.1 Properties of Vectors in 3-Space

17.2 Vector Spaces

17.3 Matrices

17.4 Some Properties of Vector Spaces

17.5 Subspaces

17.6 Spans of Vectors

17.7 Linear Dependence and Independence

17.8 Linear Transformations

17.9 Properties of Linear Transformations

Exercises for Chapter 17

18. Proofs with Real and Complex Numbers (Online)

18.1 The Real Numbers as an Ordered Field

18.2 The Real Numbers and the Completeness Axiom

18.3 Open and Closed Sets of Real Numbers

18.4 Compact Sets of Real Numbers

18.5 Complex Numbers

18.6 De Moivre's Theorem and Euler's Formula

Exercises for Chapter 18

19. Proofs in Topology (Online)

19.1 Metric Spaces

19.2 Open Sets in Metric Spaces

19.3 Continuity in Metric Spaces

19.4 Topological Spaces

19.5 Continuity in Topological Spaces

Exercises for Chapter 19

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