Journey into Discrete Mathematics is designed for use in a first course in mathematical abstraction for early-career undergraduate mathematics majors. The important ideas of discrete mathematics are included—logic, sets, proof writing, relations, counting, number theory, and graph theory—in a manner that promotes development of a mathematical mindset and prepares students for further study. While the treatment is designed to prepare the student reader for the mathematics major, the book remains attractive and appealing to students of computer science and other problem-solving disciplines. The exposition is exquisite and engaging and features detailed descriptions of the thought processes that one might follow to attack the problems of mathematics. The problems are appealing and vary widely in depth and difficulty. Careful design of the book helps the student reader learn to think like a mathematician through the exposition and the problems provided. Several of the core topics, including counting, number theory, and graph theory, are visited twice: once in an introductory manner and then again in a later chapter with more advanced concepts and with a deeper perspective.
Owen D. Byer and Deirdre L. Smeltzer are both Professors of Mathematics at Eastern Mennonite University. Kenneth L. Wantz is Professor of Mathematics at Regent University. Collectively the authors have specialized expertise and research publications ranging widely over discrete mathematics and have over fifty semesters of combined experience in teaching this subject.
Author(s): Owen Byer; Deirdre L. Smeltzer; Kenneth L. Wantz
Series: AMS/MAA Textbooks 41
Publisher: MAA Press
Year: 2018
Language: English
Pages: xii+388
Cover
Title page
Copyright
Contents
Preface
What Is Discrete Mathematics?
Goals of the Book
Features of the Book
Course Outline
Acknowledgments
Chapter 1. Convince Me!
1.1. Opening Problems
1.2. Solutions
Chapter 2. Mini-Theories
2.1. Introduction
2.2. Divisibility of Integers
2.3. Matrices
Chapter 3. Logic and Sets
3.1. Propositions
3.2. Sets
3.3. Logical Operators and Truth Tables
3.4. Operations on Sets
3.5. Truth Values of Compound Propositions
3.6. Set Identities
3.7. Infinite Sets and Paradoxes
Chapter 4. Logic and Proof
4.1. Logical Equivalences
4.2. Predicates
4.3. Nested Quantifiers
4.4. Rules of Inference
4.5. Methods of Proof
Chapter 5. Relations and Functions
5.1. Relations
5.2. Properties of Relations on a Set
5.3. Functions
5.4. Sequences
Chapter 6. Induction
6.1. Inductive and Deductive Thinking
6.2. Well-Ordering Principle
6.3. Method of Mathematical Induction
6.4. Strong Induction
6.5. Proof of the Division Theorem
Chapter 7. Number Theory
7.1. Primes
7.2. The Euclidean Algorithm
7.3. Linear Diophantine Equations
7.4. Congruences
7.5. Applications
7.6. Additional Problems
Chapter 8. Counting
8.1. What Is Counting?
8.2. Counting Techniques
8.3. Permutations and Combinations
8.4. The Binomial Theorem
8.5. Additional Problems
Chapter 9. Graph Theory
9.1. The Language of Graphs
9.2. Traversing Edges and Visiting Vertices
9.3. Vertex Colorings
9.4. Trees
9.5. Proofs of Euler’s and Ore’s Theorems
Chapter 10. Invariants and Monovariants
10.1. Invariants
10.2. Monovariants
Chapter 11. Topics in Counting
11.1. Inclusion-Exclusion
11.2. The Pigeonhole Principle
11.3. Multinomial Coefficients
11.4. Combinatorial Identities
11.5. Occupancy Problems
Chapter 12. Topics in Number Theory
12.1. More on Primes
12.2. Integers in Other Bases
12.3. More on Congruences
12.4. Nonlinear Diophantine Equations
12.5. Cryptography: Rabin’s Method
Chapter 13. Topics in Graph Theory
13.1. Planar Graphs
13.2. Chromatic Polynomials
13.3. Spanning Tree Algorithms
13.4. Path and Circuit Algorithms
Hints
List of Names
Bibliography
Index
Back Cover
