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Publisher:

Chapman & Hall/CRC

Publication Date:

2012

Number of Pages:

538

Format:

Hardcover

Price:

59.95

ISBN:

9781466504998

Category:

Textbook

[Reviewed by , on ]

Charles Ashbacher

08/22/2012

There are many textbooks that can be used in discrete mathematics classes; this one has some unique features, of which one is the many references to ducks. As I looked through the book and approached new topics, my mind drifted to wondering whether the author would use another play on words involved web-footed fowl. For example, when the topic of induction began, my mind wandered to the word “inducktion”, wondering if it would be used. Images of ducks with pencils on their shoulders are used throughout to label positions where there are exercises to work through.

Some of the worked examples and exercises contain a reference to ducks, for example, “How many ways can you select 2 ducks from a flock of 9?” Through the references to ducks, serious and effective mathematical instruction is achieved. The coverage is pretty standard for discrete mathematics courses; the only unusual addition was the section on transfinite numbers. It was presented as a play featuring a set of duck droppings and a modified Hilbert Hotel, which is quite a combination.

There are three sections, each labeled with a theme. They are:

- The basics
- Combinatorics
- Graph theory
- Other material: probability and cardinality.

Graph theory is strongly emphasized; it is the primary topic of 5 of the fifteen chapters. The sections on the basics of set theory and logic are brief, a total of 30 pages. It includes the amusing and quite “natural” use of ducks that only tell the truth and others that only lie. There are some proofs, but probably not enough to satisfy the instructor that wants to emphasize the mathematics.

This book can certainly be used to teach the standard course in discrete mathematics designed for computer science majors. The lighter touch of duck references does amuse you and does not ever overshadow or disguise the mathematical concepts.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing *The Journal of Recreational Mathematics*. In his spare time, he reads about these things and helps his daughter in her lawn care business.

**THE BASICS**Counting and Proofs

Introduction and Summary

The Sum and Product Principles

Preliminaries on Proofs and Disproofs

Pigeons and Correspondences

Where to Go from Here

** **

**Sets and Logic**

Introduction and Summary

Sets

Logic*Try This!* Problems on Sets and Logic

Proof Techniques: Not! *Try This!* A Tricky Conundrum

Where to Go from Here

Bonus: Truth Tellers

** **

**Graphs and Functions**

Introduction and Summary

Function Introduction *Try This!* Play with Functions and Graphs

Functions and Counting

Graphs: Definitions and Examples

Isomorphisms

Graphs: Operations and Uses*Try This!* More Graph Problems

Ramseyness

Where to Go from Here

Bonus: Party Tricks

Bonus 2: Counting with the Characteristic Function

** **

**Induction**

Introduction and Summary

Induction *Try This!* Induction

More Examples

The Best Inducktion Proof Ever *Try This!* More Problems about Induction

Are They or Aren’t They? Resolving Grey Ducks

Where to Go from Here

Bonus: Small Crooks

Bonus 2: An Induction Song

** **

**Algorithms with Ciphers**

Introduction and Summary

Algorithms

Modular Arithmetic (and Equivalence Relations)

Cryptography: Some Ciphers*Try This!* Encryptoequivalent Modulagorithmic Problems

Where to Go from Here

Bonus: Algorithms for Searching Graphs

Bonus 2: Pigeons and Divisibility

** **

**COMBINATORICSBinomial Coefficients and Pascal’s Triangle**

Introduction and Summary

You Have a Choice

Pascal’s Triangle

Overcounting Carefully and Reordering at Will

Try This! Play with Powers and Permutations

Binomial Basics

Combinatorial Proof

Where to Go from Here

Bonus: Sorting Bubbles in Order of Size

Bonus 2: Mastermind

** **

**Balls and Boxes and PIE—Counting Techniques**

Introduction and Summary

Combinatorial Problem Types *Try This!* Let’s Have Some PIE

Combinatorial Problem Solutions and Strategies

Let’s Explain Our PIE! *Try This!* What Are the Balls and What Are the Boxes? And Do You Want Some PIE?

Where to Go from Here

Bonus: Linear and Integer Programming

** **

**Recurrences**

Introduction and Summary

Fibonacci Numbers and Identities

Recurrences and Integer Sequences and Induction *Try This!* Sequences and Fibonacci Identities

Naive Techniques for Finding Closed Forms and Recurrences

Arithmetic Sequences and Finite Differences

Try This! Recurrence Exercises

Geometric Sequences and the Characteristic Equation *Try This!* Find Closed Forms for *These *Recurrence Relations!

Where to Go from Here

Bonus: Recurring Stories

** **

**Cutting up Food (Counting and Geometry)**Introduction and Summary

Try This! Slice Pizza (and a Yam)

Pizza Numbers

Yam, Spaghetti and Pizza Numbers

Where to Go from Here

Bonus: Geometric Gems

** **

**GRAPH THEORYTrees**

Introduction and Summary

Basic Facts about Trees

Spanning Tree Algorithms

Binary Trees

Matchings

Backtracking

Where to Go from Here

Bonus: The Branch-and-Bound Technique in Integer Programming

** **

**Euler’s Formula and Applications**

Introduction and Summary *Try This!* Planarity Explorations

Planarity

A Lovely Story

Or, Are Emus Full?: A Theorem and a Proof

Applications of Euler’s Formula

Try This! Applications of Euler’s Formula

Where to Go from Here

Bonus: Topological Graph Theory

** **

**Graph Traversals**

Introduction and Summary *Try This!* Euler Traversals

Euler Paths and Circuits

Hamilton Circuits, the Traveling Salesperson Problem, and Dijkstra’s Algorithm *Try This!—*Do This!—*Try This!*

Where to Go from Here

Bonus: Digraphs, Euler Traversals, and RNA Chains

Bonus 2: Network Flows

Bonus 3: Two Hamiltonian Theorems

** **

**Graph Coloring**

Introduction and Summary *Try This!* Coloring Vertices and Edges

Introduction to Coloring*Try This!* Let’s Think about Coloring

Coloring and Things (Graphs and Concepts) That Have Come Before

Where to Go from Here

Bonus: The Four-Color Theorem

** **

**OTHER MATERIALProbability and Expectation**

Introduction and Summary

What

High Expectations

You are Probably Expected to

Conditional Probability and Independence

Higher Expectations

Where to Go from Here

Bonus: Ramsey Numbers and the Probabilistic Method

** **

**Fun with Cardinality**

Introduction and Summary

Read This! Parasitology, The Play

How Big Is Infinite? *Try This!* Investigating the Play

How High Can We Count?

Where to Go from Here

Bonus: The Schröder–Bernstein Theorem

**
**

**Additional Problems**

Solutions to Check Yourself Problems

The Greek Alphabet and Some Uses for Some Letters

List of Symbols

Glossary

Bibliography

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