Discrete Mathematics

160 Pages·2009·1.16 MB·English

by  The Teaching Company, Professor Arthur T. Benjamin

Discrete Mathematics
Arthur T. Benjamin, Ph.D.
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Arthur T. Benjamin, Ph.D.

Professor of Mathematics, Harvey Mudd College

Arthur T. Benjamin is a Professor of Mathematics at Harvey Mudd College.
He graduated from Carnegie Mellon University in 1983, where he earned a
B.S. in Applied Mathematics with university honors. He received his Ph.D.
in Mathematical Sciences in 1989 from Johns Hopkins University, where he
was supported by a National Science Foundation graduate fellowship and a
Rufus P. Isaacs fellowship. Since 1989, Professor Benjamin has been a
faculty member of the Mathematics Department at Harvey Mudd College,
where he has served as department chair. He has spent sabbatical visits at
Caltech, Brandeis University, and the University of New South Wales in
Sydney, Australia.
In 1999, Professor Benjamin received the Southern California Section of the
Mathematical Association of America (MAA) Award for Distinguished
College or University Teaching of Mathematics, and in 2000, he received
the MAA Deborah and Franklin Tepper Haimo National Award for
Distinguished College or University Teaching of Mathematics. He was
named the 2006–2008 George Pólya Lecturer by the MAA.
Professor Benjamin’s research interests include combinatorics, game
theory, and number theory, with a special fondness for Fibonacci numbers.
Many of these ideas appear in his book (coauthored with Jennifer Quinn)
Proofs That Really Count: The Art of Combinatorial Proof, published by
the MAA. In 2006, that book received the MAA’s Beckenbach Book Prize.
From 2004 to 2008, Professors Benjamin and Quinn served as the coeditors
of Math Horizons magazine, published by the MAA and enjoyed by more
than 20,000 readers, mostly undergraduate math students and their teachers.
In 2009, the MAA published Professor Benjamin’s latest book, Biscuits of
Number Theory, coedited with Ezra Brown.
Professor Benjamin is also a professional magician. He has given more than
1000 “mathemagics” shows to audiences all over the world (from primary
schools to scientific conferences), where he demonstrates and explains his
calculating talents. His techniques are explained in his book Secrets of
Mental Math: The Mathemagician’s Guide to Lightning Calculation and
Amazing Math Tricks. Prolific math and science writer Martin Gardner calls
it “the clearest, simplest, most entertaining, and best book yet on the art of
calculating in your head.” An avid game player, Professor Benjamin was
winner of the American Backgammon Tour in 1997.
©2009 The Teaching Company. i
Professor Benjamin has appeared on dozens of television and radio
programs, including the Today show, CNN, and National Public Radio. He
has been featured in Scientific American, Omni, Discover, People, Esquire,
The New York Times, the Los Angeles Times, and Reader’s Digest. In 2005,
Reader’s Digest called him “America’s Best Math Whiz.”
ii ©2009 The Teaching Company.
Acknowledgments

It is a pleasure to thank the many people who helped me with Discrete
Mathematics. First I would like to thank the many students from Harvey
Mudd College and other Claremont Colleges who have taken discrete
mathematics from me over the last 20 years. I have learned a great deal
teaching and working with these highly motivated students. Special thanks
are due to Harvey Mudd College students Craig Burkhart, Jennifer Iglesias,
Jack Newhouse, Aaron Pribadi, and Elizabeth Reiland; Pitzer College
student Scott Garrabrant; and Harvey Mudd College Professor Geoff
Kuenning, all of whom offered valuable comments on the first draft of this
course. I was very fortunate to be able to present most of these lectures to
the students and faculty at Denison University and Roanoke College. I am
especially grateful to Professors Sarah Crown, Tom Wexler, Jan Minton,
and Roland Minton for their expertise, input, support, and hospitality.
It has been a pleasure working with the ultraprofessional staff of The
Teaching Company. Although I know there were many people working on
this course behind the scenes, I would especially like to thank Zach
Rhoades, Matt Costanza, John Levin, and most of all, Jay Tate.
Last, but not least, I thank my family for their patience and understanding
while this course was being created. I must especially thank my wife, Deena
Benjamin, who is my light, my inspiration, the love of my life, and my
typesetter. This course could not have been made without you!
Arthur T. Benjamin
©2009 The Teaching Company. iii
Table of Contents

Discrete Mathematics

Professor Biography .................................................................................... i
Acknowledgments ...................................................................................... iii
Course Scope ............................................................................................... 1
Lecture One What Is Discrete Mathematics? ......................... 3
Lecture Two Basic Concepts of Combinatorics ...................... 7
Lecture Three The 12-Fold Way of Combinatorics ................ 12
Lecture Four Pascal’s Triangle
and the Binomial Theorem............................... 19
Lecture Five Advanced Combinatorics—Multichosing. 25
Lecture Six The Principle of Inclusion-Exclusion............... 30
Lecture Seven Profs—Inductive,
Geometric, Combinatorial................................ 35
Lecture Eight Linear Recurrences
and Fibonacci Numbers ................................... 39
Lecture Nine Gateway to Number Theory—Divisibility....... 44
Lecture Ten The Structure of Numbers................................ 48
Lecture Eleven Two Principles—Pigeonholes and Parity......... 51
Lecture Twelve Modular Arithmetic—
The Math of Remainders.................................. 55
Lecture Thirteen Enormous Exponents and Card Shuffling........ 59
Lecture Fourteen Fermat’s “Little” Theorem
and Prime Testing ............................................ 62
Lecture Fifteen Open Secrets—Public Key Cryptography........ 66
Lecture Sixteen The Birth of Graph Theory .............................. 69
Lecture Seventen Ways to Walk—
Matrices and Markov Chains ........................... 73
Lecture Eighteen Social Networks and Stable Marriages ............ 76
Lecture Nineteen Tournaments and King Chickens ..................... 80
Lecture Twenty Weighted Graphs
and Minimum Spanning Trees ......................... 83
iv ©2009 The Teaching Company.
Table of Contents

Discrete Mathematics

Lecture Twenty-One Planarity—When Can a Graph
Be Untangled?.................................................. 88
Lecture Twenty-Two Coloring Graphs and Maps .............................. 92
Lecture Twenty-Three Shortest Paths and Algorithm Complexity....... 97
Lecture Twenty-Four The Magic of Discrete Mathematics .............. 101
Answers to Questions to Consider ......................................................... 104
Timeline ................................................................................................... 131
Glossary ................................................................................................... 134
Biographical Notes .................................................................................. 143
Bibliography............................................................................................ 146
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vi ©2009 The Teaching Company.
Discrete Mathematics

Scope:
Discrete mathematics can be described as an advanced look at the
mathematics that we learned as children. In elementary school, we learned
to count, did basic arithmetic, and amused ourselves with solving puzzles,
ranging from connecting the dots, to coloring, to more sophisticated
creative pursuits.
So what exactly is discrete mathematics? Perhaps it is easier to first say
what it is not. Most of the mathematics that we are taught in high school—
from geometry through calculus—is continuous mathematics. Think of the
second hand of a wristwatch or the path traveled by a ball as it is thrown in
the air. These objects are typically described by real numbers and
continuous functions. By contrast, discrete mathematics is concerned with
processes that occur in separate chunks, such as how the seconds or minutes
change on a digital watch, or the way the path of the ball would look if we
took a few snapshots of its journey. The numbers used in discrete
mathematics are whole numbers. Discrete mathematics is the foundation of
computer science, where statements are true or false, numbers are
represented with finite precision, and every piece of data is stored in a
specific place.
In this course, we concentrate on 3 major fields of discrete mathematics:
combinatorics, number theory, and graph theory. Combinatorics is the
mathematics of counting. How many ways can we rearrange the letters of
“Mississippi”? How many different lottery tickets can be printed? How
many ways can we be dealt a full house in poker? Central to the answers to
these questions is Pascal’s triangle, whose numbers contain some amazingly
beautiful patterns, which we shall explore.
Number theory, as its name suggests, is the study of the whole numbers: 0,
1, 2, 3, … . Many of their basic properties were taught to us in elementary
school without any reason given. We remedy that here and present you with
additional surprises. For instance, why can every number be factored into
primes in exactly one way? Why do the digits of a multiple of 9 always sum
to a multiple of 9? How can we tell if a number is composite, even if we do
not know any of its factors? Why are the Fibonacci numbers so beautiful?
Although some mathematicians used to boast that number theory would
have little practical value beyond arithmetic, its applications are (if you will
excuse the pun) numerous, from card shuffling, to ISBNs found in every
©2009 The Teaching Company. 1
book, to Internet security. We will see how number theory forms the basis
for public key cryptography, allowing safe and convenient financial
transactions over the Internet.
Graph theory allows us to explore relationships between objects in a most
effective way. For example, did you know that among any 6 people, there
must always be 3 mutual friends or 3 mutual strangers? Graph theory
enables us to prove this just by drawing 6 dots on a piece of paper,
connected with lines of red for friends and blue for strangers: No matter
how the lines are colored, there must exist either an all-red triangle or an
all-blue triangle. Graph theory can be used to describe networks that model
everything from transportation grids to how computers communicate and
store information. We will answer questions like, Using a network of roads,
what is the quickest way of getting from one point to another? We will see
that this question can be answered using a very efficient algorithm but that a
similar-sounding problem (the traveling salesman problem) has no known
efficient algorithm.
Throughout this course, we will see some beautiful patterns, leading to
some amazing theorems and formulas, but you will not just have to take
my word for them. Using nothing more than elementary logic (requiring
nothing more sophisticated than a first course in algebra), we will be able
to give complete and satisfying explanations to nearly everything presented
in the course. In high school, too much of the mathematics is taught as
nothing more than a collection of facts or techniques to be mastered
without any understanding. In discrete mathematics (and indeed most
college-level math courses), the real joy and mastery of the material
comes from deep understanding.
2 ©2009 The Teaching Company.