1 Introduction
For most students, the first and often only area of mathematics in college is calculus. And
it is true that calculus is the single most important field of mathematics, whose emergence
in the 17th century signalled the birth of modern mathematics and was the key to the
successful applications of mathematics in the sciences.
But calculus (or analysis) is also very technical. It takes a lot of work even to introduce
its fundamental notions like continuity or derivatives (after all, it took 2 centuries just
to define these notions properly). To get a feeling for the power of its methods, say by
describing one of its important applications in detail, takes years of study.
If you want to become a mathematician, computer scientist, or engineer, this investment
is necessary. But if your goal is to develop a feeling for what mathematics is all about,
where is it that mathematical methods can be helpful, and what kind of questions do
mathematicians work on, you may want to look for the answer in some other fields of
mathematics.
There are many success stories of applied mathematics outside calculus. A recent hot
topic is mathematical cryptography, which is based on number theory (the study of positive
integers 1,2,3,...), and is widely applied, among others, in computer security and electronic
banking. Other important areas in applied mathematics include linear programming, coding
theory, theory of computing. The mathematics in these applications is collectively called
discrete mathematics. (“Discrete” here is used as the opposite of “continuous”; it is also
often used in the more restrictive sense of “finite”.)
The aim of this book is not to cover “discrete mathematics” in depth (it should be clear
from the description above that such a task would be ill-defined and impossible anyway).
Rather, we discuss a number of selected results and methods, mostly from the areas of
combinatorics, graph theory, and combinatorial geometry, with a little elementary number
theory.
At the same time, it is important to realize that mathematics cannot be done without
proofs. Merely stating the facts, without saying something about why these facts are valid,
would be terribly far from the spirit of mathematics and would make it impossible to give
any idea about how it works. Thus, wherever possible, we’ll give the proofs of the theorems
we state. Sometimes this is not possible; quite simple, elementary facts can be extremely
difficult to prove, and some such proofs may take advanced courses to go through. In these
cases, we’ll state at least that the proof is highly technical and goes beyond the scope of
this book.
Another important ingredient of mathematics is problem solving. You won’t be able
to learn any mathematics without dirtying your hands and trying out the ideas you learn
about in the solution of problems. To some, this may sound frightening, but in fact most
people pursue this type of activity almost every day: everybody who plays a game of chess,
or solves a puzzle, is solving discrete mathematical problems. The reader is strongly advised
to answer the questions posed in the text and to go through the problems at the end of
each chapter of this book. Treat it as puzzle solving, and if you find some idea that you
come up with in the solution to play some role later, be satisfied that you are beginning to
get the essence of how mathematics develops.
We hope that we can illustrate that mathematics is a building, where results are built
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