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10 Euclidean Algorithm: Number theory is the mathematics of integer arithmetic. In this chapter we will restrict
ourselves to integers, Date Added: 8/6/2008  Visits: 31671 
8 Horner's Algorithm: Horner's algorithm is an excellent example of an algorithm. It is a simple and useful
algorithm, but it does not have much to do with discrete mathematics. Date Added: 8/6/2008  Visits: 30409 
7 Graph Searching: Algorithms for searching graphs go back at least to the nineteenth century when they
were used for mazes. Until the advent of computers they were an extremely obscure topic (as
was graph theory). We are going to look at two algorithms which represent the two main
approaches to searching graphs. Date Added: 8/6/2008  Visits: 30804 
6 Induction: The principle of induction is a one of the most fundamental laws of mathematics.
Students come across it, as here, as a method of proof. However, it is a basic property of the
natural numbers (i.e. the positive integers) and can be considered as part of the definition of the
natural numbers. Since we do not define the natural numbers here, the principle will be simply
given as an axiom.1 Date Added: 8/6/2008  Visits: 30499 
5 Recursive Algorithms: Recursion is a form of definition and of algorithms that is very important in computer
science theory as well as in practice. Recursive algorithms can be inefficient or efficient.1 A
recursive definition or a recursive algorithm is characterized by selfreference. Date Added: 8/6/2008  Visits: 32430 
16 Multinomials: Let us slightly generalize the problem of selecting r out of n objects without respect to
order. Suppose, for example, that I have 20 students. Date Added: 8/6/2008  Visits: 32518 
3 Integer Functions: The integer function that is the heart of this section is the modulo function. However,
before getting to it, let us look at some very simple functions. The first (and most important) of
these is the floor function. We denote this function by lxm although it can be denoted by floor(x)
It is also widely known as the greatest integer function and is found in some computer languages
as the integer function (sometimes denoted INT(x)). Date Added: 8/6/2008  Visits: 30290 
25 Sigma Notation: The sign 3, called sigma, is a glorified symbol for addition that can be quite useful. It
is utilized throughout mathematics, statistics, computer science and all other mathematical
disciplines. With the 3 there is usually an index that typically is an i or j. Date Added: 8/6/2008  Visits: 32401 
1 Relations: This book starts with one of its most abstract topics, so don't let the abstract nature deter
you. Relations are quite simple but like virtually all simple mathematical concepts they have
their subtle aspects. Relations are defined on collections (or sets) of objects. Date Added: 8/6/2008  Visits: 31989 
Preface: This book was begun around January 1991, as a response to a need I perceived in the
classroom. Students who have not had calculus but intend to pursue the mathematical sciences are
not ready for a formalistic proofladen treatment of the subject. Date Added: 8/6/2008  Visits: 32255 
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