A function that is defined only for a set of numbers that can be listed, such as
the set of whole numbers or the set of
integers, is called a discrete function.
This chapter explores several different discrete functions.
The first function explored is the factorial function. This is the focus of
the first section. Here, we will learn how to compute the factorial function of
a number, and how to use the factorial function to find the number of ways n
items can be arranged in an order.
The second section introduces two functions that are derived from the factorial
function -- the permutation function and the combination function.
These functions are used to compute the number of ways n items can be chosen
or arranged in n or fewer spots.
The final section deals with a different type of discrete functions:
recursively defined functions. These are functions that are defined in
terms of the same function of a smaller variable. Some can also be defined
explicitly, but others cannot. One particularly interesting function that
cannot easily be defined explicitly yields the Fibonacci numbers, which are
explored at the end of this section. These numbers have several interesting
properties which mathematicians spend much time studying. They also occur
frequently in nature.
Discrete functions comprise their own branch of mathematics. In addition, they
have many applications: the factorial, permutation, and combination functions
are used in statistics and probability, and recursively defined functions are
used to prove theorems in mathematical logic. Discrete functions are both
useful and fascinating to study.