# Discrete Functions

## Introduction and Summary

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.