Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 1

Basic Concepts

Sets, Functions, and Variables

In mathematics, sets, functions, and variables are three fundamental concepts. First, a set is a well-defined collection of objects. A set is a gathering together into a whole of definite, distinct objects of our perception, which are called elements of the set. Sets are one of the most fundamental concepts in mathematics. Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields, and rings, are sets closed under one or more operations. One of the main applications of set theory is constructing relations. Second, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Functions are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. A function can be described through its relationship with other functions, for example, as an inverse function or as a solution of a differential equation. Finally, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. ...