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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. Variables are further distinguished as being either a dependent variable or an independent variable. Independent variables are regarded as inputs to a system and may take on different values freely. Dependent variables are those values that change as a consequence of changes in other values in the system.
The concepts of sets, functions, and variables are fundamental to many areas of finance and its applications. Starting with the mean-variance portfolio theory of Harry Markowitz in 1952, then the capital asset pricing model of William Sharpe in 1964, the option pricing model of Fischer Black and Myron Scholes in 1973, and the more recent developments in financial econometrics, financial risk management and asset pricing, financial economists constantly use the concepts of sets, functions, and variables. In this chapter we discuss these concepts.
What you will learn after reading this chapter:
- The notion of sets and set operations
- How to define empty sets, union of sets, and intersection of sets.
- The elementary properties of sets.
- How to describe the dynamics of quantitative phenomena.
- The concepts of distance and density of points.
- How to define and use functions and variables.
INTRODUCTION
In this chapter we discuss three basic concepts used throughout this book: sets, functions, and variables. These concepts are used in financial economics, financial modeling, and financial econometrics.
SETS AND SET OPERATIONS
The basic concept in calculus and in probability theory is that of a set. A set is a collection of objects called elements. The notions of both elements and set should be considered primitive. Following a common convention, letās denote sets with capital Latin or Greek letters: A, B, C, Ī© ā¦ and elements with small Latin or Greek letters: a, b, Ļ. Letās then consider collections of sets. In this context, a set is regarded as an element at a higher level of aggregation. In some instances, it might be useful to use different alphabets to distinguish between sets and collections of sets.1
Proper Subsets
An element
a of a set
A is said to belong to the set
A written as
a A. If every element that belongs to a set
A also belongs to a set
B, we say that
A is contained in
B and write:
A B. We will distinguish whether
A is a
proper subset of
B (i.e., whether there is at leas...
