Numerical Methods and Optimization

Chapter 1

Preliminaries

1.1 Sets and Functions

A set is a collection of objects of a certain kind, which are called elements

of the set. We typically denote a set by a capital Greek or Latin letter, i.e.,

A, B, Σ,..., and an element of a set by a small letter, such as a, b, c, . . ..A

set may be given by the list of its elements, for example, A = {a, b, c} or

B = {1, 2}, or by some clearly defined rules, which can be used to determine

whether a given object is an element of the given set. For example, the set of

all real numbers between 0 and 1, inclusive, can be equivalently represented

in several different ways as follows:

X = {x : x

2

− x ≤ 0} = {x :0