ABSTRACT
Preliminaries Here we present some of the basic concepts and terminology that will be used in the following chapters of this book.
The introduction of the concept of a “set” by the nineteenth century German mathematician Georg Cantor may have been the most significant development in mathematics since calculus was invented in the seventeenth century. While calculus was used as a tool for solving problems involving motion, set theory was first used to study the properties of the real numbers, then it was developed, together with logic, to investigate the foundations of mathematics (see [CAN], [KAM], and [TAR], for example). Here we are mainly interested in set theory as a symbolic, or supplementary, language for expressing mathematical ideas more precisely and concisely. According to Cantor, a set is a “collection into a whole of definite
and separate objects (called the elements of the set) of our intuition or our thought”, such as the odd numbers from 1 to 9, or the points on a given line segment. We shall not attempt to define what a set is, but rather accept it as a primitive concept which can then be used to define other concepts in the larger mathematical structure. But it is helpful, for practical purposes, to keep in mind this intuitive notion of a set as a collection of objects, whether physical or abstract. Though we shall have occasion to talk about sets of such entities as points or lines or intervals, we shall mainly be interested here in sets of numbers. A set may be finite, such as the odd integers from 1 to 9, which is
made up of five elements, and denoted by
{1, 3, 5, 7, 9} (1.1)
in conventional set notation; or it may be infinite, such as the set of natural numbers
N = {1, 2, 3, 4, · · · } . (1.2)
write we write a /∈ A to indicate that a does not belong to A. Thus 105 ∈ N, while −1 /∈ N. The essential feature of a (well-defined) set A is that, for any element
a, either a ∈ A or a /∈ A. A set is defined either by explicitly writing down its elements, as in (1.1) and (1.2), or by giving a rule for obtaining them. For example, ©
x : x ∈ N, x2 − 4x+ 3 = 0 ª
(1.3)
is the set of natural numbers x such that x2−4x+3 = 0. This can also be expressed more briefly as
© x ∈ N : x2 − 4x+ 3 = 0
ª .
