1550251515-Classical_Complex_Analysis__Gonzalez_

2 Introduction

There is a relation between the elements of a set and that set, which is
expressed by such phrases as "a is an element of the set A," "a is a member
of A,'' "the element a belongs to the set A," or "a lies in A." This relation
is denoted by the symbol E, and we write a E A to express that a is an
element of the set A. The negation of this relation is denoted by the symbol
~- Thus b ~A means that the object b does not belong to the set A.

If x E A implies that x E B, then set A is said to be a subset of B,

or that B is a superset of A. It is also said that A is contained in B, or

that B contains A. Set inclusion is denoted by the symbol C. Thus we
write A C B (equivalently, B ::::> A) to indicate that every element of A
is also an element of B. ·

If A C B and B C A, then A and B contain the same elements. They

are said to be equal sets, and we write A = B. If the sets do not contain

the same elements, we write A -:/=-B. If A C B and A -:/=-B, A is called

a proper subset of B.
In every mathematical discussion the sets under consideration are always
subsets of a certain fixed set called the master set, the universal set, or the
space, the latter term being applied mainly when the master set is endowed
with an algebraic or topological structure: for example, a vector space or
a Euclidean space. The elements of the space are sometimes called points,
but this term is not to be taken literally in most cases.
To specify a particular set, we may exhibit its elements in a tabulation,

customarily enclosed in braces. Thus we write M = {1, 2, 3, 4} to indicate

that M is the set of the integers from 1 to 4. However, this notation, called

the roster notation, is often impractical. In the so-called rule method a

property or condition that characterizes the elements of a set (among those

of a given master set) is used to specify the set, with the understanding that

those elements that have the property, and only those, belong to the set.

This is sometimes called the axiom of specification. The set A of elements

X such that x E X, with the x having a certain property P, is denoted by

A={x:xEX,P(x)}

In this notation the colon is read "such that" and the comma as "and." For

instance, an open interval (a, b ), a < b, of the real numbers JR is defined

as follows:

(a, b) = { x : x E IR, a < x < b}

A set consisting of just one element is called a singleton) denoted {a}.

For convenience, the concept of a set is extended to include the so-called

empty, or null, or void set, which contains no elements. The empty set is

usually denoted by the symbol 0. This set may be thought as defined by