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2 / Basic Structures: Sets, Functions, Sequences, and Sums
22
deﬁnition of a set, and the use of the intuitive notion that any property whatever there is a set
consisting of exactly the objects with this property, leads to
paradoxes,
or logical inconsistencies.
This was shown by the English philosopher Bertrand Russell in 1902 (see Exercise 38 for a
description of one of these paradoxes). These logical inconsistencies can be avoided by building
set theory beginning with axioms. We will use Cantor’s original version of set theory, known
as
naive set theory,
without developing an axiomatic version of set theory, because all sets
considered in this book can be treated consistently using Cantor’s original theory.
DEFINITION 2
The objects in a set are called the
elements,
or
members,
of the set. A set is said to
contain
its elements.
We will now introduce notation used to describe membership in sets. We write
a
∈
A
to
denote that
a
is an element of the set
A
. The notation
a
±∈
A
denotes that
a
is not an element of
the set
A
. Note that lowercase letters are usually used to denote elements of sets.
There are several ways to describe a set. One way is to list all the members of a set, when
this is possible. We use a notation where all members of the set are listed between braces. For
example, the notation
{
a
,
b
,
c
,
d
}
represents the set with the four elements
a
,
b
,
c
, and
d
.
EXAMPLE 1
The set
V
of all vowels in the English alphabet can be written as
V
={
a
,
e
,
i
,
o
,
u
}
.
▲
EXAMPLE 2
The set
O
of odd positive integers less than 10 can be expressed by
O
1
,
3
,
5
,
7
,
9
}
.
EXAMPLE 3
Although sets are usually used to group together elements with common properties, there is
nothing that prevents a set from having seemingly unrelated elements. For instance,
{
a
,
2
,
Fred,
New Jersey
}
is the set containing the four elements
a
, 2, Fred, and New Jersey.
Sometimes the brace notation is used to describe a set without listing all its members. Some
members of the set are listed, and then
ellipses
(
...
) are used when the general pattern of the
elements is obvious.
EXAMPLE 4
The set of positive integers less than 100 can be denoted by
{
1
,
2
,
3
,...,
99
}
.
Another way to describe a set is to use
set builder
notation. We characterize all those
elements in the set by stating the property or properties they must have to be members. For
instance, the set
O
of all odd positive integers less than 10 can be written as
O
x

x
is an odd positive integer less than 10
}
,
or, specifying the universe as the set of positive integers, as
O
x
∈
Z
+

x
is odd and
x
<
10
}
.
We often use this type of notation to describe sets when it is impossible to list all the elements
of the set. For instance, the set
Q
+
of all positive rational numbers can be written as
Q
+
x
∈
R

x
=
p
/
q
,
for some positive integers
p
and
q
.