4
CHAPTER 1.
SET THEORY
1.1
Sets
A set
is a collection of objects. These objects are called elements
of the set.
We often write a set by displaying its elements within braces. For exam-
ple,
{
a, b, c
}
is the set whose elements are
a
,
b
, and
c
.
It is important to note that the set
{
a, b, c
}
is the same as the set
{
b, c, a
}
;
these being just two different ways of displaying the same set.
Similarly,
{
a, b, b
}
is the same set as
{
a, b
}
, since both sets really have the exact same
elements
a
and
b
.
A set
x
is said to be equal
to a set
y
, written
x
=
y
, if
x
and
y
have
exactly the same elements. For example,
{
a, a, b, b, c
}
=
{
b, a, c
}
because both sets have exactly the same elements:
a
,
b
, and
c
.
To prove that two sets are not equal we need only
produce an object which
is an element of one set but not an element of the other
.
The statement “
x
is an element of
y
” is written symbolically as:
x
∈
y
For example,
a
∈ {
a, b
}
The simplest set of all is the empty set
, denoted
∅
.
This set has no
elements at all. Thus, using the braces notation to specify the empty set,
∅
=
{ }
It is an amazing fact that