Stat 225 Lecture Notes
“Set Theory and Fundamentals of Probability”
Ryan Martin
Spring 2008
1
Set Theory
Set theory is concerned with mathematical properties of very abstract collections of ob
jects. In this course, we will need only the very basics, which we discuss below. This
material is taken primarily from Weiss, Section 1.2.
1.1
Set Notation
A
set
is just a collection of objects, called
elements
. Some common notations follow.
•
Universal and Empty Sets:
It is assumed that there is a universal set, denoted by
U
or Ω, that contains everything. There is also an empty set, denoted by
∅
, that
contains nothing.
•
Containment:
If
x
is an element of a set
A
, we write
x
∈
A
. If
x
is
not
an element
of
A
, we write
x /
∈
A
.
•
Subsets:
If for two sets
A
and
B
, all elements in
A
are also in
B
, then we
A
a subset
of
B
and write
A
⊂
B
. If
A
is
not
a subset of
B
, we write
A
n⊂
B
. To verify that
A
⊂
B
, one must show that if
x
∈
A
then
x
∈
B
.
Sets are often deFned by some property that its elements satisfy. If
P
(
x
) is some statement
regarding
x
, then
{
x
∈
Ω :
P
(
x
)
}
is the set of all
x
∈
Ω such that
P
(
x
) is true. ±or
example,
P
(
x
) could be [
x
≤
2]. In this case,
{
x
∈
Ω :
P
(
x
)
}
= (
∞
,
2].
Defnition 1.1.
Let Ω be a set with subsets
A
and
B
.
(i) The
complement
of
A
is
A
C
=
{
x
∈
Ω :
x /
∈
A
}
(ii) The
union
of
A
and
B
is
A
∪
B
=
{
x
∈
Ω :
x
∈
A
or
x
∈
B
}
.
(iii) The
intersection
of
A
and
B
is
A
∩
B
=
{
x
∈
Ω :
x
∈
A
and
x
∈
B
}
.
Exercise 1.2.
Let Ω =
{
1
,
2
,
3
,
4
,
5
,
6
}
,
A
=
{
1
,
2
,
3
,
4
}
and
B
=
{
2
,
4
,
6
}
. Write out the
elements in each of
A
∪
B
,
A
∩
B
and
B
C
.
1
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View Full DocumentExercise 1.3.
If
A
=
{
x
:
x
≤
r
}
and
B
=
{
x
:
x > s
}
for
s
≤
r
, describe
A
C
and
A
∩
B
.
Those of you familiar with formal logic might see some similarities between the nota
tion there and the set notation introduced above. In particular,
∪ ⇐⇒
“or”
,
∩ ⇐⇒
“and”
,
(
·
)
C
⇐⇒
“not”
To illustrate this relationship, let
P
(
x
) and
Q
(
x
) be statements concerning an element
x
and let
A
=
{
x
:
P
(
x
)
}
and
B
=
{
x
:
Q
(
x
)
}
. Then
A
∪
B
=
{
x
∈
Ω :
P
(
x
) or
Q
(
x
)
}
A
∩
B
=
{
x
∈
Ω :
P
(
x
) and
Q
(
x
)
}
A
C
=
{
x
∈
Ω : not
P
(
x
)
}
Defnition 1.4.
To sets
A
and
B
are said to be
disjoint
if
A
∩
B
=
∅
. In other words,
the sets
A
and
B
have no elements in common.
Exercise 1.5.
For an arbitrary set
A
, give an example of a set
B
such that
A
∩
B
=
∅
.
1.2
Venn Diagrams
Venn diagrams are an easy way to visualize sets and set operations. Let Ω be the universal
set with subsets
A
,
B
,
C
, etc. Figure 1 shows an example of a Venn diagram for three
sets
A
,
B
and
C
.
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 Spring '08
 MARTIN
 Set Theory, Probability, Probability theory

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