ARE 210
Course Notes
page 2
Date of last Update:
8/29/2003
page 2
B
ASIC
C
ONCEPTS FOR THE
A
LGEBRA OF
S
ETS
Many types of study may be characterized by repeated experimentation under
the same set of conditions.
Each
experiment
has an
outcome
, but prior to the experiment the outcome
cannot be predicted with certainty.
If every possible outcome can be described
a priori
, and the experiment can
be repeated, we have a
random experiment
.
The collection of every possible outcome is the
outcome set
, or
sample
space
.
Generically,
space
means the totality of all elements, or outcomes.
A sample
space is a
set
C
, a collection of members or elements,
e
, called events.
Let
C
be a
subset
of
C
.
That is,
C
is a collection of some of the possible out-
comes.
This is denoted
C
⊂
C
.
Then
C
is an
event
.
The outcomes,
e
, are
elements
(members) of the set of all possible outcomes,
C
.
This is denoted
e
∈
C
.
The theory of probability is defined and developed axiomatically in terms of
sets and an algebra system defined over sets.
To develop an algebra of sets, we must have notions of order, equality, zero,
addition, and multiplication similar to the algebra of numbers.
Definitions:
1.
Subset
- A set
C
1
is a
subset
of the set
C
2
, denoted
CC
12
⊂
, if each
element of
C
1
is also an element of
C
2
.
C C
eC eC
1
2
⊂⇔
∈
⇒
∈
lq
2.
Equality
- If
⊂
and
21
⊂
, then
=
.
Two sets are
equal
if and only if they have exactly identical elements.
3.
Null set
- If
C
has no elements it is
empty
, the
null set
.
This is de-
noted
C
=∅
.
The null set is a subset of every other set.
Logical note:
If the hypothesis of a statement is false, then the conclusion
always follows (the statement is vacuous).
If
e
∈∅
(there are
no
such
e
's), then
eC
∈
for all
C
⊂
C
.
If
11 0
+
=
,
then the world is flat.
(Same type of statement.)
4.
Union
- The union of the sets
C
1
and
C
2
, denoted
∪
, is the set
of all elements in at least one of
C
1
or
C
2
.
5.
Intersection
- The
intersection
of two sets
C
1
and
C
2
is the set of all
elements belonging to both of the sets,
CC e
1
2
∩
≡∈
∈
:
and
Note:
Intersections of sets satisfy commutative and associative laws.
CC C C CC C CC
3 1
23
2
13
∩∩
bg
bgbg
==
.
Similarly unions are commutative and associative.
2
∪∪
Theorem:
If
⊂
, then
CC C
2
∪
=
and
1
∩
=
.
Proof:
Suppose
⊂
.
Then
eC C
∈
⇒
∈
∈
1
2
&
∪
.
Hence
C
2
⊂
∪
.
∈
⇒
∈
⊂
2
1
2
∪
since
.
Hence
2
∪
⊂
.
Therefore,
2
∪
=
.
∈⇒
∈
∈
1
2
&
∩
.
Hence
CCC
112
⊂
∩
.
eC C eC
∈
1
∩
(by definition).
Hence
1
∩
⊂
.
Therefore,
1
∩
=
.
Q.E.D.