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Set theory
Set algebra
Example: Let
S
=
{
a,b
}
with
P
S
=
{∅
,
{
a
}
,
{
b
}
,
{
a,b
}}
{
a
}
is a subset of
S
that contains a single element (
a
) of
S
;
{
a
} ∈ P
S
(avoid writing
a
∈ P
S
)
ECSE 305  Winter 2012
40
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View Full Document Set theory
Set algebra
Deﬁnition
A set
F
of subsets of
S
, that is,
F ⊆ P
S
, is an algebra iﬀ
1
S
∈ F
2
A
∈ F ⇒
A
c
∈ F
3
A,B
∈ F ⇒
A
∪
B
∈ F
∅ ∈ F
since
∅
=
S
c
.
The algebra
F
is closed under the operations of complementation and
union.
ECSE 305  Winter 2012
41
Set theory
Set algebra
Example: Let
S
=
{
a,b
}
with
P
S
=
{∅
,
{
a
}
,
{
b
}
,
{
a,b
}}
,and
F
=
{∅
,
{
a
}
,
{
a,b
}}
F ⊂ P
S
{
b
} 6∈ F
.
{
b
} ⊂ {
a,b
} ∈ F 6⇒ {
b
} ∈ F
.
Is
F
an algebra?
ECSE 305  Winter 2012
42
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View Full Document Set theory
Set algebra
Example (contd):
F
=
{∅
,
{
a
}
,
{
a,b
}}
1
S
=
{
a,b
} ∈ F
? YES
2
A
∈ F ⇒
A
c
∈ F
? NO
{
a
}
c
=
{
b
} 6∈ F
3
A,B
∈ F ⇒
A
∪
B
∈ F
? YES
{
a
} ∪ {
a,b
}
=
{
a,b
} ∈ F
ECSE 305  Winter 2012
43
Set theory
Set algebra
An algebra
F
is also closed under the operation of intersection, that
is:
A,B
∈ F ⇒
A
∩
B
∈ F
.
Indeed, we have
A,B
∈ F ⇒
A
c
,B
c
∈ F
⇒
A
c
∪
B
c
∈ F
⇒
(
A
c
∪
B
c
)
c
∈ F
⇒
A
∩
B
∈ F
ECSE 305  Winter 2012
44
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View Full Document Set theory
σ
algebra
Deﬁnition
A set
F
of subsets of
S
, that is,
F ⊆ P
S
, is a
σ
algebra iﬀ
1
S
∈ F
2
A
∈ F ⇒
A
c
∈ F
3
If the sets
A
1
,A
2
,A
3
,...
belong to
F
, so does
∪
∞
i
=1
A
i
.
The algebra
F
is closed under under any countable combination of
complement, union and intersection operations.
Any ﬁnite algebra (i.e.
F
contains a ﬁnite number of elements) is
also a
σ
algebra.
ECSE 305  Winter 2012
45
Set theory
R
is uncountable (simpliﬁed proof)
It suﬃces to show that
A
= [0
,
1]
⊂ R
is uncountable.
Suppose that
A
is countable:
A
=
{
r
1
,r
2
,...,r
n
,...
}
ECSE 305  Winter 2012
46
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View Full Document Set theory
R
is uncountable (simpliﬁed proof)
Decimal expansion:
r
1
= 0
.d
11
d
12
d
13
...d
1
n
...
r
2
= 0
.d
21
d
22
d
23
... d
2
n
...
.
.
.
r
n
= 0
.d
n
1
d
n
2
d
n
3
... d
nn
...
.
.
.
ECSE 305  Winter 2012
47
Set theory
R
is uncountable (simpliﬁed proof)
Decimal expansion:
r
1
= 0
.
d
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This note was uploaded on 02/12/2012 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.
 Spring '09
 Champagne

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