Geography Reference
In-Depth Information
Exercise A.2 (Power set).
If
n
is the number of elements of a set
A
for which we write
|
A
|
=
n
,then
=2
n
,
|
power(
A
)
|
(A.10)
namely the power set of
A
has exactly 2
n
elements. The result motivates the name
power set
.The
proof is based on complete induction:
A
∅
123 4 5 6
...n
(A.11)
12 48163264
...
2
n
.
power(
A
)
End of Exercise.
Example
A.3
and Exercise
A.2
have already used the following definition.
Definition A.3 (Power set).
The power set of a set
A
, shortly written power (
A
), is by definition the set of all subsets
M
of
A
:
power(
A
):=
{
M
|
M
⊆
A
}
(A.12)
End of Definition.
power(
A
) is a set sytem whose elements are just all subsets of
A
.If
A
is a set of
first kind
,power(
A
)
is a set of
second kind
. Inclusions of the above type can be illustrated by
Hasse diagrams
(Fig.
A.9
),
also called
order diagrams
(H. Hasse 1896-1979). In such a diagram, two sets
M
1
and
M
2
are identi-
fied by two points and are connected by a straight line if the
lower
set
M
2
is a subset of
M
1
or M
2
⊆
M
1
.Inthisway,aset
M
is contained in any set which is
above
of
M
, illustrated by all upward
line.
Example A.4 (Hasse diagram).
For the set
A
=
{
1
,
2
,
3
}
,
|
A
|
=3:
power(
A
)=
{∅
,
{
1
}
,
{
2
}
,
{
3
}
,
{
12
}
,
{
13
}
,
{
23
}
,
{
123
}}
,
(A.13)
|
power(
A
)
|
=8
.
End of Example.
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