Geography Reference
In-Depth Information
For the Cartesian product, alternative notions are
direct product, product set, cross set, pair set
,
or
union set
.
Exercise A.1 (Cartesian product).
The set of theoretical operations
, we shortly call
intersection, union, symmetric
difference, difference
,and
Cartesian product
. They are illustrated by Figs.
A.2
,
A.3
,
A.4
,
A.5
,
A.6
,
A.7
,and
A.8
. With respect to these operations, draw the Cartesian product
A × B
of the
following sets
A
and
B
:
∩
,
∪
,Δ,
\
,and
×
(i)
A
:=
{
x
∈
N
|
x
∈
[1; 3]
∨
x
=4
}
,
B
:=
{y ∈
N
|y ∈
[1; 2]
∨ y
=3
} ,
(ii)
A
:=
{
1
,
2
,
3
} ,
B
:=
{y ∈
N
|y ∈
[1; 2[
∪{
3
}} ,
(iii)
A
:= [1; 2]
∪
]3; 4[
,
B
:= [0; 1]
∪
[3; 4[
.
(A.6)
Here, we have applied the definitions of a closed, left and right open intervals:
[
x
;
y
]:=
x
≤•≤
y,
]
x
;
y
]:=
x<
y,
[
x
;
y
[:=
x ≤•<y,
•≤
(A.7)
]
x
;
y
[:=
x<• <y.
End of Exercise.
Fig. A.2.
Venn diagram/Euler circles
A
∩
B
: the intersection
A
∩
B
of two sets
A
and
B
is the set of all elements
which are elements of the set
Aand
the set
B
:
A
∩
B
:=
{
x
|
x
∈
A
∧
x
∈
B
}
B
as a set of third kind, we have to understand
better the
power set P
(
A
) of a set, the intersection and union of
set systems
, and the partitioning
of a set system into subsets called
fibering
.
In order to interpret the Cartesian product
A
×
Example A.3 (Power set).
The
power set
as the set of all subsets of a set
A
may be demonstrated by the set
A
=
{
1
,
2
,
3
}
,
whose complete list of subsets read
M
1
=
∅
,
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