Geography Reference
In-Depth Information
A
×
B
=
{
(red,yellow)
,
(red,pink)
,
(green,yellow)
,
(green,pink)
,
(blue,yellow)
,
(green,pink)
}
.
(A.3)
End of Example.
Definition A.1 (Reflexive partial order).
Let
M
be a non-empty set. The binary relation
R
2
on
M
is called
reflexive partial order
if for all
x, y, z
∈
M
the following three conditions are fulfilled:
(i)
x
≤
x
(reflexivity)
,
(ii)
if
x
≤
y
and
y
≤
x,
then
x
=
y
(antisymmetry)
,
(A.4)
(iii)
if
x
≤
y
and
y
≤
z,
then
x
≤
z
(transitivity)
.
End of Definition.
Obviously, in a reflexive relation
R
, any element
x
∈
M
is in relation
R
to itself. But a relation is
not symmetric if at least one element
x
M
, which in turn is
notinrelationto
x
.If
xRy
, but by no means
yRx
applies, we speak of an
asymmetric relation
.
This notion should not be confused with
antisymmetry
:if
xRy
and
yRx
applies for all
x, y
∈
M
is in relation to an element
y
∈
∈
M
,
M,xRy
and
yRz
implies
xRz
.
Now we are prepared for to introduce more strictly the method of a
Cartesian product
.
The elements of the
Cartesian product
can be illustrated as point set if
A
and
B
are bounded
subsets of
then
x
=
y
is implied. And
R
is
transitive
in
M
if, for all
x, y, z
∈
. Actually, we consider the ordered pair (
a, b
)as(
x, y
) coordinate of the point
P
(
a, b
)
in a
Cartesian coordinate system
such that all ordered pairs of
A
R
B
are represented as points
within a rectangle. Example
A.2
and Fig.
A.1
have indeed prepared the following definition.
×
Fig. A.1.
Cartesian product, Cartesian coordinate system
Definition A.2 (Cartesian product).
The Cartesian product
A×B
of arbitrary sets
A
and
B
is the set of all ordered pairs (
a, b
)whose
left element is
a
∈
A
and whose right element is
b
∈
B
. Symbolically, we write
A
×
B
:=
{
(
a, b
)
|
a
∈
A, b
∈
B
}
.
(A.5)
End of Definition.
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