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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