A Law and order
Relation preserving maps. Symmetric relations, asymmetric relations, and antisymmetry. The Cartesian
product and the Venn diagram. Euler circles, power sets, partitioning or fibering.
A-1 Law and order: Cartesian product, power sets
In daily life, we make comparisons of type . . . is higher than . . . is smarter than . . . is faster than . . . .
Indeed, we are dealing with order. Mathematically speaking, order is a binary relation of type
xR 1 y : x < y (smaller than) ,
xR 2 y : x ≤ y (smaller-equal) ,
xR 3 y : x > y (larger than, R 3 = R
−1
1 ) ,
xR 4 y : x ≥ y (larger-equal, R 4 = R
−1
2 ) .
(A.1)
Example A.1 (Real numbers).
x and y, for instance, can be real numbers: x, y ∈ R.
End of Example.
Question.
Question: “What is a relation?” Answer: “Let us explain the term relation in the frame of
the following example.”
Example A.2 (Cartesian product).
Define the left set A = {a 1 , a 2 , a 3 } as the set of balls in a left basket, and {a 1 , a 2 , a 3 } = {red,green,blue}.
In contrast, the right set B = {b 1 , b 2 } as the set of balls in a right basket, and {b 1 , b 2 } = {yellow,pink}.
Sequentically, we take a ball from the left basket as well as from the right basket such that we are led
to the combinations
A × B =
(a 1, b 1 ) , (a 1 , b 2 ) , (a 2 , b 1 ) , (a 2 , b 2 ) , (a 3 , b 1 ) , (a 3 , b 2 )
,
(A.2)
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) ,
(iii)
if x ≤ y and y ≤ z, then x ≤ z (transitivity) .
(A.4)
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 is in relation to an element y ∈ M , which in turn is
not in relation to 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 , then x = y
is implied. And R is transitive in M if, for all x, y, z ∈ M , xRy and yRz implies xRz. Now we are
prepared for to introduce more strictly the method of a Cartesian product.
Relation preserving maps. Symmetric relations, asymmetric relations, and antisymmetry. The Cartesian
product and the Venn diagram. Euler circles, power sets, partitioning or fibering.
A-1 Law and order: Cartesian product, power sets
In daily life, we make comparisons of type . . . is higher than . . . is smarter than . . . is faster than . . . .
Indeed, we are dealing with order. Mathematically speaking, order is a binary relation of type
xR 1 y : x < y (smaller than) ,
xR 2 y : x ≤ y (smaller-equal) ,
xR 3 y : x > y (larger than, R 3 = R
−1
1 ) ,
xR 4 y : x ≥ y (larger-equal, R 4 = R
−1
2 ) .
(A.1)
Example A.1 (Real numbers).
x and y, for instance, can be real numbers: x, y ∈ R.
End of Example.
Question.
Question: “What is a relation?” Answer: “Let us explain the term relation in the frame of
the following example.”
Example A.2 (Cartesian product).
Define the left set A = {a 1 , a 2 , a 3 } as the set of balls in a left basket, and {a 1 , a 2 , a 3 } = {red,green,blue}.
In contrast, the right set B = {b 1 , b 2 } as the set of balls in a right basket, and {b 1 , b 2 } = {yellow,pink}.
Sequentically, we take a ball from the left basket as well as from the right basket such that we are led
to the combinations
A × B =
(a 1, b 1 ) , (a 1 , b 2 ) , (a 2 , b 1 ) , (a 2 , b 2 ) , (a 3 , b 1 ) , (a 3 , b 2 )
,
(A.2)
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) ,
(iii)
if x ≤ y and y ≤ z, then x ≤ z (transitivity) .
(A.4)
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 is in relation to an element y ∈ M , which in turn is
not in relation to 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 , then x = y
is implied. And R is transitive in M if, for all x, y, z ∈ M , xRy and yRz implies xRz. Now we are
prepared for to introduce more strictly the method of a Cartesian product.