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A Law and Order Question ALawandorder 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 −1 xR1y : x<y (smaller than) , xR3y : x>y (larger than, R3 = R ) , 1 (A.1) ≤ ≥ −1 xR2y : x y (smaller-equal) , xR4y : x y (larger-equal, R4 = R2 ) . Example A.1 (Real numbers). x and y, for instance, can be real numbers: x, y ∈ R. End of Example. Question: “What is a relation?” Answer: “Let us explain the term relation in the frame of the following example.” Question. Example A.2 (Cartesian product). Define the left set A = {a1,a2,a3} as the set of balls in a left basket, and {a1,a2,a3} = {red,green,blue}. In contrast, the right set B = {b1,b2} as the set of balls in a right basket, and {b1,b2} = {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 = (a1,b1) , (a1,b2) , (a2,b1) , (a2,b2) , (a3,b1) , (a3,b2) , (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 R2 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,thenx = y (antisymmetry) , (A.4) (iii) if x ≤ y and y ≤ z,thenx ≤ 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 is in relation to an element y ∈ M, which in turn is notinrelationtox.IfxRy, but by no means yRx applies, we speak of an asymmetric relation.This notion should not be confused with antisymmetry:ifxRy and yRx applies for all x, y ∈ M,thenx = y is implied. And R is transitive in M if, for all x, y, z ∈ M, xRy and yRz implies xRz.Nowweare prepared for to introduce more strictly the method of a Cartesian product. 498 A Law and order b2 b1 a1 a2 a3 Fig. A.1. Cartesian product, Cartesian coordinate system. The elements of the Cartesian product can be illustrated as point set if A and B are bounded subsets of R. 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 × B are represented as points within a rectangle. Example A.2 and Fig. A.1 have indeed prepared the following definition. 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. 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 ∩, ∪,∆,\,and×, we shortly call intersection, union, symmetric difference, difference,andCartesian product. They are illustrated by Figs. A.2–A.8. With respect to these operations, draw the Cartesian product A × B of the following sets A and B: (i) A := {x ∈ N | x ∈ [1;3] ∨ x =4} , B := {y ∈ N | y ∈ [1;2] ∨ y =3} , (ii) A := {1, 2, 3} , (A.6) B := {y ∈ N | y ∈ [1;2[ ∪{3}} , (iii) A := [1;2] ∪ ]3;4[ , B := [0;1] ∪ [3;4[ . Here, we have applied the definitions of a closed, left and right open intervals: [x; y]:= x ≤•≤y, ]x; y]:= x<•≤y, (A.7) [x; y[:= x ≤•<y, ]x; y[:= x<• <y. End of Exercise. A-1 Law and order: Cartesian product, power sets 499 AB A ∩ B 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 A and the set B: A ∩ B := {x | x ∈ A ∧ x ∈ B}. AB A ∪ B Fig. A.3. Venn diagram/Euler circles A ∪ B: the union A ∪ B of two sets A and B is the set af all elements which are in the set A or the set B: A ∪ B := {x | x ∈ A ∨ x ∈ B}. AB A∆B Fig. A.4. Venn diagram/Euler circles: the symmetric difference A∆B of two sets A and B is the set af all elements which are either in set A or in set B: A∆B := {x | x ∈ A∨˙ x ∈ B}. AB A \ B Fig. A.5. Venn diagram/Euler circles: the difference set A \ B of two sets A and B is the set af all elements of A which are not in B: A \ B := {x | x ∈ A ∧ x ∈ B}. 500 A Law and order A × B y x Fig. A.6. Cartesian product A × B; A := {x ∈ N | x ∈ [1; 3] ∨x =4} and B := {y ∈ N | y ∈ [1; 2] ∨ y =3}. A × B y x Fig. A.7. Cartesian product A × B; A := {1, 2, 3} and B := [1; 2[∪{3}. y A × B x Fig. A.8. Cartesian product A × B; A := [1; 2]∪]3; 4[ and B := [0; 1] ∪ [3; 4[. In order to interpret the Cartesian product A × 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. A-1 Law and order: Cartesian product, power sets 501 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 M1 = ∅ , M2 = {1},M3 = {2},M4 = {3} , (A.8) M5 = {1, 2},M6 = {2, 3},M7 = {3, 1} , M8 = {1, 2, 3} , namely built on 8 elements: power(M)= ∅, {1}, {2}, {3}, {12}, {13}, {23}, {123} . (A.9) End of Example. Exercise A.2 (Power set). If n is the number of elements of a set A for which we write |A| = n,then |power(A)| =2n , (A.10) namely the power set of A has exactly 2n elements. The result motivates the name power set.The proof is based on complete induction: A ∅ 123 4 5 6 ... n . (A.11) power(A) 1248163264...2n 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.IfA 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, also called order diagrams (H. Hasse 1896–1979). In such a diagram, two sets M1 and M2 are identified by two points and are connected by a straight line if the lower set M2 is a subset of M1 or M2 ⊆ M1.Inthis way, a set M iscontainedinanysetwhichisabove of M, illustrated by an 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. 502 A Law and order M8 M5 M7 M6 M3 M2 M4 M1 Fig. A.9. Hasse diagram for power(A), |A| =3,|power(A)| =8. A-2 Law and order: Fibering For a set system (which we experienced by power( A), for instance) M = {A1,A2, ..., An},wecall ∩ M ∩n ∪ M ∪n = i=1Ai and = i=1Ai intersection and union of the set system. The inverse operation of the union of a set system, namely the partioning or fibering of a set system into specific subsets is given by the following definition. Definition A.4 (Fibering). ∗ A set system M = {M1,M2,...,Mn} (n ∈ N ) of subsets M1,...,Mn is called a partioning or a fibering of M if and only if (i) Mi = ∅ for any i ∈{1, 2, ..., n} , (ii) Mi ∩ Mj = ∅ for any i, j ∈{1, 2, ..., n} , ∪ ∪ ∪ ∪M ∪n (iii) M = M1 M2 ... Mn = = i=1Mi holds. These subsets of M, the elements of M, are called fibres of M or of M, respectively. End of Definition. In other words, M1,...,Mn are non-empty subsets of M, their paired intersection Mi ∩ Mj is the empty set and their ordered union is M again. axiom analysis algorithm algebra absolute value abacus Fig. A.10. Hasse diagram, lexicographic order. A-2 Law and order: Fibering 503 Example A.5 (Fibering). N∗ { } P | ∈ P Let M = , M1 = 1 , M2 = (set of prime numbers) and M3 = x ab = x for any a and for b ∈ N∗ \{1} the set of compound numbers. Then M = {M1,M2,M3} (A.14) is a partitioning or a fibering of M.
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