シナジーのメタファー１

(8) Unification set
A ⋃ B = {(x; μA ⋃B (x))} ∀x∈ G

(9) Intersecting set
A ⋂ B = {( x;μA ⋂ B(x))} ∀x ∈ G

(10) Distributive law
a. A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)
b. A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C)

(11) Complement
A = {( x); μA (x)}∀x ∈ G with μA (x):=1 - μA (x) ∀x ∈ G

(12) Theorem von De Morgan
a. A⋃B// = A/⋂B/
b. A⋂B// = A/⋃B/

(13) Contained
A in B contained ⇔ μA(x) ≤μB(x) ∀x ∈ G

(14) Product of two sets
A・B = {(x; μA.B(x))} ∀x ∈ G with μA.B(x) := μA(x)・B(x) ∀x ∈ G
The product image of the normalized fuzzy set is commutative and associative.

(15) Sum
A+B = {(x;μA+B(x))} ∀x ∈ G mit μA+B(x) := μA(x) + μB(x) - μA(x).μB(x) ∀x∈G
The Sum image of the normalized fuzzy set is commutative and associative.

(16) Implication
When (A) then (B)
Mathematics: (x ∈ A) ⇒ (y ∈ B)
or short A⇒B
where (x), (y) are individual elements
X basic set to x, therefore x ∈ X
Y basic set to y, therefore y ∈ Y
A subset from X, therefore A ⊂ X
B subset from Y, therefore B ⊂Y

Zum Beispiel starben die Eltern von Hans Castorp in der kurzen Frist zwischen seinem fünften und siebenten Lebensjahr, zuerst die Mutter....
Da sein Vater sehr innig an seiner Frau gehangen hatte, auch seinerseits nicht der stärkste war, so wußte er nicht darüber hinwegzukommen. Sein Geist war verstört und geschmälert seitdem; in seiner Benommenheit beging er geschäftliche Fehler, so daß die Firma Castorp &Sohn empfindliche Verluste erlitt; im übernächsten Frühjahr holte er sich bei einer Speicherinspektion am windigen Hafen die Lungenentzündung, und da sein erschüttertes Herz das hohe Fieber nicht aushielt, so starb er trotz aller Sorgfalt.... (Der Zauberberg: 32)

x: momentary work
y: momentary health status
X: generally work = {easy, hard, boring, interesting,...}
Y: generally health status = {healthy, good, tired,...}
A: hard work = {too much, complicate,...}
B: bad health status = {painful, disordered, sick,...}

Implication: When the work is hard, then the body is disordered.
μhard (momentary) =1
μhard (momentary) = 0.8
μhard, disordered (momentary) = min (1; 0.8) = 0.8
The body is disordered by working too hard on a constant basis.
The assignment of the elements (x0) to the membership value μA(x0) mag be fuzzy. That is, the membership function μA(x) is fuzzy by itself. The case is called “ultra-fuzzy”. For example, one can determine whether a fixed child (Hans Castorp) is tolerant. To put it another way, to which extent does it belong to the fuzzy set “tolerant” (with parents, with father or mother, without parents etc.)?

(17) Ultra-fuzzy
An intervall [μA,1(x0); μA, 2 (x0)] is assigned to the value (x0) and μA(x0) represents “tolerant”. Here it is identified as to what extent a person (Typ 1) belongs to fuzzy set “tolerant”.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura