(7) Unification set
A ⋃ B = {(x; μA ⋃B (x))} ∀x∈ G
(8) Intersecting set
A ⋂ B = {( x;μA ⋂ B(x))} ∀x ∈ G
(9) Distributive law
a. A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)
b. A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C)
(10) Complement
A = {( x); μA (x)}∀x ∈ G with μA (x):=1 – μA (x) ∀x ∈ G
(11) Theorem von De Morgan
a. A⋃B// = A/⋂B/
b. A⋂B// = A/⋃B/
(12) Contained
A in B contained ⇔ μA(x) ≤μB(x) ∀x ∈ G
(13) 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.
(14) 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.
(15) 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
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