シナジーのメタファー１

　A fuzzy set is the extension of the classical set theory. Hanamura (2005) exhibits it through linguistic variables such as “young” and “big” or by modifiers such as “very” or “most”. For example, how tall is Joachim Ziemßen? In the magic mountain Joachim Ziemßen could belong to the set of the bigger people than Hans Castorp. But how does he fulfill the feature of the fuzzy set? Here is a quantitative measure called the membership value and the membership function.

(2) μA (x) = 0.7
　t means that (x) has a membership value of 0.7 to the set A.
(3) a. μbig (J. Ziemßen)= 0.7
b. μbig (H. Castorp) = 0.3
　There are three kinds of the notations about the fuzzy set; Graphic notation (4), sum notation (5), and paired notation (6). A is a fuzzy set and xi is the element with the membership value μi.

(4) Graphic notation is the clearest and most often used (curve form is selected arbitrarily).

(5) The notation as a sum is used extremely rarely.
A =　μ1/x1 + μ2/x2+ … = �買ﾊi/xi∀x ∈ G
　This is only a possible notation of set (A). The membership value (μi) isn’t divided by the elements (xi), and the pairs (μi/ xi) are also not added.

(6) Notation as ordered pairs set
A = {( x1,μ1), (x2,μ2)，…} ∀x ∈ G
　When (G) is a selection of objects (x), then (A) is a fuzzy set with A = {(x; μA(x))|x ∈G}.
　The elements (xi) whose membership value is μi = 0 is not used to see the things clearly. Set operations and set combinations are also possible in the fuzzy set theory, because they include the sharp and classic set theory.

(7) A, B = fuzzy, normalized set
μA(x), μB(x) = Membership value of the elements (x) to the fuzzy set (A) or (B)
x = considered element
G = Set of all elements (x), therefore the primary set (sharp set involves all [x] completely)
min {...} = Minimum operator selects the minimum from the following curly bracket.
max {...} = Maximum operator selects the maximum from the following curly bracket.
∀ = Allquantor, read as “for all”.
∀x ∈ G means“for all elements x from the set G”.

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