
## Iv. Interpretation Of Fuzzy Sets

We want to prove that fuzzy sets satisfy the definition of mnesors. But we first substitute the ck (x) = e k ln(x ) for function x →1− x in the definition of the complement. Like 1− x , 
ck (x) takes values in 
[01] , decreases monotonously, exchanges 0 and 1 (that is, ck (0) =1 and ck (1) = 0) and is involutive. Moreover ck (x) values are very close to 1− x if k ≈ 0,4 as you can see below.

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![0_image_0.png](0_image_0.png)

$\left(\mathrm{a}\right)$ 6sec. 
ck ( (x))
n= *ck×n* (x) and ck x n
( ) = c k n
(x) .

Note that 
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The reunion of fuzzy sets and the external multiplication satisfy the linear properties and the complement properties.

€ € µA1(⋅) = µA ( (⋅)) 1 −1 = µA ( (⋅)) 1= µA (⋅) PROOF. (1)  µ(A∪B ) λ(⋅) = µA∪B ( (⋅)) λ−1 = µA (⋅) ⊕ µB ( (⋅)) λ −1 = µA ( (⋅)) λ −1 ⊕ µB ( (⋅)) λ−1 = µA λ(⋅) ⊕ µB λ(⋅) (2)  µA (λ ⊕δ )(⋅) = µA ( (⋅)) (λ ⊕δ ) −1 = µA ( (⋅)) λ−1 ⊕ µA ( (⋅)) δ −1 = µA λ(⋅) ⊕µAδ (⋅) (3)  € µ(A λ)δ (⋅) = µA λ ( (⋅)) δ −1 = µA ( (⋅)) λ −1      δ−1 = µA ( (⋅)) (λ⊗δ ) −1 = µA (λ ⊗δ )(⋅) (4)  € (5)  ck is involutive (6)  µ A (⋅) = µA (⋅) because  € µs 1  (⋅) = ck µs  1 ( (⋅)) = ck (1) = 0 = µs 0 (⋅)

µA λ(⋅) = ck µA λ ( (⋅)) = ck µA ( (⋅))
λ

−1 
 
 = ck×λ µA ( (⋅)) = ck µA ( ( (⋅)))
λ

= µA ( (⋅))
λ

= µA λ
−1 (⋅)
(7) 
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(8) If µA (⋅) ≤ µB (⋅) , then ck µA ( (⋅)) ≥ ck µB ( (⋅)), since the ck function decreases. Hence µA (⋅) ≥ µB (⋅)
REFERENCES
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1. ZADEH L. (1965), Fuzzy Sets