Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 44

·

Proof The operations min, and max are obviously idempotent. Let us show that if

·= μ · ˃ (μ, ˃)

is idempotent, it must be min. By Theorem 2.2.6 , it is always min ,

and the idempotency of · implies

min (μ, ˃) · min (μ, ˃) = min (μ, ˃).

But min (μ, ˃) μ, min (μ, ˃) ˃ , imply min (μ, ˃) · min (μ, ˃) μ · ˃ , that is

min (μ, ˃) μ · ˃,

and, by Theorem 2.2.6 ,min (μ, ˃) = μ · ˃ . A similar proof applies to + and max .

Theorem 2.2.8 (Absortion laws)
• μ · (μ + ˃) = μ

X

holds for all μ, ˃ ∈[ 0 , 1 ] ⃔·= min

X

• μ + (μ · ˃) = μ holds for all μ, ˃ ∈[ 0 , 1 ] ⃔+= max

·= (μ, μ + ˃) = μ μ μ + ˃

Proof If min, the formula min does hold, since .

μ · (μ + ˃) = μ ˃ = μ 0 follows μ · (μ + μ 0 ) = μ · μ = μ

Provided it is always , taking ,

that holds if and only if ·= min. If += max, the formula max (μ, μ · ˃) = μ does

hold since μ · ˃ μ . Provided it is always μ + (μ · ˃) = μ , taking ˃ = μ 1 follows

μ + (μ · μ 1 ) = μ + μ = μ , that holds if and only if += max.

(Duality, or De Morgan laws) Provided the complement Theorem 2.2.9 is invo-

((μ ) = μ

X X

lutive = μ , for all μ ∈[ 0 , 1 ] ) , the algebra of fuzzy sets ( [ 0 , 1 ] ,

, ) min , max is a dual algebra.

Proof If is involutive, from μ + ˃ = (μ · ˃) it follows μ · ˃ = (μ + ˃) since

μ + ˃ = μ + ˃ = (μ · ˃ ) , hence (μ + ˃) = μ · ˃ . The converse is proven in

the same way. And the two De Morgan laws

(μ · ˃) = μ + ˃ , μ + ˃) = μ · ˃

(μ , ˃ )) , for all

result equivalent. Then, it is enough to prove max (μ, ˃) = ( min

X . Since

μ, ˃ in [ 0 , 1 ]

(μ , ˃ ) μ , (μ , ˃ ) ˃ , min min

it follows

μ = μ ( (μ , ˃ )) , ˃ = ˃ ( (μ , ˃ )) , min min

and

(μ , ˃ ))

max (μ, ˃) ( min (2.1)

On the other hand,

(μ, ˃)) μ

μ (μ, ˃) ⃒ (

max max