Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 43

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μ ˃ ʳ · μ ʳ · ˃ μ · ʳ ˃ · ʳ ʳ ∈[ , ]

(a) If , then , and , for all 0 1

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μ ˃ μ + ʳ ˃ + ʳ ʳ + μ ʳ + ˃ ʳ ∈[ , ]

(b) If , then , and for all 0 1

˃ μ

(c) If μ ˃ , then

X ,

(d) For any μ ∈[ 0 , 1 ] μ · μ 1 = μ 1 · μ = μ , μ + μ 0 = μ 0 + μ = μ

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, μ A

(e) For all μ A , μ B ∈{ 0 , 1 } = μ A c , μ A · μ B = μ A ∩ B , μ A + μ B = μ A ∪ B

(preservation of the classical case).

Remark 2.2.3 It is not difficult to prove that no general algebra of fuzzy sets is a
Boolean algebra. The proof comes from the fact that to be a Boolean algebra would
imply μ · μ = μ 0 and μ + μ = μ 1 for all

X , and consists in finding some

μ ∈[ 0 , 1 ]

μ for which these equalities are not satisfied.

Remark 2.2.4 It is immediate to prove that μ · μ 0 = μ 0 · μ = μ 0 , μ + μ 1 = μ 1 + μ =

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μ 1 for all μ ∈[ 0 , 1 ]

Remark 2.2.5 Notice that the laws μ · ˃ = ˃ · μ (commutativity of the intersection),

μ μ + ˃ = ˃ + μ (involution of the

complement) are not supposed to be always verified. Nor it is supposed that the
algebras

(commutativity of the union), and = μ

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( [ 0 , 1 ] · , + , ) are dual ones , that is, the so-called De Morgan laws,

(μ + ˃) = μ · ˃ ,(μ · ˃) = μ + ˃ ,

are not supposed to hold in general. It is neither supported μ · μ = μ , and ˃ + ˃ = ˃ ,

μ · (˃ · ʻ) = (μ · ˃) · ʻ μ + (˃ + ʻ) = (μ + ˃) + ʻ · +

nor and , nor that and are

μ · (˃ · ʻ) = (μ · ˃) · ʻ μ + (˃ + ʻ) = (μ + ˃) + ʻ

associative, , and .

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Theorem 2.2.6 For any μ, ˃ ∈[ 0 , 1 ] : μ · ˃ min (μ, ˃) max (μ, ˃) μ + ˃ .

Proof From μ μ 1 (μ( x ) 1, for all x ∈ X ), follows μ · ˃ μ 1 · ˃ = ˃ .From

˃ μ 1 , follows μ · ˃ μ 1 · μ = μ . Thus,

(μ · ˃)( x ) ˃( x ), (μ · ˃)( x ) μ( x )

⃒ (μ · ˃)( x ) min (μ( x ), ˃( x )) = min (μ, ˃)( x ),

or . Hence, the operation min is the greatest possible intersection

of fuzzy sets. Analogously,

μ · ˃ min (μ, ˃)

μ 0 μ, μ 0 ˃ ⃒ μ 0 + ˃ = ˃ μ + ˃ , μ + μ 0 = μ

μ + ˃ , and max (μ, ˃) μ + ˃ : The operation max is the smallest possible union of

fuzzy sets.

X :

Obviously, for all μ, ˃ ∈[ 0 , 1 ] μ · ˃ μ μ + ˃, μ · ˃ ˃ μ + ˃ .

X X X

Theorem 2.2.7 An operation ∗:[ 0 , 1 ] ×[ 0 , 1 ] ₒ[ 0 , 1 ] is called idempotent

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if and only if μ ∗ μ = μ , for all μ ∈[ 0 , 1 ]

• The intersection · is idempotent if and only if ·= min

• The union + is idempotent if and only if += max