Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 66

Table 2.1 Basic boolean properties

T S N

Lattice Min Max All

Identity
T ( a , 1 ) = a , T ( a , 0 ) = 0 All - -

S ( a , 0 ) = a , S ( a , 1 ) = 1 - All -

Commutativity
T ( a , b ) = T ( b , a ) All - -

S ( a , b ) = S ( b , a ) - All -

Associativity
T ( a , T ( b , c )) = T ( T ( a , b ), c ) All - -

S ( a , S ( b , c )) = S ( S ( a , b ), c ) - All -

Involution
N ( N ( a )) = a

- - All

B1. Idempotency
T

( a , a ) = a Min - -

S ( a , a ) = a

- Max -

Distributivity
T ( a , S ( b , c )) = S ( T ( a , b ), T ( a , c )) All Max -

S ( a , T ( b , c )) = T ( S ( a , b ), S ( a , c )) Min All -

Absorption
T ( a , S ( a , b )) = a Min All -

S ( a , T ( a , b )) = a All Max -

Non-contradiction
T ( a , N ( a )) = 0 W ˕ - N

N ˕

Excluded-middle
S W ˕ ( a , N ( a )) = 1 - N ≥ N ˕

De Morgan's laws
N ( T ( a , b )) = S ( N ( a ), N ( b )) T = N ⓦ S ⓦ N × N

N ( S ( a , b )) = T ( N ( a ), N ( b )) T = N ⓦ S ⓦ N × N

A c B c A c B c A c A c that follows from B ∪ ( ∩ ) = ( B ∪ ) ∩ ( B ∪ ) = ( B ∪ ) ∩ X = B ∪ =

B c

c . This law, in fuzzy set theory, is translated by

( A ∩ )

(μ · ˃ ) = ˃ + μ · ˃

or

N ( T ( a , N ( b ))) = S ( b , T ( N ( a ), N ( b ))),

W ˕ ,

that holds with T .

Nevertheless, not all derived law has solution within the standard algebras of fuzzy
sets, as it is the case with

= pr od

˕ ,

S = N = N

˕

A c ( A ∪ A ) ∩ ( A ∩ ) = ∅ (or A ∩ ∅ = ∅ ), since for