On the Roots of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 21

( ) ∗ ( ) ( ), ( ( ) ∗ ( )) = ↕

Analogously, from N a N b N a it follows a N N a N b a b ,

↕ ,

and b a b , for all a b .

Then, μ P ( x ) ↕ μ Q ( x ) is an L-degree for P or Q , since (remember that it is

P ↂ − 1 ):

P

x Por Q y ⃔ x ( P and Q ) y ⃔ y P and Q x ⃒ μ P and Q ( y )

μ P and Q ( x ) ⃔ μ P ( y ) ∗ μ Q ( y ) μ P ( x ) ∗ μ Q ( x ) ⃒ N (μ P ( y )) ∗ N (μ Q ( y ))

N (μ P ( x )) ∗ N (μ Q ( x )) ⃒ N ( N (μ P ( x )) ∗ N (μ Q ( x )) N ( N (μ P ( y )) ∗ N (μ Q ( y ))

⃔ μ P ( x ) ↕ μ Q ( x ) μ P ( y ) ↕ μ Q ( y ).

Hence, μ P ( x ) ↕ μ Q ( x ) can be taken as an L-degree for ' P or Q 'in X , and the

operation ↕ can be called an or-operation or a disjunction. Notice that

μ P ( ) μ P or Q ( ), μ Q ( ) μ P or Q ( ).

x x x x

In the case ( L , · , + ) is a lattice with the maximum operation += max, and + ∗ ,

it implies μ P ( x ) + μ Q ( x ) μ P or Q ( x ) .

Then, if ( L , · , + , ) is a lattice, at least it is the operation ↕=+ (max), with

which we have the L-degree

μ Por Q ( x ) = μ P ( x ) + μ Q ( x ), ∀ x ∈ X .

With such degree it holds the 'duality' law

μ P or Q = μ PandQ .

Remark 1.3.1 The lattice operation · ( + ) is not necessarily the only operation ∗ ( ↕ ) ,

that can exist. In the case ( L , ) is not a lattice for · ( + ) , there can also exist other

operations ∗ and ↕ . For example, L =[ 0 , 1 ] is not a lattice with ∗= pr od ,but

a b , c d ⃒ pr od ( a , c ) pr od ( b , d ),

and pr od ( a , b ) a , pr od ( a , b ) b . Hence, pr od can be eventually used to

pr od ∗ ( model the use of and . Analogously, ↕= a , b ) = 1 − pr od ( 1 − a , 1 −

pr od ∗ ( pr od ∗ (

b ) = a + b − a · b verifies a b , c d ⃒ a , c ) b , d ), and

pr od ∗ ( pr od ∗ ( . Hence, pr od ∗ can be eventually used to model

, ), , )

a a b b a b

the use of or .

Remark 1.3.2 The existence of operations ∗ and ↕ in L , warrants the existence of

L-degrees for and , or , respectively.

Remark 1.3.3 Since a ∗ b ( a ∗ b ) ↕ ( a ∗ b ) , and a ∗ b a , a ∗ b b it follows

( a ∗ b ) ↕ ( a ∗ b ) a ↕ b ,

and a ∗ b a ↕ b , for all a , b in L , and all pair of operations ∗ and ↕ .