Reasoning and Fuzzy Logic - Fuzzy Logic: An Introductory Course for Engineering Students - page 99

From this representations should follow a representation of Q ∗ , that is,

μ Q ∗ ∈

Y , by taking into account that it should be a logical consequence of the set of
premises

[ , ]

0 1

{ μ P ∗ ,μ P ₒ μ Q } by the Generalized Modus Ponens (GMP). That is, μ Q ∗

does verify

0 = T 0 (μ P ∗ ( x ), J (μ P ( x ), μ Q ( y ))) μ Q ∗ ( y ), (3.1)

for all x ∈ X , y ∈ Y , and a continuous t-norm T 0 verifying

0 = T 0 (μ P ( x ), J (μ P ( x ), μ Q ( y ))) μ Q ( y ),

Y , stating that if P ∗ P , then Q ∗ for all x Q . This is done by

preserving the Modus Ponens (MP) in the occasion in which ' x is P ' is observed.
The fuzzy set

∈ X , y ∈ = =

μ P ∗ is called the input

Q*

P*

and μ Q ∗ is the output .

Obviously,

μ Q ∗ ( y ) = sup

x

T 0 (μ P ∗ ( x ), J (μ P ( x ), μ Q ( y ))), ∀ y ∈ Y ,

∈ X

is the greatest function verifying the GMP ( 3.1 ). This formula is known as the Com-
positional Rule of Fuzzy Inference (CRI, for short), and was introduced by Lotfi A.
Zadeh as the output fuzzy logic considers in the systems that are described by rules. It
is not to be forgotten that T 0 is the continuous t-norm that makes J a T 0 -conditional.
Sometimes, the input is just numerical, crisp in the form ' x

= x 0 ', that is, ' x is

P ∗ 'is' x is x 0 ', or ' x ∈{ x 0 } ' and then μ P ∗ = μ { x 0 } , with

1

, if x = x 0

μ { x 0 } ( x ) =

0 , if x = x 0 .

In this case,

μ Q ∗ ( y ) = sup

x

T 0 (μ { x 0 } ( x ), J (μ P ( x ), μ Q ( y ))) = J (μ P ( x 0 ), μ Q ( y )), ∀ y ∈ Y ,

∈ X

is a simpler expression that does not force to compute sup x ∈ X .
Sometimes, in addition to the input, the rule's consequent μ Q is also numerical,

that is ' y is Q 'is' y = y 0 ', or ' y is y 0 ',or' y ∈{ y 0 } '. In this case