Reasoning and Fuzzy Logic - Fuzzy Logic: An Introductory Course for Engineering Students - page 113

{5}

μ Q ∗ μ { 5 } μ Q ∗ NC μ { 5 } μ { 5 }

it is clear that is not comparable with ( ), and that

μ Q ∗ = 1 − μ Q ∗ . Hence,

μ Q ∗ ∈ Conj ( { μ { 5 } } ) , and namely μ Q ∗ ∈ Sp ( { μ { 5 } } ).

1

Notice that μ Q ∗ ( 5 ) =

4 =

1.

Example . Rule, “If x is big, then y is very big”, with X = Y =[ 0 , 10 ] and the

observation “ x is constantly equal to 0 . 8” for all x ∈[ 0 , 1 ] . Taking

y 2

μ B ( x ) = x ,μ v B ( y ) = , J ( a , b ) = min ( a , b ) (Mamdani),

follows

y 2 μ Q ∗ ( y ) = min (μ P ∗ ( 0 . 8 ), μ v B ( y )) = min ( 0 . 8 , ).

Graphically,

= √ 0

since y 2 μ P ∗ = μ Q ∗ = = 0 . 8 means y . 8. Hence, it is μ 0 = μ Q ∗ μ P ∗ , and

1 .

Notice that this second example contains the observation that the input is a con-
stant.
Last Remark
For any continuous t-norm T , the function

− μ Q ∗ , that imply μ Q ∗ ∈ Conj ( { μ P ∗ } ) , and namely μ Q ∗ ∈ Hyp ( { μ P ∗ } )

μ : Y ₒ[ 0 , 1 ] , defined by

μ( ) = (μ P ∗ ( ), (μ P ( ), μ Q ( ))), ∀ ∈ ,

y Sup

x ∈ X

T x J x y y Y

does verify