A First Course in FUZZY and NEURAL CONTROL

3.4. SENSITIVITY OF FUNCTIONS 105

Proof. Using the fact thatxand 1 −xhave the same range of values for
x∈[0,1], we have the equalities

ρ 4 (δ)=

_

|x−u|∨|y−v|≤δ

|x 4 y−u 4 v|

=

_

|(1−x)−(1−u)|∨|(1−y)−(1−v)|≤δ

|(1−x) 4 (1−y)−(1−u) 4 (1−v)|

=

_

|(1−x)−(1−u)|∨|(1−y)−(1−v)|≤δ

| 1 −(1−x) 4 (1−y)−(1−(1−u) 4 (1−v))|

=

_

|x−u|∨|y−v|≤δ

|x 5 y−u 5 v|

= ρ 5 (δ)

Thusρ 4 (δ)=ρ 5 (δ).

Theorem 3.4The functionsx∧y,x∨y,andα(x)=1−xare the least sensitive
among all continuous Archimedean t-norms and t-conorms, and all negations,
respectively.

Proof. We showed in the first example above thatρ∧(δ)=δ.If 4 is any
t-norm, then

| 141 −(1−δ) 4 (1−δ)| = | 1 −(1−δ) 4 (1−δ)|

≤ ρ 4 (δ)

so(1−δ)≥(1−δ) 4 (1−δ)> 1 −ρ 4 (δ).Thus,ρ 4 (δ)≥δ=ρ∧(δ).
Note that∧is the only t-norm 4 such thatρ 4 (δ)=δ. Indeed, for∧ 6 = 4 ,
there arex,ysuch thatx 4 y 6 =x∧y, and we may assume thatx 4 y<x.Now

|x 41 − 141 |=1−x 4 y> 1 −x

so thatρ 4 (1−x) 6 =1−x.Weleavetherestoftheproofasexercises.

To consider sensitivity of membership functions, let us look at the triangular
functionAwith endpoints(a,0)and(b,0)and high point(c,α)(wherea≤c≤b
and 0 ≤α≤ 1 ). This function is defined by

A(x)=















α

μ

x−a

c−a

∂

ifa≤x≤c

α

μ

x−b

c−b

∂

ifc≤x≤b

0 otherwise

Using the fact that if eitherx≤c≤yory≤c≤x,and|x−y|<δ,then
|x−c|<δand|y−c|<δ, we get an upper bound for the sensitivity of this