A First Course in FUZZY and NEURAL CONTROL

124 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Anα-cutfor a functionAis the setAαof points for which the value of the
function is greater than or equal toαñthatis,

Aα={x∈X:A(x)≥α} (3.10)

Example 3.14For the Gaussian functionA(x)=e−
x 22
theα-level set is the

set{x 1 ,x 2 }such thate−

x^2 i

(^2) =αfor 0 <α< 1 , the single point 0 forα=1,
and empty forα=0.
Theα-cut is the interval[x 1 ,x 2 ]such thate−
x^2 i
(^2) =αwhen 0 <α< 1 ,as
depicted in thefigure above, the single point 0 forα=1,andtheentiredomain
forα=0.
The notion ofα-cut is important in fuzzy set theory. Theα-cut of a function
A:X→[0,1]is a (crisp) subset ofX, and there is one such subset for each
α∈[0,1]. A fundamental fact about theα-cutsAαis that they determineA.
This fact follows immediately from the equation
A−^1 (α)=Aα

T

(

S

β>α

Aβ)^0 (3.11)

This equation just says that the left side,{x:A(x)=α}, namely those elements
thatAtakes toα, is exactly the set of points that the two sets{x:A(x)≥α}
and{u:A(u)≯α}have in common. But the two sets on the right are given
strictly in terms ofα-cuts, and knowing the setsA−^1 (α)for allαdeterminesA.
So knowing all theα-cuts ofAisthesameasknowingAitself. We can state
this as follows.

Theorem 3.6LetAandBbe fuzzy sets. ThenAα=Bαfor allα∈[0,1]if
and only ifA=B.

3.10.1Extensionprinciple......................

Whenf:X→Yis a function between ordinary setsXandY,andA:X→
[0,1]is a fuzzy subset ofX, we can obtain a fuzzy subsetBofYby

B(y)=

_

f(x)=y

A(x) (3.12)