A First Course in FUZZY and NEURAL CONTROL

3.10. LEVEL CURVES AND ALPHA-CUTS 125

This procedure is called theextension principle. The extension principle can
be viewed as a function

Map(X,Y)→Map(F(X),F(Y)) :f 7 →E(f)

whereMap(X,Y)denotes the set of all functions fromXtoY,F(X)andF(Y)
denote the set of all fuzzy subsets ofXandY, respectively,Map(F(X),F(Y))
is the set of all functions fromF(X)toF(Y),andE(f)(A)=Bas defined in
Equation 3.12.^2
The fuzzy setE(f)(A)=Bobtained above can be viewed as a composition
of functions

B=∨Af−^1

where

f−^1 (y)={x∈X|f(y)=x}

∨is the function from the set of subsets of[0,1]to[0,1]that takes a set to its
supremum

∨:2[0,1]→[0,1] :S 7 →

_

{s:s∈S}

andAis identified with the set function induced byA

A:2X→ 2 [0,1]:S 7 →{A(s)|s∈S}

That is, we have the composition

B:Y

f−^1

→ 2 X→A 2 [0,1]→∨ [0,1]

3.10.2Imagesofalpha-levelsets ..................

Given a functionf:X→Y, there is a connection between theα-level sets, or
theα-cuts, of a fuzzy setAand the fuzzy set∨Af−^1. This relation is important
in the calculus of fuzzy quantities. Note that the functionfinduces a partition
ofXinto the subsets of the formf−^1 (y).

Theorem 3.7LetXandY be sets,A:X→[0,1]andf:X→Y.Then

1.f(Aα)⊆(∨Af−^1 )αfor allα.

2.f(Aα)=(∨Af−^1 )αforα> 0 if and only if for eachy∈Y,∨A(f−^1 (y))≥

αimpliesA(x)≥αfor somex∈f−^1 (y).

3.f(Aα)=(∨Af−^1 )αfor allα> 0 if and only if for eachy∈Y,∨A(f−^1 (y))

=A(x)for somex∈f−^1 (y).

(^2) In appropriate categorical settings, the mapEis functorial.