A First Course in FUZZY and NEURAL CONTROL

116 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Observing that the truth values for the expressionsp(x)andq(x)are identical,
we can conclude that

A(x)∧A^0 (x)=(A(x)∧A^0 (x)∧B(x))∨(A(x)∧A^0 (x)∧B^0 (x))

for allx∈[0,1], or in terms of the fuzzy sets,

A∧A^0 =(A∧A^0 ∧B)∨(A∧A^0 ∧B^0 )

You will be asked to verify this identity for specific fuzzy sets in the exercises.
A proof of the general result is contained in[26].

3.7 Fuzzypartitions ...........................

There is no standardized interpretation of the term ìfuzzy partition.î In prac-
tice, the term is often used to mean any collection{Ai:Xi→[0,1]}of fuzzy
sets for which∪iXiis the desired universe. It may also be assumed that for
eachi,Ai(x)=1for somex,thatis,Aiisnormal. Sometimes the term is used
in one of several ways that more accurately generalize the notion of partition
of ordinary sets. We describe some of these other notions after reviewing the
classical definition of partition.
For ordinary sets, the term ìpartitionî has a universally accepted meaning.
A partition of an ordinary set is a division of the set into nonoverlapping non-
empty pieces. There are many occasions, for example, when we divide a set of
people into nonoverlapping sets, such as age groups or gender groups. Here is
the formal definition.

Definition 3.10 Afinite set{A 1 ,A 2 ,...,An}of nonempty subsets of a setX
is apartitionofXif the following two conditions are satisfied:

1.A 1 ∪A 2 ∪∑∑∑∪An=X

2.Ai∩Aj=∅ifi 6 =j

If we regardAias a characteristic function, this is equivalent to the two
conditions

1.

Pn

i=1Ai(x)=1

2.Ai(x)Aj(x)=0(or equivalently,Ai(x)∧Aj(x)=0) for allx∈X,i 6 =j

In extending the notion of partition to fuzzy sets, we cannot simply use prop-
erties 1 and 2 since having the two conditions (1)max{Ai(x):i=1,...,n}=1
and (2)min{Ai(x),Aj(x)}=0fori 6 =j, would imply that for alliandx,
Ai(x)=0or 1 and thus that theAiare crisp sets. Call a fuzzy setAnormal
ifA(x)=1for somex. This condition together with

Pn
i=1Ai(x)=1does lead
to a good definition of partition for fuzzy sets.