A First Course in FUZZY and NEURAL CONTROL

3.8. FUZZY RELATIONS 117

Definition 3.11 Afinite set of normal fuzzy subsets{A 1 ,A 2 ,...,An}ofUis a
finite fuzzy partitionof a setUif

1.

Pn

i=1Ai(x)=1for allx∈U

2. EachAiisnormal;thatis,foreachi,Ai(xi)=1for somexi

This definition captures the meaning of properties 1 and 2 above in the
following sense. Eachxhas a nonzero membership value for someAi.Also,if
Ai(x)=1for somei,thenitis 0 for all others. A typical picture of a fuzzy
partition is as follows.

0

0.2

0.4

0.6

0.8

1234

Here, the setX=[0,4]is partitioned into three fuzzy (triangular) subsets.
For afinite setX={x 1 ,...,xm},withm≥n≥ 2 , condition 2 is sometimes
replaced by condition 20 :

20 .Foreachi, 0 <

Pm

k=1Ai(xk)<m.

See [86] for example. This is a weaker condition. If 1 and 2 are satisfied, takej 6 =
i.ThereisanxswithAj(xs)=1, and henceAi(xs)=0,sothat

Pm
k=1Ai(xk)<
m. Also there is anxtfor whichAi(xt)=1,sothat

Pm

k=1Ai(xk)>^0 .Thus^2

0

is satisfied. However, this weaker condition does not force theAito be normal,
and does not force eachxto have a nonzero membership value for anyAi.

3.8 Fuzzyrelations ............................

Arelationis a mathematical description of a situation where certain elements
of sets are related to one another in some way. There are many special kinds
of relations ñ products, functions, equivalence relations, partial orders ñ to
name a few.

Definition 3.12 LetX 1 ,...,Xnbe ordinary sets. Ann-ary relation inX 1 ◊
X 2 ◊∑∑∑◊Xnis a subsetR⊆X 1 ◊X 2 ◊∑∑∑◊Xn.IfX 1 =∑∑∑=Xn=X,a
subsetR⊆X◊X◊∑∑∑◊Xis ann-ary relation onX.A 2 -ary relation is
called abinary relation.
Afuzzy n-ary relation in X 1 ◊X 2 ◊∑∑∑◊Xnis a fuzzy subset R:
X 1 ◊X 2 ◊∑∑∑◊Xn→[0,1],andafuzzyn-ary relation onXis a fuzzy
subsetR:X◊X◊∑∑∑◊X→[0,1].