118 CHAPTER 3. FUZZY LOGIC FOR CONTROL
Example 3.11IfX={ 2 , 3 , 4 , 6 , 8 }andRis the relation ì(x,y)∈Rif and
only ifxdividesy,î then
R={(2,2),(2,4),(2,6),(2,8),(3,3),(3,6),(4,4),(4,8),(6,6),(8,8)}Example 3.12IfXis the set of real numbers andQis the fuzzy binary relation
onXdescribed by ìQ(x,y)is the degree to whichxis close toy,î then one
possibility forQis the function
Q(x,y)=1
|x−y|+1so thatQ(x,x)=1for allx, whileQ(2,8) =^17.
IfXisfinite, sayX={x 1 ,x 2 ,...,xn}, a fuzzy relation can be represented
as a matrix withijentryR(xi,xj). This matrix will have entries all 0 sand
1 s if and only if the relation is an ordinary relation. For example, the relation
Rabove onX={ 2 , 3 , 4 , 6 , 8 }would be represented as
23468
2
3
4
6
8
10000
01000
10100
11010
10101
while the relationQ, restricted to the same domain, would be represented as
23468
2
3
4
6
8
(^112131415)
1
2 1
1
2
1
3
1
4
1
3
1
2 1
1
2
1
3
1
4
1
3
1
2 1
1
2
1
5
1
4
1
3
1
2 1
Since fuzzy relations are special kinds of fuzzy sets, all of the methods of
combining fuzzy sets can be applied to fuzzy relations. In addition, however,
there is a composition for fuzzy relations.
Definition 3.13 IfRis a fuzzy binary relation inX◊YandQis a fuzzy binary
relation inY◊Z,thesup-min compositionormax-min compositionof
RandQis defined as
(R◦Q)(x,z)=W
y∈Y(R(x,y)∧Q(y,z))