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§3.3 Fuzzy Relation 71
It’s corresponding fuzzy matrix is as follows.
a b c d
A
a 0.0 0.0 0.8 0.0
b 1.0 0.0 0.0 0.0
c 0.0 0.9 0.0 1.0
d 0.0 0.0 0.0 0.0
Fuzzy relation is mainly useful when expressing knowledge. Generally,
the knowledge is composed of rules and facts. A rule can contain the
concept of possibility of event b after event a has occurred. For instance,
let us assume that set A is a set of events and R is a rule. Then by the rule
R, the possibility for the occurrence of event c after event a occurred is 0.8
in the previous fuzzy relation.
When crisp relation R represents the relation from crisp sets A to B, its
domain and range can be defined as,
dom(R) {x | x A, y A, P (x, y) 1}
R
ran(R) {y | x A, y A, P (x, y) 1}
R
Definition (Domain and range of fuzzy relation) When fuzzy relation R
is defined in crisp sets A and B, the domain and range of this relation are
defined as :
P (x) max P (x, y)
dom(R) R
yB
P (y) max P (x, y)
ran(R) R
xA
Set A becomes the support of dom(R) and dom(R) A. Set B is the support
of ran(R) and
ran(R) B. Ƒ
3.3.3 Fuzzy Matrix
Given a certain vector, if an element of this vector has its value between 0
and 1, we call this vector a fuzzy vector. Fuzzy matrix is a gathering of
72 3. Fuzzy Relation and Composition
such vectors. Given a fuzzy matrix A (a ) and B (b ), we can perform
ij ij
operations on these fuzzy matrices.
(1) Sum
A + B Max [a , b ]
ij ij
(2) Max product
AxB AB Max [ Min (a ,b ) ]
k ik kj
(3) Scalar product
OA where 0 dOd 1
Example 3.4 The followings are examples of sum and max product on
fuzzy sets A and B.
a b c a b c
A = a 0.2 0.5 0.0 B = a 1.0 0.1 0.0
b 0.4 1.0 0.1 b 0.0 0.0 0.5
c 0.0 1.0 0.0 c 0.0 1.0 0.1
a b c a b c
A + B = a 1.0 0.5 0.0 AxB a 0.2 0.1 0.5
b 0.4 1.0 0.5 b 0.4 0.1 0.5
c 0.0 1.0 0.1 c 0.0 0.0 0.5
Here let's have a closer look at the product A x B of A and B. For
instance, in the first row and second column of the matrix C A x B, the
value 0.1 (C 0.1) is calculated by applying the Max-Min operation to
12
the values of the first row (0.2, 0.5 and 0.0) of A, and those of the second
column (0.1 , 0.0 and 1.0) of B.
0.2 0.5 0.0
0.1 0.0 1.0
Min
0.1 0.0 0.0 0.1
Max
In the same manner C13 0.5 is obtained by applying the same
procedure of calculation to the first row (0.2, 0.5, 0.0) of A and the third
column of B (0.0, 0.5, 0.1).
§3.3 Fuzzy Relation 73
0.2 0.5 0.0
0.0 0.5 0.1
Min
0.0 0.5 0.0 0.5 Ƒ
Max
And for all i and j, if a d b holds, matrix B is bigger than A.
ij ij
aij d bij AdB
Also when A d B for arbitrary fuzzy matrices S and T, the following
relation holds from the Max-Product operation.
AdB SAdSB,ATdBT
Definition (Fuzzy relation matrix) If a fuzzy relation R is given in the
form of fuzzy matrix, its elements represent the membership values of this
relation. That is, if the matrix is denoted by M , and membership values by
R
P (i, j), then M (P (i, j)) Ƒ
R R R
3.3.4 Operation of Fuzzy Relation
We know now a relation is one kind of sets. Therefore we can apply
operations of fuzzy set to the relation. We assume R A u B and
S A u B.
(1) Union relation
Union of two relations R and S is defined as follows :
(x, y) A u B
P (x, y) Max [P (x, y), P (x, y)]
RS R S
P (x, y) P (x, y)
R S
We generally use the sign for Max operation. For n relations, we
extend it to the following.
PRR R R x, y PRi x,y
1 2 3 n R
i
If expressing the fuzzy relation by fuzzy matrices, i.e. M and M ,
R S
matrix M concerning the union is obtained from the sum of two
R S
matrices M + M .
R S
M M + M
RS R S
(2) Intersection relation
The intersection relation R S of set A and B is defined by the
following membership function.
74 3. Fuzzy Relation and Composition
P (x) =Min [P (x, y), P (x, y)]
R S R S
= P (x, y) P (x, y)
R S
The symbol is for the Min operation. In the same manner, the
intersection relation for n relations is defined by
PR R R R x, y PRi x, y
1 2 3 n
Ri
(3) Complement relation
Complement relation R for fuzzy relation R shall be defined by the
following membership function.
(x, y) A u B P (x, y) 1 - P (x, y)
R R
Example 3.5 Two fuzzy relation matrices M and M are given.
R S
M a b c M a b c
R S
1 0.3 0.2 1.0 1 0.3 0.0 0.1
2 0.8 1.0 1.0 2 0.1 0.8 1.0
3 0.0 1.0 0.0 3 0.6 0.9 0.3
Fuzzy relation matrices M and M corresponding R S and R S
RS R S
yield the followings.
MRS a b c MRS a b c
1 0.3 0.2 1.0 1 0.3 0.0 0.1
2 0.8 1.0 1.0 2 0.1 0.8 1.0
3 0.6 1.0 0.3 3 0.0 0.9 0.0
Also complement relation of fuzzy relation R shall be
MR a b c
1 0.7 0.8 0.0
2 0.2 0.0 0.0
3 1.0 0.0 1.0
(4) Inverse relation
When a fuzzy relation R A u B is given, the inverse relation of R-1 is
defined by the following membership function.
For all (x, y) A u B, P -1 (y, x) P (x, y)
R R
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