CHAPTER-5
INFORMATION MEASURE OF FUZZY MATRIX AND
FUZZY BINARY RELATION
5.1. Introduction
The basic concept of the fuzzy matrix theory is very simple and can be applied to
social and natural situations. A branch of fuzzy matrix theory uses algorithms and
algebra to analyse data. It is used by social scientists to analyse interactions between
actors and can be used to complement analyses carried out using game theory or other
analytical tools. The concepts of fuzzy matrix have been defined in chapter 1.
A standard fuzzy matrix is the fuzzy matrix of the following form where all the
entries are less than or equal to 0.5:
0.1 0 0.1
0.5 0.2 0.5
0.4 0.2 0.3
X=
(5.1.1)
Operations on Two Fuzzy Matrices
Let us define two fuzzy matrices
X and Y of order 4 x 4 as
0.3 0.7 0.8 0.9
X
0.4 0.5
1
0.3
(5.1.2)
0.6 0.1 0.4 0.8
0.9 0.4 0.6 0.2
110
1
0.3
0.8
0.3
Y
0.2
0.6
0.9
0.2
0.4
0.1
0.5
0.5
0.9
0.4
0.6
0.7
(5.1.3)
Addition of Two Fuzzy Matrices
Two fuzzy matrices X and
Y are compatible under matrix addition if they are of
same order. For instance for the fuzzy matrices
X and Y given by (5.1.2) and (5.1.3) we
have,
1.3 0.9 1.2 1.8
X
Y
0.7 1.1 1.1 0.7
1.4
1
(5.1.4)
0.9 1.4
1.2 0.6 1.1 0.9
Clearly
X
Y is a matrix, but not a fuzzy matrix. Hence we can conclude that
addition of two fuzzy matrices compatible under addition need not be a fuzzy matrix.
However, addition of two standard fuzzy matrices is a fuzzy matrix.
Maximum Operation of Two Fuzzy Matrices
Two fuzzy matrixes are conformable for maximum operation if they are of the
same order. Hence for two matrices X
these two matrices is a matrix
cij
xij and Y
Max( X , Y )
cij
yij of order m x n, maxima of
of order m x n, where
max( xij , yij ) .
Hence for the matrices
X and Y given by (5.1.2) and (5.1.3) we have,
1
Max X ,Y
0.7 0.8 0.9
0.4 0.6
1
0.4
0.8 0.9 0.5 0.8
0.9 0.4 0.6 0.7
111
(5.1.5)
Minimum Operation of Two Fuzzy Matrices
Two fuzzy matrixes are conformable for minimum operation if they are of same
order. Hence for two matrices X
xij and Y
these two matrices is a matrix Min( X , Y )
where cij
yij of order m x n, fuzzy minima of
cij of order m x n
min( xij , yij ) .Hence for the matrices X and Y given by (5.1.2) and (5.1.3)
we have,
0.3 0.2 0.4 0.9
Min (X, Y)
0.3 0.5 0.1 0.3
(5.1.6)
0.6 0.1 0.4 0.6
0.3 0.2 0.5 0.2
In case of fuzzy matrices, we have seen that the addition is not defined, where as the
maxima and minima operations are defined. Clearly under the maximum and minimum
operations the resultant matrix is again a fuzzy matrix of the same order and thus is in
someway analogous to our usual addition.
Product of Two Fuzzy Matrices
X with Y given by (5.1.2) and (5.1.3),
where X and Y are compatible under multiplication; i.e. the number of column of X
To find the product of two fuzzy matrices
equal to the number of row of
Y ; still we may not have the product X Y to be a fuzzy
matrix.
XY
Clearly
1.42 1.38 1.04 1.60
1.24 0.60 0.86 1.37 .
0.74 0.7 0.85 1.38
0.75
1
0.8 1.47
(5.1.7)
X Y is not a fuzzy matrix.
Thus we need to define a compatible operation analogous to multiplication of two
fuzzy matrices so that the product again happens to be a fuzzy matrix. However, even for
this new operation if the product
X Y is to be defined we need the number of columns
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of
X is equal to the number of rows of Y i.e. the fuzzy matrix should be compatible for
multiplication. The two types of operations which we can have are max-min operation
and min-max operation.These operations are defined below:
To find
X Y using max-min operations, we have
C
c11
c21
c31
c12
c22
c32
c13
c23
c33
c14
c24
c34
c41
c42
c43
c44
(5.1.8)
where,
c11 = max {min (0.3, 1), min (0.7, 0.3), min (0.8, 0.8),min (0.9, 0.3)}
= max {0.3, 0.3, 0.8, 0.3}= 0.8.
c12 = max {min (0.3, 0.2), min (0.7, 0.6), min (0.8, 0..9),min (0.9, 0.2)
= max {0.2, 0.6, 0.8, 0.2}= 0.8 and so on. Thus, we get
C
0.8 0.8 0.5 0.7
0.8 0.9 0.5 0.6
0.6 0.4 0.5 0.7
(5.1.9)
0.9 0.6 0.5 0.9
Now suppose that the fuzzy matrices
X and Y are given by (5.1.2) and (5.1.3), on
applying min-max operation, we get
XY
D11
D21
D31
D41
D12
D22
D32
D42
D13
D23
D33
D43
D14
D24
D34
D44
(5.1.10)
where,
D11 = min {max (0.3, 1), max (0.7, 0.3), max (0.8, 0.8), max (0.9, 0.3)}
= min {1, 0.7, 0.8, 0.9} = 0.7.
D12 = min {max (0.3, 0.2), max (0.7, 0.6), max (0.8, 0.9), max (0.9, 0.2)}
= min {0.3, 0.7, 0.9, 0.9} = 0.3 and so on. Thus, we have
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0.7 0.3 0.4 0.7
0.3 0.3 0.4 0.5
D
(5.1.11)
0.3 0.6 0.1 0.4
0.3 0.2 0.4 0.4
D.
From (5.1.9) and (5.1.11), it is clear that C
Some experts may like to work with maxima-minima value and some with the
minima-maxima value and accordingly they can adopt it. Hence we can have the product
of two fuzzy matrices
XY = C or D as per our requirement. It is also observed that
XY is defined but YX may not be defined.
Conjugate of Fuzzy Matrix
Let
X = ( xij ) m x n
element xij of
be a fuzzy matrix, the matrix obtained by replacing each
X by its dual (1 - xij ) is called conjugate fuzzy matrix of X and is
denoted as X Conjugate of fuzzy matrices
X and Y
fuzzy matrix product of max-min and min-max operations as given below:
(i)
max min XY
(ii)
min max XY
(5.1.12)
min - max X Y
(5.1.13)
max - min X Y
(5.1.12) and (5.1.13) can be illustrated by the following example:
Illustration(i) Let
X and Y are two fuzzy matrices given by (5.1.2) and (5.1.3)
respectively. Then,
0.2 0.2 0.5 0.3
max min XY
0.2 0.1 0.5 0.4
(5.1.14)
0.4 0.6 0.5 0.3
0.1 0.4 0.5 0.1
0.2 0.2 0.5 0.3
min - max X Y
0.2 0.1 0.5 0.4
0.4 0.6 0.5 0.3
(5.1.15)
0.1 0.4 0.5 0.1
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From (5.1.14)and (5.1.15) we conclude (i) holds.
Illustration: (ii) Let
X and Y are two fuzzy matrices given by (5.1.2) and (5.1.3)
respectively. Then,
0.3
0.7
0.7
0.7
min max XY
0.7
0.7
0.4
0.8
0.6
0.6
0.9
0.6
0.3
0.5
0.6
0.6
0.3 0.7 0.6 0.3
max - min X Y
0.7 0.7 0.6 0.5
0.7 0.4 0.9 0.6
0.7 0.8 0.6 0.6
From (5.1.16) and (5.1.17) we conclude (ii) holds.
Fuzzy matrix Theory (FST) has been applied to many fields such as control, signal
and image processing, medicine, the economy, etc. The results show that FMT yields
efficient solutions to various problems. In crisp set theory, a member of a set is
represented by 0 or 1. So, in a crisp matrix, a member either belongs or doesn't belong to
a class. However, in FMT, a member of a set is represented by a degree between 0 and 1.
The degree is called membership degree which shows belonging degree of the member to
the class. The membership degree is computed using the membership function obtained
by the experts on the subject or a priori knowledge.
In the present chapter we prove that the set of fuzzy matrices forms a lattice under
matrix minima
section 5.3 we
and matrix maxima
binary fuzzy operations in section 5.2. In
introduce and characterize a new fuzzy information measure on fuzzy
matrix and its properties have been studied in section 5.4. In section 5.5 we define a new
measure of information on fuzzy binary relation.
5.2. A Lattice of Fuzzy Matrices
A partial ordered set in which every pair of elements has both least upper bound
and greatest lower bound is called a lattice refer to Tremblay and Manohar (1997) or in
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L, ,
other words Algebra
are binary operations on
(A-1)
A = A or A
(A-2) Commutative ( A
B=B
(B
(A-4) Absorption law ( A
Theorem 1 Let A
Ai
under matrix minima
L is a non empty set,
and
L , satisfying
Idempotent ( A
(A-3) Associative ( A
is called a lattice if
i J
A = A ),
A or A
C)= (A
(A
B=B
A)
B ) C or A ( B
B ) = A or A
(A
C )=( A
B ) = A ).
is a set of m x n fuzzy matrices where J
and matrix maxima
Proof To prove that the given set
operations
B) C)
1,2,......... , then
A forms a lattice.
A of fuzzy matrices forms a lattice, we shall show that
A satisfies the four properties (A-1) to (A-4).
(A-1)
Idempotent Law
For any fuzzy matrix Ai
min( Ai , Ai )
, the following holds:
Ai and max( Ai , Ai )
Ai .
Hence Idempotent Law is satisfied
(A-2)
Commutative Law
It can be easily verified that for all the fuzzy matrices Ai andAj
A , the following
holds:
min( Ai , A j ) min( A j , Ai ) and max( Ai , Aj ) max( Aj , Ai ) .
This proves that Commutative Law is satisfied.
(A-3)
Associative Law
For any fuzzy matrices Ai , A j , Ak
min ( Ai , Aj ), Ak
A , it can be proved that
min( Ai , Aj , Ak ) and max ( Ai , A j ), Ak
Hence we can conclude that
max( Ai , A j , Ak ) .
A is associative under matrix maxima and matrix minima
operations.
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(A-4) Absorption Law
For any fuzzy matrices Ai , A j , we can prove that
min Ai , max Ai , Aj
Ai and max Ai , min Ai , A j
Ai .
Hence Absorption Law holds.
Since it satisfies all the four properties (A-1) to (A-4) of the lattices, therefore the set of
the matrices is a lattice under matrix maxima and matrix minima.
Max(Ai,Aj,Ak)
Max(Ai, Aj)
Ai
Min(Ai, Aj)
Max(Aj, Ak)
Max(Aj, Ak)
Aj
Ak
Min(Aj, Ak
Min(Aj, Ak)
Min(Ai,Aj, Ak)
Figure 5.2.1 A lattice of fuzzy matrices under matrix maxima and matrix minima
Figure 2.1 is the pictorial representation of the lattice with the fuzzy matrices
and
A i, A j
A k with the least element minima ( A i, A j, A k) and the greatest element
maxima ( A i,
A j, A k).
5.3. Information Measure on Fuzzy Matrix
Fuzzy information measures the degree of fuzziness of a fuzzy set. It is peculiar to
mathematics, information theory and computer science. It is an important concept in
fuzzy set theory and has been successfully applied to pattern recognition, image
processing, classifier design and neural network structure etc.
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The concept of information measures was developed by Shannon (1948) to
measure the uncertainty of a probability distribution. The concept of fuzzy set was
introduced by Zadeh (1966) who also developed his own theory to measure the ambiguity
of a fuzzy set.
Let X
x1 , x2 ,..., xn
A ( xi )
be the universe set of discourse and
membership function defined on A
X . Then
A ( x1 ),
A ( x 2 ),
A ( xn )
be
lie between
(0, 1) and these are not probabilities because their sum is not unity. However,
A ( xi )
A ( xi )
, i
n
(5.3.1)
1,2,...,n,
A ( xi )
i 1
is a probability distribution. Thus Kaufman (1980) defined entropy of a fuzzy set A
having n support points as
1 n
A ( xi ) log
log n i 1
H ( A)
A ( xi ).
(5.3.2)
Corresponding to entropy due to Shannon (1948), Deluca and Termini (1971)
suggested the following measure of fuzzy entropy:
n
H ( A)
A ( xi ) log
A ( xi )
1
A ( xi )
log 1
A ( xi )
.
(5.3.3)
i 1
Corresponding to (5.3.3) we propose the following information measure defined on fuzzy
matrices:
m
n
H(X )
xij log xij
1 xij
log 1 xij
,
(5.3.4)
i 1 j 1
where
xij is the
( i , j )th
element of the standard fuzzy matrix
X.
Theorem 2 The fuzzy information measure given by (5.3.4) is a valid measure.
Proof To prove that the given measure is a valid measure of fuzzy information, we shall
show that (5.3.4) satisfies the following four properties (P-1) to (P-4):
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(P-1) H ( X )
0 if only if X is non fuzzy matrix or crisp matrix.
We know that
xij
xij log xij
1, i = 1,2
0 and 1 xij log 1 xij
It implies H ( X )
1,2
0 if only if either xij
0 or
0 if only if X is non fuzzy or
crisp matrix.
(P-2) H ( X ) is maximum if and only if
all i = 1,2
1,2
X is the most fuzzy matrix, i.e. xij
0.5 for
Differentiating H ( X ) with respect to xij , we have
m n
dH X
log xij log 1 xij
dxij
i 1 j 1
which vanishes at xij =0.5
(5.3.5)
Again differentiating H ( X ) with respect to xij , we have
d 2H X
dxij2
m
n
i 1j 1
xij 1
1 xij
1
(5.3.6)
2
0.5 in (5.3.6), we have d H ( X ) 0
2
Putting xij
dxij
Hence H ( X ) is maximum if and only if
1,2
X is most fuzzy matrix i.e. xij 0.5 : i =
1,2
(P-3) Sharpening reduces the value of information measure
Let us consider 0
xij
0.5 , then
m n
dH X
dxij
log xij
log 1 xij
(5.3.7)
0
i 1j 1
It implies H ( X ) is an increasing function of xij in the region 0
xij
Similarly, we can prove that H ( X ) is a decreasing function of
0.5
xij
1
Hence we can conclude that H ( X ) is a concave function.
119
0.5 .
xij in the region
Let
X = (xij ) m x n such that xij
x *ij [0,1] such that be x *ij
Then
0.5 then, X* = (x *ij ) m x n :
xij for all i=1,2
1,2
X * is called the sharpened version of fuzzy matrix X and since H ( X ) is
increasing function of xij in the region 0
(i) if x *ij
xij
Similarly if
H ( X *)
0.5 , therefore
xij
H ( X ) in [0,0.5)
X = (xij ) m x n such that xij
(5.3.8)
0.5 then, X * = (x *ij ) m x n :
x *ij
[0,1] such that be x *ij
xij for all i=1,2
Then
X * is called the sharpened version of fuzzy matrix X and since H ( X ) is
decreasing function of xij in the region 0.5
(ii) if xij
x *ij
xij
1,2
1, therefore
H ( X *) H ( X )
(5.3.9)
(5.3.8) and (5.3.9) together gives H ( X *)
H(X ) .
Thus (P-3) is proved.
(P-4) Dual property i.e. H ( X )
H(X ) .
It is evident from the definition that H ( X )
H(X ) .
Hence H ( X ) satisfies all the essential four properties of fuzzy information
measure. Thus it is a valid measure of fuzzy information. We can call this measure as
fuzzy matrix information measure.
5.4. Properties of Fuzzy Matrix Information Measure
Property 5.4.1 For any standard fuzzy matrices
X and Y compatible under maxima
and minima operations, the following holds:
min { X , Y } X
Proof Let
max { X , Y } and H min { X , Y }
H X
H max { X , Y } .
X and Y are the two standard fuzzy matrices. Since fuzzy matrix information
measure H ( X ) is an increasing function of xij in the region 0
120
xij
0.5 . Therefore
min { X , Y } X
max { X , Y } and H min { X , Y }
H min { X , Y }
HY
H X
H max { X , Y } or
H max { X , Y } .
Property 5.4.2 For any two fuzzy matrices
X and Y , such that XY and YX exists,
we have
XY YX and
H XY
H YX
H X H Y .
Proof of this is evident, which can be illustrated in table 5.4.1, considering different
X and Y .
pairs of fuzzy matrices
Table 5.4.1 H XY
H XY
H YX
H X H Y
H X H Y
H YX
86.2
73.6
5750
78.2
66.5
3355
21.5
23.5
3621
11.2
15.3
1825
23.5
21.5
2221
Property 5.4.3 For any fuzzy matrix X we have,
H X
H X T , where X T is the transpose of X .
The following table 5.4.2 illustrates the proof for different fuzzy matrix
Table 5.4.2 H X
X:
H XT
H X
90.2
86.2
73.6
57
45.3
25.39
H XT
90.2
86.2
73.6
57
45.3
25.39
Property 5.4.4 For any fuzzy matrix
sub matrices, such that
X
X1
X of order n x n, let X i
X2
n
i 1
be the set of square
X 3 ........ , i.e. the nested form of the sub
121
matrices, where A
H X
H X1
B means order of B is less than order of A. We have
H X2
H X 3 ........ ,which forms a line lattice with the least
element H X n and the greatest element H X .
The following table.5.4.3 verifies the result empirically:
Table 5.4.3 H X
X
X1
X2
X3
X4
X5
X6
X7
X8
X9
X 10
X 11
H X1
H X2
H X 3 ........
98.3061
97.55
98.313
97.321
99
90.21977
86.23
87.71
89.333
88.5
73.69292
72.22
73.1231
74.256
73.27
57.52026
56.323
56.421
59.654
58.25
45.39076
45.454
45.356
47.22
45.68
34.56448
33.23
33.32
37.253
35.69
25.39433
26.32
25.64
28.35
26.7
18.97071
17.123
17.87
19.35
20.59
10.68046
10.232
10.132
11.51
10.673
3.883802
3.2525
3.7878
4.865
4.865
1.970951
2.3131
2.0121
2.33
2.045
1
0
1.1
1.26
1.65
122
Figure 5.4.2 is the pictorial representation of the line lattice / column lattice of the
fuzzy information measures on fuzzy matrix of the nested matrices
H
X
H
X1
H
X
2
.
.
.
.
H
X
n
Figure 5.4.2. The pictorial representation of the line lattice
Property 5.4.5 For any fuzzy matrix
between 0 and 1, we have H aX
X of order m x n and any arbitrary element a lies
aH X .
5.5. Fuzzy Binary Relation Information Measure
The word relation suggests some familiar examples of relations such as the
relation of father to son, mother to son, etc. Familiar examples in arithmetic are the
relations such as greater than or less than and so on. These examples suggest that
relationships exist among two objects. Such type of relations is known as binary
relations, between a pair of objects. i.e. any set of order pairs defines a binary relations.
The relationship between three coincident lines or a point between two given points is
examples of relations among three objects. Similarly relations can exist among four or
more objects.
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Fuzzy Binary Relation
Let X
binary relation
each element of
x1 , x 2 , x3 , x 4 , x5 and Y
ij
y1 , y 2 , y3 , y 4 , y5 be two finite sets, fuzzy
which may assign two or more elements of
R
X . Let
ij
Y corresponding to
R be the fuzzy binary membership relation such as
x1 , y1
.9 , x 2 , y1
.1 , x 2 , y 2
.4 , x 2 , y3
x4 , y 4
1 , x5 , y 4
.4 , x 5 , y 5
.5 .
.5 , x 3 , y 3
1 , x4 , y3
.2 ,
(5.5.1)
Operations on Fuzzy Binary Relations
Fuzzy binary relation is very important because they can describe interactions between
variables.
(i)
Intersection of fuzzy binary relations
The intersection of two fuzzy binary relations
R S
(ii)
xi , y j
min
ij
R,
ij
ij
R and
ij
S is defined by
S for each i and j
(5.5.2)
Union of fuzzy binary relations
The union of two fuzzy binary relations
R S
xi , y j
max
ij
R,
ij
ij
R and
ij
S is defined by
S for each i and j
(5.5.3)
Fuzzy binary relations play an important role in fuzzy modelling, fuzzy diagnosis,
and fuzzy control. These also have wide applications in fields such as psychology,
medicine, economics, and sociology.
Fuzzy Binary Relation Matrix
A Fuzzy binary relation
ij
R from a finite set X to a finite set Y can also be
represented by a fuzzy matrix called the fuzzy binary relation matrix of
R . Fuzzy binary
relation matrix is obtained using the membership function and the membership degree.
Let X
x1 , x 2 , x3 , x 4 , x5 and Y
y1 , y2 , y3 , y4 , y5 be two sets and the fuzzy binary
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relation defined on ( X , Y ) is given by (5.5.1), then the fuzzy binary relation matrix is as
follows:
y1
x1 0.9
R
x2 1
x3 0
X ,Y
x4 0
x5 0
y2
y3
0
0
0.4 0.5
y4
y5
0
0
0
0
0
1
0
0
0
0. 2
1
0
0
0
(5.5.4)
0. 4 0. 5
The inverse fuzzy binary relation of
R is the relation
ij
R
ij
and the corresponding matrix is equal to the transpose of the matrix
x1
x2
x3
x4
x5
1
0
0
0
0.4
0
0
0
y3 0
y4 0
0 .5
1
0.2
0
0
0
1
0 .4
y5 0
y6 0
0
0
0
0.5
0
0
0
.2
y1 0.9
y2 0
R 1
Y ,X
R
1
from
Y to X ,
X , Y , i.e.
(5.5.5)
R
Y ,X
T
Analogous to (5.3.4) we can define the following fuzzy information measure on
fuzzy binary relation matrix (5.5.4):
m
n
H R X ,Y
ij (
R ) ln
ij (
R)
1
ij ( R )
ln 1
ij ( R )
,
(5.5.6)
i 1 j 1
where,
ij (R )
is the i , j th element of the binary fuzzy relation matrix
R
X ,Y .
On the lines of proof of theorem 2, it can easily be verified that (5.5.6) is a valid
measure of fuzzy information. However, this measure will be called as fuzzy binary
relation information measure.
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5.6. Conclusion
In the present chapter we have defined two binary operations on fuzzy matrices.
We have also proved that the set of fuzzy matrices forms a lattice under these binary
fuzzy operations. Further we have introduced and characterized a new information
measure on fuzzy matrix. Properties of this proposed measure have also been studied.
Fuzzy relation plays an important role in fuzzy modelling, fuzzy diagnosis and
fuzzy control. They also have applications in fields such as psychology, medicine,
economics and sociology. A fuzzy binary relation R from a finite set
X to a finite set Y
can also be represented by a fuzzy matrix called the fuzzy binary relation matrix of R. The
fuzzy information measure thus defined on fuzzy binary relation matrix is a valid measure
of fuzzy information and can further be generalized. We can also study its application;
however, it is an open problem.
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