Vol 3, No 7 (2012)

Available online throug

ISSN 2229 – 5046

SOME NEW OPERATORS ON INTERVAL FUZZY NUMBERS

AND INTERVAL FUZZY NUMBER MATRICES

1

Dr. A. KALAICHELVI* &

2

K. JANOFER

1

Associate Professor, Department of Mathematics, Avinashilingam Institute for Home Science and

Higher Education for Women University, Coimbatore – 641 043, India

2

M. Phil. Scholar, Department of Mathematics, Avinashilingam Institute for Home Science and Higher

Education for Women University, Coimbatore – 641 043, India

(Received on: 02-07-12; Accepted on: 20-07-12)

ABSTRACT

I

n this paper, some new elementary operators on Interval Fuzzy Numbers (IFNs) and some new operators on Interval Fuzzy Number Matrices (IFNMs) are defined. Using these operators, some important properties are proved.

INTRODUCTION

Real world decision making problems are very often uncertain (or) vague in a number of ways. In 1965, Zadeh [6] introduced the concept of fuzzy set theory to meet those problems. The fuzzyness can be represented by different ways. One of the most useful representation is the membership function. Depending on the nature of the membership function the fuzzy numbers can be classified in different forms, such as Triangular Fuzzy Numbers (TFNs), Trapezoidal Fuzzy Numbers, Interval Fuzzy Numbers etc. Fuzzy matrices play an important role in scientific development. Fuzzy matrices were introduced by M.G. Thomason [5]. A.K. Shymal and M. Pal introduced Triangular Fuzzy Number Matrices [2]. Interval Fuzzy Number Matrices are introduced by M. Pal, Gobinda Murmu and Anita Pal [4]. Two new operators and some properties of fuzzy matrices over these new operators are given in [1]. Some new operators on Triangular Fuzzy Numbers (TFNs) and Triangular Fuzzy Number Matrices (TFNMs) are discussed in [3].

In this paper, some new elementary operators on Interval Fuzzy Numbers (IFNs) and some new operators on Interval Fuzzy Number Matrices (IFNMs) are defined. Using these operators, some important properties are proved.

1. BASIC DEFINITIONS

Definition: 1.1 An Interval number is defined as

A

ˆ

=

[

a a

_L

,

_R

]

={a:

a

_L

≤

a

≤

a

_R} where

a

_L and

a

_R are the real

numbers called the left end point and right end point respectively of the interval

A

ˆ

.

Example: 1.2

A

ˆ

= [3, 6] is an interval number.

Definition: 1.3 A matrix of order nxn is said to be an interval matrix if all its elements are the interval numbers.

Definition: 1.4 An interval fuzzy number is defined as

A

ˆ

=[

a

_L

,

a

_R]={a:

a

_L

≤

a

≤

a

_R} where

a

_L ,

a

_R∈[0,1] and

are called the left end point and right end point respectively of the interval

A

ˆ

.

Example: 1.5

M

ˆ

= [0.65, 0.90] is an interval fuzzy number.

Definition: 1.6 A matrix of order nxn is said to be an interval fuzzy number matrix (IFNM) if all its elements are the interval fuzzy numbers.

Corresponding author:1_{Dr. A. KALAICHELVI*,} 1_{Associate Professor, Department of Mathematics,}

Example: 1.7 M =













]

4

.

0

,

6

.

0

[

]

2

.

0

,

3

.

0

[

]

7

.

0

,

1

.

0

[

]

1

.

0

,

5

.

0

[

is an interval fuzzy number matrix.

2. SOME NEW OPERATORS ON INTERVAL FUZZY NUMBERS AND INTERVAL FUZZY NUMBER MATRICES

Definition: 2.1 Let

M

ˆ

= [

m

_L

,

m

_R] and

N

ˆ

= [

n

_L

,

n

_R] be two interval fuzzy numbers where

R L R

L

m

n

n

m

,

,

,

∈[0,1]. Then the following operators are defined.

(1)

M

ˆ

⊕

N

ˆ

=

[

m

_L

+

n

_L

−

m n m

_{L L}

,

_R

+

n

_R

−

m n

_{R R}

]

,

(2)

M

ˆ

⊙

N

ˆ

=

[

m n m n

_{L L}

,

_{R R}

]

(3)

M

ˆ

∨

N

ˆ

_{= [}

m

_L ∨

n

_L ,

m

_R

∨

n

_R] where x ∨ y means max {x, y}. That is, x ∨ y = max {x, y}.

(4)

M

ˆ

^

N

ˆ

= [

m

L ^

n

L ,

m

R

∧

n

R] where x ^ y means min {x, y}. That is, x ^ y = min {x, y}.

(5)

M

ˆ

N

ˆ

_{= [}

m

_L

n

L,

m

R

n

R] where x y =







≤

>

y

x

if

0,

y

x

if

,

x

(6)

M

ˆ

≥

N

ˆ

if

m

L ≥

n

_L and

m

_R ≥

n

_R.

(7)

M

ˆ

±

N

ˆ

_{= [}

m

L

±

n

_L,

m

_R

±

n

_R].

Definition: 2.2 Let M =

(

M

ˆ

_ij

)

_mxn and N =

(

N

ˆ

_ij

)

_{mxn be two interval fuzzy number matrices of the same order}

where

M

ˆ

_ij = [

m

L_ij ,

m

R_ij ] and

N

ˆ

ij = [

n

Lij,

n

Rij]. Then the following operators are defined. (1) M ⊕ N = (

M

ˆ

_ij ⊕

N

ˆ

_ij)

(2) M ⊙ N = (

M

ˆ

_ij⊙

N

ˆ

_ij)

(3) M ∨ N = (

M

ˆ

_ij ∨

N

ˆ

_ij)

(4) M ^ N = (

M

ˆ

ij ^

N

ˆ

ij)

(5) M N = (

M

ˆ

_ij

N

ˆ

_ij)

(6) M ≥ N iff

M

ˆ

_ij≥

N

ˆ

_ij ∀ i = 1 to m, j = 1 to n.

Property: 2.3 For any interval fuzzy number matrix M,

(i) M ⊕ M ≥ M (ii) M ⊙ M ≤ M

Proof:

(i) The ijth element of M ⊕ M is

M

ˆ

_ij

⊕

M

ˆ

_ij which is equal to

[

m

L_ij +

m

L_ij −

m

L_ij .

m

L_ij ,

m

R_ij +

m

R_ij −

m

R_ij .

m

R_ij ]

Consider

ij L

m

+

ij L

m

₋

ij L

m

.

ij L

m

=

ij L

m

+

ij L

m

(1−

ij L

m

) ≥ ij

L

m

(1)

ij R

m

+

ij R

m

−

m

R_ij .

m

R_ij =

m

R_ij +

m

R_ij (1−

m

R_ij ) ≥

m

R_ij (2)

From (1) and (2), we get that

(ii) The ijth element of M⊙M is

M

ˆ

_ij⊙

M

ˆ

ij which is equal to

[

m

L_ij .

m

L_ij ,

m

R_ij .

m

R_ij ].

Since,

2

ij L

m

_≤

ij L

m

,

2

ij R

m

_≤

ij R

m

.

∴[

2

ij L

m

,

2

ij R

m

] which is less than or equal to [

m

L_ij ,

m

R_ij ].

Therefore, M ⊙ M ≤ M.

Definition: 2.4 Let M = (

M

ˆ

_ij) be an nxn interval fuzzy number matrix. Then M is Nearly irreflexive iff

M

ˆ

_ii≤

M

ˆ

_ij

for all i,j=1,2,……,n.

Property: 2.5 Let M and N be two interval fuzzy number matrices, then

(i) M ⊕ N ≥ M ⊙ N

(ii) If M and N are nearly irreflexive then M ⊕ N and M ⊙ N are nearly irreflexive.

Proof: Let M = (

M

ˆ

_ij) where

M

ˆ

_ij= [

m

L_ij ,

m

R_ij ] and N = (

N

ˆ

ij) where

ij

N

ˆ

= [

ij L

n

,

ij R

n

].

(i) The ijth element of M ⊕ N is

M

ˆ

_ij

⊕

N

ˆ

_ij which is equal to

[

m

L_ij +

n

L_ij −

m

L_ij .

n

L_ij ,

m

R_ij +

n

R_ij −

m

R_ij .

n

R_ij ] and that of

M ⊙ N is

M

ˆ

_ij ⊙

N

ˆ

ij = [

m

Lij .

n

Lij ,

m

Rij .

n

Rij ].

Now consider

ij L

m

+

ij L

n

₋

ij L

m

.

ij L

n

₋

ij L

m

.

ij L

n

=

ij L

m

. (1−

ij L

n

) +

ij L

n

.(1−

ij L

m

) ≥ 0 (since 0 ≤ ij

L

m

_{≤1 &}

0 ≤ ij

L

n

≤1)

∴

m

L_ij +

n

L_ij −

m

L_ij .

n

L_ij ≥

m

L_ij .

n

L_ij .

Similarly,

m

R_ij +

n

R_ij −

m

R_ij .

n

R_ij ≥

m

R_ij .

n

R_ij

Thus, M ⊕ N ≥ M ⊙ N .

(ii) Since M and N are nearly irreflexive,

M

ˆ

_ii≤

M

ˆ

ij and

N

ˆ

ii≤

N

ˆ

ij for all i,j.

∴[

ii L

m

,

ii R

m

] ≤ [

ij L

m

,

ij R

m

]

∴

m

L_ii ≤

m

L_ij and

m

R_ii≤

m

R_ij (1)

Similarly, [

ii L

n

,

ii R

n

] ≤ [

n

L_ij ,

n

R_ij ]

∴

ii L

n

_≤

ij L

n

and

ii R

n

_≤

ij R

n

(2)

Let

C

ˆ

_ij and

D

ˆ

_ij be the ijth elements of M ⊕ N and M ⊙ N respectively.Then

ij

C

ˆ

-

C

ˆ

_ii=

.

.

.

.

.

.

ij ij ij ij ij ij ij ij ii ii ii ii ii ii ii ii

L L L L R R R R L L L L R R R R

m

n

m n m

n

m n

m

n

m n m

n

m n



+

−

+

−

 

−

_

+

−

+

−



_





.

= [

ij L

m

+

ij L

n

₋

ij L

m

.

ij L

n

₋

ii L

m

₋

ii L

n

+

ii L

m

.

ii L

ij R

m

+ ij R

n

₋ ij R

m

. ij R

n

₋ ii R

m

₋ ii R

n

+ ii R

m

. ii R

n

]

= [(1−

ii L

m

).(1−

ii L

n

) − (1−

ij L

m

).(1−

ij L

n

), (1−

ii R

m

).(1−

ii R

n

) − (1−

ij R

m

).(1−

ij R

n

)]

From (1) and (2), we get

(1−

ii L

m

).(1−

ii L

n

) − (1−

ij L

m

).(1−

ij L

n

) ≥ 0 as 1−

ii L

m

_{≥ 1−}

ij L

m

and 1−

ii L

n

_≥1−

ij L

n

(3)

Similarly, (1−

ii R

m

).(1−

ii R

n

) − (1−

m

R_ij ).(1−

n

R_ij ) ≥ 0 (4)

From (3) and (4), we get

C

ˆ

_ij ≥

C

ˆ

_ii

Hence, M

⊕

N is nearly irreflexive.

Now consider,

ij

D

ˆ

−

D

ˆ

_ii = [

ij L

m

. ij L

n

, ij R

m

. ij R

n

] − [

ii L

m

. ii L

n

, ii R

m

. ii R

n

]

= [

m

L_ij .

n

L_ij −

m

L_ii .

n

L_ii,

m

Rij .

n

Rij −

m

Rii.

n

Rii ]

But

m

L_ij .

n

L_ij −

m

L_ii.

n

L_ii ≥ 0,

m

Rij .

n

Rij −

m

Rii.

n

Rii ≥ 0 (By (1))

∴

D

ˆ

_ij ≥

D

ˆ

_ii

Hence, M ⊙ N is nearly irreflexive.

Property: 2.6 Let M, N and Q be three interval fuzzy number matrices. If M ≤ N, then (1) M

⊕

Q ≤ N

⊕

Q

(2) M ⊙ Q ≤ N ⊙ Q

Proof: Let M = (

M

ˆ

_ij) where

M

ˆ

_ij=[

ij L

m

,

ij R

m

] , N = (

N

ˆ

_ij) where

N

ˆ

_ij= [

ij L

n

,

ij R

n

] and Q=(

Q

ˆ

_ij) where

ij

Q

ˆ

=[

ij L

q

, ij R

q

].

(1) Let

D

ˆ

_ij,

E

ˆ

_ij,

F

ˆ

_ij and

G

ˆ

_ij be the ijth elements of M

⊕

Q,N

⊕

Q,M ⊙ Q and N ⊙ Q respectively.

But the ijth element of M

⊕

Q , N

⊕

Q , M ⊙ Q and N ⊙ Q is

M

ˆ

_ij

⊕

Q

ˆ

_ij,

ij

N

ˆ

⊕

Q

ˆ

ij,

M

ˆ

ij ⊙

Q

ˆ

ij and

N

ˆ

ij⊙

Q

ˆ

ij.Then

ij

M

ˆ

⊕

Q

ˆ

_ij =

D

ˆ

_ij = [

ij L

m

+

ij L

q

−

m

L_ij .

q

L_ij ,

m

R_ij +

q

R_ij −

m

R_ij .

q

R_ij ]

ij

N

ˆ

⊕

Q

ˆ

ij =

E

ˆ

ij = [

n

Lij +

q

Lij −

n

Lij .

q

Lij ,

n

Rij +

q

Rij −

n

Rij .

q

Rij ]

ij

M

ˆ

⊙

Q

ˆ

ij =

F

ˆ

ij = [

m

Lij .

q

Lij ,

m

Rij .

q

Rij]

ij

N

ˆ

⊙

Q

ˆ

ij =

G

ˆ

ij = [

n

Lij .

q

Lij ,

n

Rij .

q

Rij ]

Since M ≤ N,

ij L

m

≤

n

L_ij .

∴

m

L_ij (1−

q

L_ij ) ≤

n

L_ij (1−

q

L_ij )

∴

m

L_ij −

m

L_ij .

q

L_ij ≤

n

L_ij −

n

L_ij .

q

L_ij

Thus, we get

[

m

L_ij +

q

L_ij −

m

L_ij .

q

L_ij ,

m

R_ij +

q

R_ij −

m

R_ij .

q

R_ij] ≤ [

n

L_ij +

q

L_ij −

n

L_ij .

q

L_ij ,

n

R_ij +

q

R_ij −

n

R_ij .

q

R_ij ]

i.e.,

D

ˆ

_ij ≤

E

ˆ

ij for all i,j.

∴

M

ˆ

_ij

⊕

Q

ˆ

_ij≤

N

ˆ

_ij

⊕

Q

ˆ

_ij for all i,j.

Hence, M

⊕

Q ≤ N

⊕

Q.

(2) Also [

ij L

m

.

ij L

q

,

ij R

m

.

ij R

q

] ≤ [

ij L

n

.

ij L

q

,

ij R

n

.

ij R

q

]

i.e.,

F

ˆ

_ij≤

G

ˆ

_ij for all i,j.

∴

M

ˆ

_ij⊙

Q

ˆ

ij ≤

N

ˆ

ij⊙

Q

ˆ

ij

∀

i,j.

Hence, M ⊙ Q ≤ N ⊙ Q.

Property: 2.7 For any two interval fuzzy number matrices M and N, (a) M

⊕

N ≥ M ∨ N ≥ M N

(b) (M ∨ N) ∨ (M N) = M ∨ N (c) (M ∨ N) (M N)

≤

N (d) M

⊕

N ≥ (M ∨ N) ∨ (M N) (e) M

⊕

N ≥ (M ∨ N) (M N)

Proof:

(a) M

⊕

N ≥ M ∨ N ≥ M N

Let

C

ˆ

_ij,

D

ˆ

_ij and

E

ˆ

_ij be the ijth elements of M

⊕

N, M ∨ N and M N respectively.

But the ijth elements of M

⊕

N, M ∨ N and M N are

M

ˆ

_ij

⊕

N

ˆ

_ij,

M

ˆ

_ij∨

N

ˆ

_ij and

M

ˆ

_ij

N

ˆ

_ij. Then

ij

M

ˆ

⊕

N

ˆ

ij =

C

ˆ

ij

= [

ij L

m

+

ij L

n

−

m

L_ij .

n

L_ij ,

m

R_ij +

n

R_ij −

m

R_ij .

n

R_ij ]

=























≥

≥

−

+

−

+

≥

≥

−

+

−

+

ij R L

R R

R L L

L ij

R L

R R

R L L

L

N

n

n

n

m

n

n

m

n

M

m

m

m

n

m

m

n

m

ij ij

ij ij

ij ij ij

ij

ij ij

ij ij

ij ij ij

ij

ˆ

]

,

[

)]

1

(

),

1

(

[

ˆ

]

,

[

)]

1

(

),

1

(

[

(1)

ij

M

ˆ

∨

N

ˆ

ij =

D

ˆ

ij = max{

M

ˆ

ij,

N

ˆ

ij}

≤ [

ij L

m

+

ij L

n

−

m

L_ij .

n

L_ij ,

m

R_ij +

n

R_ij −

m

R_ij .

n

R_ij ]

=

C

ˆ

_ij(from (1))

Thus,

C

ˆ

_ij ≥

D

ˆ

ij for all i,j.

i.e.,

M

ˆ

_ij

⊕

N

ˆ

ij ≥

M

ˆ

ij∨

N

ˆ

ij

∀

i,j.

Now consider

ij

E

ˆ

=

M

ˆ

_ij

N

ˆ

ij = [

m

Lij

n

Lij ,

m

Rij

n

Rij ]

where

m

L_ij

n

L_ij =









≤

>

ij ij

ij ij

L L

L L

n

m

0,

n

m

,

ij

L

m

i.e.,

ij L

m

ij L

n

_≤

ij L

m

≤ max{

ij L

m

,

ij L

n

}

ij R

m

n

R_ij ≤

m

R_ij

≤ max{

m

R_ij ,

n

R_ij }

∴

E

ˆ

_ij ≤ max{[

ij L

m

,

ij R

m

] , [

ij L

n

,

ij R

n

]}

≤ max{

M

ˆ

_ij,

N

ˆ

_ij}

≤

M

ˆ

_ij∨

N

ˆ

ij

Thus, M N ≤ M ∨ N (3)

From (2) and (3), we get M

⊕

N ≥ M ∨ N ≥ M N.

Let

D

ˆ

_ij,

E

ˆ

_ij and

H

ˆ

_ijbe the ijth elements of M ∨ N, M N and (M ∨ N) ∨ (M N) respectively.

But the ijth elements of M ∨ N, M N and (M ∨ N) ∨ (M N) is

M

ˆ

_ij∨

N

ˆ

_ij,

ij

M

ˆ

N

ˆ

_ij and (

M

ˆ

_ij∨

N

ˆ

_ij) ∨ (

M

ˆ

_ij

N

ˆ

_ij). Then

ij

D

ˆ

=

M

ˆ

_ij∨

N

ˆ

_ij = [

ij L

m

∨

ij L

n

,

ij R

m

∨

ij R

n

] where

ij L

m

∨

ij L

n

= max{

ij L

m

,

ij L

n

}

ij

E

ˆ

=

M

ˆ

_ij

N

ˆ

_ij= [

ij L

m

ij L

n

,

ij R

m

ij R

n

] where

ij L

m

ij L

n

=









≤

>

ij ij ij

L

m

L L

L L

n

m

0,

n

m

,

ij ij

(a) (M ∨ N) ∨ (M N) = M ∨ N

Case (i):

m

L_ij >

n

L_ij ,

m

R_ij >

n

R_ij .

∴

M

ˆ

_ij

N

ˆ

_ij = [

ij L

m

,

ij R

m

] and

M

ˆ

_ij∨

N

ˆ

_ij= [

ij L

m

,

ij R

m

]. Then

ij

H

ˆ

= [

ij L

m

,

ij R

m

] ∨ [

ij L

m

,

ij R

m

]

= [

m

L_ij ,

m

R_ij ]

=

M

ˆ

_ij∨

N

ˆ

ij

Case (ii) :

ij L

m

>

n

L_ij ,

m

R_ij

≤

n

R_ij .

∴

M

ˆ

_ij

N

ˆ

_ij = [

ij L

m

,0] and

M

ˆ

_ij∨

N

ˆ

_ij= [

ij L

m

,

ij R

n

]. Then

ij

H

ˆ

= [

m

L_ij ,0] ∨ [

m

L_ij ,

n

R_ij ]

= [

ij L

m

,

ij R

n

]

=

M

ˆ

_ij∨

N

ˆ

_ij ∴

H

ˆ

_ij =

D

ˆ

_ij

Case (iii):

m

L_ij

≤

n

L_ij ,

m

R_ij >

n

R_ij

∴

M

ˆ

_ij

N

ˆ

ij = [0,

m

Rij ] and

M

ˆ

ij∨

N

ˆ

ij= [

n

Lij ,

m

Rij ]. Then

ij

H

ˆ

= [0,

ij R

m

] ∨ [

ij L

n

,

ij R

m

]

= [

ij L

n

,

ij R

m

]

=

M

ˆ

_ij∨

N

ˆ

_ij

=

D

ˆ

_ij

Case (iv):

m

L_ij

≤

n

L_ij ,

m

R_ij

≤

n

R_ij

∴

M

ˆ

_ij

N

ˆ

_ij = [0,0] and

M

ˆ

_ij∨

N

ˆ

_ij= [

ij L

n

,

ij R

n

]. Then

ij

H

ˆ

= [0,0] ∨ [

ij L

n

,

ij R

n

]

= [

n

L_ij ,

n

R_ij ]

=

M

ˆ

_ij∨

N

ˆ

ij

=

D

ˆ

_ij ∴

H

ˆ

_ij =

D

ˆ

_ij

∴ In all cases,

D

ˆ

_ij =

H

ˆ

_ij.

i.e.,

M

ˆ

_ij∨

N

ˆ

ij = (

M

ˆ

ij∨

N

ˆ

ij) ∨ (

M

ˆ

ij

N

ˆ

ij)

Hence M ∨ N = (M ∨ N) ∨ (M N).

(c) (M ∨ N) (M N)

≤

N

Let

F

ˆ

_ij be the ijth elements of (M ∨ N) (M N).But the ijth element of

(M ∨ N) (M N) is (

M

ˆ

_ij∨

N

ˆ

_ij) (

M

ˆ

_ij

N

ˆ

_ij).

Case (i):

ij L

m

>

ij L

n

,

ij R

m

>

ij R

n

∴

M

ˆ

_ij

N

ˆ

_ij = [

ij L

m

,

ij R

m

] and

M

ˆ

_ij∨

N

ˆ

_ij= [

ij L

m

,

ij R

m

].Then

= [

ij L

m

ij L

m

,

ij R

m

ij R

m

]

= [0,0].

Case (ii):

ij L

m

≤

n

L_ij ,

m

R_ij >

n

R_ij

∴

M

ˆ

_ij

N

ˆ

_ij = [0 ,

ij R

m

] and

M

ˆ

_ij∨

N

ˆ

_ij= [

ij L

n

,

ij R

m

].Then

ij

Fˆ

= [0,

m

R_ij ] [

n

L_ij ,

m

R_ij ]

= [0

n

L_ij ,

m

R_ij

m

R_ij ]

= [

ij L

n

, 0].

Case (iii) :

ij L

m

>

ij L

n

,

ij R

m

≤

n

R_ij

∴

M

ˆ

_ij

N

ˆ

_ij = [

m

L_ij ,0] and

M

ˆ

ij∨

N

ˆ

ij= [

m

Lij ,

n

Rij ].Then

ij

Fˆ

= [

m

L_ij ,0] [

m

L_ij ,

n

R_ij ]

= [

ij L

m

ij L

m

,0

ij R

n

]

= [0,

ij R

n

]

Case (iv):

m

L_ij

≤

n

L_ij ,

m

R_ij

≤

n

R_ij

∴

M

ˆ

_ij

N

ˆ

ij = [0,0] and

M

ˆ

ij∨

N

ˆ

ij= [

n

Lij ,

n

Rij ]. Then

ij

Fˆ

= [0,0] [

ij L

n

,

ij R

n

]

= [0

ij L

n

,0

ij R

n

]

= [

n

L_ij ,

n

R_ij ]

=

N

ˆ

_ij

From case (i) to case (iv), we see that the ijth element of

Fˆ

_ij is either 0 or

N

ˆ

_ij according as [

ij L

m

,

ij R

m

] > or

≤

to

[

n

L_ij ,

n

R_ij ].

Thus, (M ∨ N) (M N)

≤

N.

(d) M

⊕

N ≥ (M ∨ N) ∨ (M N)

(M ∨ N) ∨ (M N) = M ∨ N

≤

M

⊕

N (By Property 2.6)

Hence, M

⊕

N ≥ (M ∨ N) ∨ (M N).

(e) M

⊕

N ≥ (M ∨ N) (M N)

It is obvious that N

≤

M ∨ N

REFERENCES

[1] Amiya K.Shyamal and Madhumangal Pal, “Two new operators on fuzzy matrices”, J. Appl. Math. and Computing, 15(2004) 91-107.

[2] Amiya K.Shyamal and Madhumangal Pal, “Triangular fuzzy matrices”, Iranian Journal of Fuzzy Systems, Vol. 4, No. 1, (2007) 75-87.

[3] Kalaichelvi, A and Janofer, K., “Some new operators on triangular fuzzy numbers and triangular fuzzy number matrices”, Proceedings of International Conference on Mathematics and its Applications – A New Wave (ICMANW – 2011) 161-167.

[4] Madhumangal Pal, Gobinda Murmu and Anita Pal., “Orthogonality of Imprecise Matrices”, International Conference On E-Business Intelligence, (2010) 560-565.

[5] Thomason, M.G., “Convergence of powers of a fuzzy matrix”, J. Math Anal. Appl., 57(1977), 476-480.

[6] Zadeh, L.A., Fuzzy Sets, “Information and Control”, 8(1965) 338-353.