Significant Characters of Fuzzy Soft Matrices

Significant Characters of Fuzzy Soft Matrices

Dr. N. Sarala

Department of Mathematics,

1. college for women(Autonomous), Nagapattinam,

   Tamil Nadu, India.

   S. Rajkumari

   Department of Mathematics, Government college for women(Autonomous),

   Kumbakonam, Tamil Nadu, India.

   Abstract-The reference function will play an important role in the definition of fuzzy soft matrices. In this paper, we intend to deal union, intersection, complement, addition, multiplication and trace of fuzzy soft matrices and its character by using reference function with example.

   Keywords-Soft set, Fuzzy soft set(FSS), Fuzzy soft matrices(FSM), Union, Intersection, Complement, Addition, Multiplication and Trace of fuzzy soft matrices.

   1. INTRODUCTION

      In 1965, Zadeh [17] introduced the notion of fuzzy set theory. In 1999, soft set theory was firstly introduced by Molodtsov [7] as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In 2001, Maji, Biswas and Roy [5] studied the theory of soft sets initiated by Molodtsov [7] and developed several basic notions of soft set theory. In 2009,Ahmad and Kharal [1] developed the result of Maji [5].

      In 2010, Cagman and Enginoglu [2] defined soft matrices which were a matrix representation of the soft sets and constructed a soft max-min decision making method. In 2011, Yong yang and Chenli Ji [16] successfully applied the proposed notion of fuzzy soft matrix in certain decision making problems. In 2011, Neog and Sut [14] have defined the addition operation for fuzzy soft matrices and an attempt has been made to apply our notion in solving a decision problem.

      In 2012, Cagman and Enginoglu [3] defined fuzzy soft matrices and constructed decision making problem. In 2012,Rajarajeswari and Dhanalakshmi [9] introduced the application of similarity between two fuzzy soft sets based on distance. In 2012,Neog, Bora and Sut[15] Combined fuzzy soft set based on reference function with soft matrices. In 2012, Borah, Neog, Sut [6] extended fuzzy soft matrix theory and its application.

      In 2013, Mamoni Dhar [8] applied this concept to fuzzy square matrix and developed some interesting properties as determinant, trace and so on. In 2013,Said Broumi, Florentin Smarandache and M.Dhar [10] introduced an application of fuzzy soft matrix based on reference function in decision making problem is given. In 2013, Md.Jalilul and T.K.Roy [4] have introduced different types of fuzzy soft matrices and some properties with proof.

      In 2014, Dr.N.Sarala and Rajkumari [11] introduced intuitionistic fuzzy soft matrices in agriculture and issued[12] intuitionistic fuzzy soft matrices in medical

      diagnosis and also introduced [13] role model service rendered to orphans by using fuzzy soft matrices.

      In this paper, we proposed fuzzy soft matrix theory and extended its characters in the field of fuzzy soft matrices.

   2. PRELIMINARIES

      In this we section, We recall some basic significant notion of fuzzy soft set and defined different types of fuzzy soft set.

      1. Soft Set [7]

         Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Let AE. A pair ( ,E) is called a soft set over U, where is a

         mapping given by :E P(U) Such that (e) = if e

         A.

         Here is called approximate function of the soft set ( ,E). The set (e) is called e-approximate value set which consist of related objects of the parameter eE. In other words, a soft set over U is a parameterized family of subsets of the universe U.

         Example 2.1:

         Let U={1, 2 , 3, 4} be a set of four types of rice and E={Costly(1),Ordinary(2),Cheap(3)} be the set of parameters. If A={ 1 , 2 }E. Let

         (1)={1 , 2, 3, 4} and (2)={ 1 , 3,4 }. Then we write the soft set

         ( ,E) ={(1 ,{ 1 , 2 , 3, 4 }),(2 ,{ 1 , 3 ,4 })} over U which describe the Quality of rice Which Mr.Y is going to buy.

         We may represent the soft set in the following form:

         U	Costly(1)	Ordinary(2)	Cheap(3)
         1	1	1	0
         2	1	0	0
         3	1	1	0
         4	1	1	0

         TABLE 2.1.1

      2. Fuzzy Soft Set [5]

         Let U be an initial universe set and E be a set of parameters. Let AE. A pair ( ,E) is called a fuzzy soft set (FSS)over U, where is a mapping given by, :E

         , where denotes the collection of all fuzzy subsets of

         U.

         Example 2.2:

         Consider the example 2.1., here we cannot express with only two real numbers 0 and 1, we can characterized it by a membership function instead of crisp

         number 0 and 1, which associate with each element a real number in the interval [0,1]. Then

         ( ,E) ={ (1) ={( 1,0.6),( 2,0.4),( 3,0.2),( 4,0.1)},

         ( 2 ) ={( 1 ,0.8),( 3 ,0.3),( 4 ,0.5)}} is the fuzzy soft set representing the Quality of rice which Mr.Y is going to buy.

         We may represent the fuzzy soft set in the following form:

         U	Costly(1)	Ordinary(2)	Cheap(3)
         1	0.6	0.8	0.0
         2	0.4	0.0	0.0
         3	0.2	0.3	0.0
         4	0.1	0.5	0.0

         TABLE 2.2.2

      3. Fuzzy Soft Class[9]

      Let U be an initial universe set and E be the set of attributes. Then the pair (U,E) denotes the collection of all fuzzy soft sets on U with attributes from E and is called a

      2.9 Intersection of Fuzzy Soft Sets Redefined[15]

      Let ( , )and ( ,E) be two fuzzy soft sets in a soft class (U,E) with AB . Then intersection of two fuzzy soft sets ( , )and ( ,E) in the fuzzy soft class

      (U,E) is a fuzzy soft set ( , ) where C=AB and e

      C, (e) = (e) () .

   3. CHARACTERIZATION OF FUZZY SOFT MATRICES

      In this section, we introduce the notion of fuzzy soft matrices with several types based on reference function.

      3.1 Fuzzy Soft Matrices:

      Let U = {1, 2, 3, } be the universal set and E be the set of parameters given by E =

      {1 , 2 , 3 ,, }. Then the fuzzy soft set ( , ) can be

      fuzzy soft class.

      expressed in matrix form as = [ ]

      m×n

      or simply by

      1. Fuzzy Soft Subset[9] [ ], = 1,2,3, , ; = 1,2,3, , and [ ] =

         For two fuzzy soft sets ( , ) and ( , ) over a

         [( , )]; where and represent the fuzzy

         common universe U, we have ( , ) ( ,E) if A B and eA, (e) is a fuzzy subset of (e). i.e, ( , )is

         membership function and fuzzy reference function U in the fuzzy set (ej) so that = gives the fuzzy

         a fuzzy soft subset of ( ,E).

      2. Fuzzy Soft Complement set[14]

         The complement of fuzzy soft set ( , ) denoted by ( , )° is defined by ( , )° = ( °, ) , where ° :

         membershipvalue of U. We shall identify a fuzzy soft set with its fuzzy soft matrix and use these two concepts interchangeable. The set of all m×n fuzzy soft matrices over U will be denoted by FSM m×n. For usual fuzzy sets

         with fuzzy reference function 0, it is obvious to see that

         E is a mapping given by °(e) = [ (e)]°, eE.

      3. Fuzzy Soft Null Set[9]

         = [( ,0)] i , j.

         Example 3.1:

         A fuzzy soft set ( , ) over U is said to be null fuzzy soft set with respect to the parameter set E, denoted by , if (e) = , eE.

      4. Union of Fuzzy Soft Sets[15]

         Let U = {1, 2, 3} be the universal set and E be the set of parameters given by E = {1, 2, 3}

         we consider a fuzzy soft set

         Union of two fuzzy soft sets ( , )and ( ,E) in

         (

         a soft class (U,E) is a fuzzy soft set ( , ) Where C=AB and eC,

         , ) = { (1) ={(1,0.7,0),( 2,0.3,0),( 3,0.4,0)},

         ( ) ={( ,0.8,0),( ,0.6,0),( ,0.9,0)},

         ,

         2 1 2 3

         (e) = ,

         ,

         and is written as ( , ) ( , ) = ( , ).

         (3) ={(1,0.5,0),( 2,0.2,0),( 3,0.1,0)}}

         We would represent this fuzzy soft set in matrix form as

      5. Intersection of Fuzzy Soft Sets[15]

      Intersection of two fuzzy soft sets ( , ) and

      (0.7,0)

      [ ]3×3 = (0.3,0)

      (0.8,0)

      (0.6,0)

      (0.5,0)

      (0.2,0)

      (

      ,E) in a soft class (U,E) is a fuzzy soft set ( , )

      (0.4,0)

      (0.9,0)

      (0.1,0)

      Where C=AB and eE, (e) = (e) or ()and is written as ( , ) ( , )=( , ).

      Ahmad and Kharal [1] pointed out that generally

      and may not be identical. Moreover in order

      1. Membership Value Matrix:

         3×3

         to avoid the degenerate case, he proposed that AB must be non-empty and thus revised the above definition as

         The membership value matrix corresponding to

         the matrix as MV( ) = [ ] m×n, where =

         follows.

         = 1,2,3, , = 1,2,3, , , where and

         represent the fuzzy membership function and fuzzy reference function respectively of U in the fuzzy set (ej).

      2. Zero Fuzzy Soft Matrices:

         Let = [ ]FSMm×n, where =( , ). Then

         complement of the fuzzy matrix which is denoted by ° and then ° is called fuzzy soft complement matrix if °=

         is called a fuzzy soft zero (or Null) matrix denoted by

         [(1, )]m×n for all [0,1]. Then the matrix obtained

         [0 ]m×n, or simply by[0 ], if =0 for all i and j. For usual

         from so called membership value would be the following

         °= [ ] = [(1- )] for all i and j.

         fuzzy sets, = i, j.

      3. Identify Fuzzy Soft Matrices:

      Let = [ ]FSMm×n, where =( , ). Then

      3.13 Addition of Fuzzy Soft Matrices:

      Let U = {1, 2, 3, , } be the universal set and E be the set of parameters given by E =

      { , , ,, }. Let the set of all m×n fuzzy soft matrices

      is called a fuzzy soft unit or identify matrix denoted by

      [1 ], if m=n, =( , ) for all ij and = (1,0) i.e,

      1 2 3

      over U be FSMm×n. Let ,

      FSMm×n, Where =

      [ ]m×n, =( , ) and = [ ]m×n, = ( , ). To

      =1 i=j.

      3.5 Fuzzy Soft Sub Matrix:

      avoid degenerate cases we assume that min(( , )max(( , ) for all i and j. We define the

      Let = [ ], = [ ]FSMm×n. Then is said to

      operation addition(+) between and as + = ,

      be a fuzzy soft sub matrix of denoted by [ ] [ ] if

      where = [ ]m×n, = (max( , ),min( , )).

      i and j.

      3.14 Product of Fuzzy Soft Matrices:

      Let = [ ]m×n, =( , ); where and

      1. Fuzzy Soft Square Matrix:

         Let = [ ]FSMm×n, where =( , ). If m

         represent the fuzzy membership function and fuzzy

         reference function of , so that = gives the

         = n, then is called a fuzzy soft square matrix.

         fuzzy membership value of . Also let = [ ]n×p, =

      2. Fuzzy Soft Rectangular Matrix:

         ( , ); where and represent the fuzzy

         Let = [ ]FSMm×n, where =( , ).If m

         membership function and fuzzy reference function of , so

         that = gives the fuzzy membership value of .

         n, then is called a fuzzy soft rectangular matrix.

      3. Fuzzy Soft Diagonal Matrix:

      We now define , the product of and as =

      [ ]m×p = [max min( , ), min max( , )]m×p,

      Let = [ ]FSMm×n, where =( , ). Then

      1 m, 1 p for j = 1,2,3,,n.

      is called fuzzy soft diagonal matrix if m = n and

      =( , ) for all ij. In other words for a fuzzy soft

      3.15 Scalar Multiplication of Fuzzy Soft Matrix:

      Let = [ ] FSM

      . Then scalar multiplication

      diagonal matrix , =0 ij.

      m×n

      of fuzzy soft matrix [ ] by a scalar k denoted by k[ ] is

      3.9 Fuzzy Soft Equal Matrix:

      defined as k[ ] = [ ], 0k1.

      Let the fuzzy soft matrices corresponding to the

      fuzzy soft sets ( , ) and (

      , E) be = [ ], =

      3.16 Transpose of Fuzzy Soft Matrices:

      [ ] FSMm×n; =( , ) and =( , ),

      Let = [ ]m×n, =( ,

      ); where

      and

      represent the fuzzy membership function and fuzzy

      i=1,2,3,m; j=1,2,3,,n. Then and are called fuzzy

      soft equal matrices denoted by = . If = and =

      reference function of U, so that = gives the

      i, j.

      fuzzy membership value of U. Then we define =

      T FSMn×m where T = [ ].

      1. Union of Fuzzy Soft Matrices:

         nm

         Let = [ ], = [ ] FSMm×n, where

         3.17 Symmetric Fuzzy Soft Matrix:

         =( , ) and =( , ). Then the union of and Let = [ ]m×n, =( , ). Then is said to

         denoted by is defined as = max{ , } for

         be a fuzzy soft symmetric matrix if = .

         all i and j.

      2. Intersection of Fuzzy Soft Matrices:

      3.18 Trace of Fuzzy Soft Matrix:

      Let be a square matrix. Then the trace of the

      Let = [ ], = [ ] FSMm×n, where

      matrix is denoted by tr and is defined as:

      =( , ) and =( , ). Ten the intersection of

      tr =(max( ), min( ))

      and denoted by is defined as = min{ , }

      for all i and j.

      3.12 Complement of Fuzzy Soft Matrices:

      Where stands for the membership functions lying along the principal diagonal and refers to the reference

      Let = [( , 0)]FSM

      , where =( , )

      function of the corresponding membership functions.

      m×n

      according to the definition in [8], the representation of the

      Example 3.18:

      (0.5,0)

      (0.6,0)

      (0.2,0)

      Proposition 4.3:

      Let = [ ], = [ ], = [ ] FSMm×n. Then

      Let = (0.7,0) (0.4,0) (0.8,0)

      (0.3,0) (0.1,0) (0.6,0)

      tr =(max(0.5,0.4,0.6), min(0,0,0)) = (0.6,0).

      (i) ( ) = ( ) ( ) , (ii) ( ) = ( ) ( ).

      Proof:

      (i) ( ) = [ ] [(min{ , })]

      3.19Principal Diagonal of Fuzzy Soft Matrices:

      = [max( , min{ , })]

      Let

      = [ ]FSMm×m, where =( , ), then

      the elements 11 ,22 ,33 ,, are called the diagonal

      ( ) ( ) = [(max{ , })] [(max{ , })]

      elements and the line along which they lie is called the

      principal diagonal of fuzzy soft matrices.

      = [min(max{ , },max{ , })]

   4. IDEALISTIC PROPERTIES OF FUZZY SOFT

      = [min( ,max{ , })]

      MATRICES

      In this section, we see the properties of fuzzy soft

      = [max( ,min{ , })]

      matrices with some example.

      Proposition 4.1:

      Hence ( ) = ( ) ( ).

      (ii) ( ) = [ ] [(max{ , })]

      Let = [ ]FSMm×n. Then (i) [[]°]° = [], (ii)

      = , (iii) = , (iv) [0 ] = , (v) [0 ] =

      = [min( , max{ , })]

      [0 ], (vi) [0 ]°=[1 ].

      Proof:

      ( ) ( ) = [(min{ , })] [(min{ , })]

      (i) [[ ]°]° = [1 ]° = [1 (1 )] = [ ] =

      = [max(min{ , },min{ , })]

      (ii) = [max{ , }] = [ ] =

      = [max( ,min{ , })]

      (iii) = [min{ , }] = [ ] =

      = [min( ,max{ , })]

      (iv) [0] = [max{ ,0}] = [ ] =

      (v) [0 ] = [min{ ,0}] = [0 ]

      (vi) [0 ]° = [10] = [1 ]

      Hence ( ) = ( ) ( ).

      Example 4.3:

      (i) Let = [ ], = [ ], = [ ] FSM2×2,

      Proposition 4.2:

      Let = [ ], = [ ]FSM

      . Then

      where =

      (0.6,0)

      (0.5,0)

      , =

      m×n

      (0.2,0)

      (0.4,0)

      (i) [ ]° = [ ]° [B]°, (ii) [ ]° = [ ]° [B]°.

      (0.4,0)

      (0.7,0)

      (0.8,0)

      (0.2,0)

      Proof:

      (0.9,0)

      (0.1,0) and = (0.3,0)

      (0.6,0)

      1. For all i and j,

         [ ] ° = [max{ , }] ° = [1 max{ , }] =

         =

         (0.4,0)

         (0.3,0)

         (0.2,0)

         (0.1,0)

         ; =

         [min{1 ,1 }] = [ ]° [B]°

      2. For all i and j,

      (0.6,0)

      (0.9,0)

      (0.7,0)

      (0.4,0)

      [ ] ° = [min{ , }] ° = [1 min{ , }] =

      (0.8,0) (0.5,0)

      [max{1 ,1 }] = [ ]° [B]°.

      =

      (0.3,0)

      (0.6,0)

      (0.6,0)

      ( ) = (0.3,0)

      (0.5,0)

      (0.4,0) ;

      (0.6,0)

      ( ) ( ) = (0.3,0)

      (0.5,0)

      (0.4,0)

      Proof:

      1. For all i and j,

         = [(min{ , })] = [(min{ , })] =

         Hence ( ) = ( ) ( ).

         (ii) ( ) = [(min{ , })] [ ]

      2. The above said matrices may be considered for the

      = ( ) ( ).

      = [(min(min{ , }, ))]

      = [(min( ,min{ , }))]

      (0.8,0)

      =

      (0.7,0)

      ;

      =

      (0.9,0)

      (0.6,0)

      ( )

      =

      (0.6,0)

      (0.4,0)

      (0.2,0)

      (0.2,0)

      (0.5,0)

      (0.1,0)

      ; =

      (iii) [1 ] = [(min{ ,1 })] = .

      Proposition 4.6:

      Let = [ ], = [ ] , = [ ] FSMm×n,

      where [ ] = [( , )], [ ] = [( , )], [ ] =

      (0.2,0)

      (0.4,0)

      [( , )]. Then the following results hold.

      (0.6,0)

      ( ) = (0.2,0)

      (0.5,0)

      (0.4,0) ;

      (i) + = + (commutative law)

      (ii) ( + )+ = +( + )(Associative law).

      Proof:

      (0.6,0)

      (0.5,0)

      (i) Let = [( , )], = [( , )]

      ( ) ( ) =

      (0.2,0)

      (0.4,0)

      Now + = [(max( , ), min( , ))]

      Hence = ( ) ( ). = [(max( , ), min( , ))]

      Proposition4.4:

      Let = [ ], = [ ], = [ ] FSMm×n. Then

      = + .

      (ii) Let = [( , )], = [( , )], = [( , )]

      (i) = , (ii) ( ) = ( ) , (iii)

      [1 ] = [1 ] .

      Now ( + )+ = [(max( , ), min( , ))] +

      Proof:

      [( , )]

      (i) For all i and j,

      = [(max{ , })] = [(max{ , })] =

      = [(max(( , ), ) ,

      min(( , ), ))]

      (ii) ( ) = [(max{ , })] [ ]

      = [(max( , ( , )) ,

      = [max(max{ , }, )]

      min( , ( , )))]

      = [max( ,max{ , })]

      = ( )

      Example 4.6:

      = +( + ).

      (iii) [1 ] = [(max{ ,1 })] = [1 ].

      1. Let =

         (0.6,0)

         (0.5,0)

         (0.3,0)

         (0.2,0)

         , =

         Proposition 4.5:

         Let = [ ], = [ ], = [ ] FSMm×n. Then

         (0.1,0)

         (0.7,0)

         (0.9,0)

         (0.2,0)

         and =

         (i) = , (ii) ( ) = ( ) , (iii)

         [1 ] = .

         (0.4,0) (0.8,0)

         (0.6,0)

         (0.6,0)

         (0.7,0)

         (0.5,0)

         + =

         (0.5,0)

         (0.8,0)

         , + =

         (0.6,0)

         (0.5,0)

         (0.7,0)

         (0.8,0)

         Hence + = +

      2. The above said matrices may be considered for the

      Then . =

      (0.3,0)

      (0.3,0)

      (0.4,0)

      (0.6,0)

      . =

      ,

      ( + )+ = +( + ).

      (0.6,0)

      (0.7,0)

      (0.3,0)

      (0.7,0)

      (0.3,0)

      (0.6,0)

      + =

      , + =

      (0.9,0)

      (0.6,0)

      (0.5,0)

      (0.7,0)

      (0.8,0) ,

      (0.8,0)

      (0.6,0)

      (0.7,0)

      From the example we obtained that . .

      Property 4.8:

      If the product . is defined then . may not be

      defined.

      Let the following illustration may be considered.

      ( + )+ =

      (0.5,0)

      (0.8,0) +

      Example 4.8:

      (0.9,0)

      (0.2,0)

      Let =

      (0.6,0)

      (0.5,0)

      (0.9,0)

      = (0.6,0)

      (0.7,0)

      (0.8,0)

      (0.6,0)

      (0.3,0)

      (0.6,0)

      (0.3,0)

      (0.5,0)

      (0.4,0)

      (0.6,0)

      (0.1,0)

      (0.7,0)

      (0.4,0)

      (0.5,0)

      (0.1,0)

      (0.5,0)

      (0.2,0)

      (0.8,0)

      (0.9,0)

      (0.4,0)

      (0.3,0)

      and =

      + ( + ) =

      (0.5,0)

      (0.2,0) +

      (0.3,0)

      (0.7,0)

      (0.2,0)

      (0.8,0)

      (0.9,0)

      (0.6,0)

      (0.7,0)

      (0.8,0)

      (0.7,0)

      (0.8,0)

      (0.9,0)

      = (0.6,0)

      (0.7,0)

      (0.8,0)

      We have . = (0.6,0)

      (0.4,0)

      (0.4,0)

      (0.5,0)

      Hence ( + )+ = +( + ).

      Property 4.7:

      The product of two fuzzy soft matrices and

      But here . is not defined as number of columns in = 2, Whereas number of row in = 3.

      Proposition 4.9:

      Let = [ ], = [ ]FSMm×n, where [ ] =

      representing fuzzy soft sets over the same initial universe is

      [( , )], [ ] = [( , )]. Then the following results

      defined only when the matrices are square matrices. Then

      . and . exist and matrices are of same type, it is not

      hold.

      T

      T

      T T

      necessary that

      (i) ( ) = , (ii) ( + )

      = + .

      . = . . Let the following illustration may be taken into account for proof.

      Example 4.7:

      Proof:

      (i) Let = [( , )]

      (0.5,0) (0.4,0)

      Here T = [( , )]T

      Let = (0.7,0) (0.6,0) ,

      = [( , )]

      (0.3,0)

      =

      (0.2,0)

      ( T)T = [( , )]T

      (0.1,0)

      (0.8,0)

      be two fuzzy soft square matrices representing two fuzzy soft sets defined over the same initial universe.

      = [( , )]

      = Hence ( )T =

      (ii) = [( , )], = [( , )]

      Here + = [(max( , ), min( , ))]

      (0.3,0)

      (0.5,0)

      (0.7,0)

      ( T

      T

      T+ T = (0.6,0)

      (0.8,0)

      (0.7,0)

      +)

      = [(max( , ), min( , ))]

      = [(max( , ), min( , ))]

      (0.5,0)

      (0.8,0)

      (0.6,0)

      Hence ( + )T = T+ T.

      = [( , )]+ [( , )]

      Proposition 4.10:

      = [( , )]T+[( , )]T

      Let and be two fuzzy soft square matrices

      each of order n. Then tr( + ) = tr( )+tr( ).

      = T+ T

      Hence ( + )T = T+ T.

      Example 4.9:

      Proof:

      From the proposed definition of trace of fuzzy soft matrices, we have obtained

      (0.3,0)

      (0.6,0)

      (0.4,0)

      tr = (max( ),min( )) and tr =(max( ),min( ))

      (i) Let = (0.5,0)

      (0.8,0)

      (0.1,0) and

      (0.7,0)

      (0.2,0)

      (0.6,0)

      Then + = , where = [ ]

      (0.2,0)

      = (0.3,0)

      (0.4,0)

      (0.1,0)

      (0.5,0)

      (0.8,0)

      Adopting the definition of addition of two fuzzy soft matrices, we have

      = (max( , ),min( , ))

      (0.6,0)

      (0.7,0)

      (0.4,0)

      (0.3,0)

      ( )= (0.6,0)

      (0.5,0)

      (0.8,0)

      (0.7,0)

      (0.2,0) ;

      According to definition 3.18 the trace of fuzzy soft matrix based on reference function would be :

      tr( ) = (max{max( , )},min{min( , )})

      (0.4,0)

      (0.1,0)

      (0.6,0)

      = (max{max( ),max( )},min{min( ),min( )})

      (0.3,0)

      ( T)T = (0.5,0)

      (0.7,0)

      (0.6,0)

      (0.8,0)

      (0.2,0)

      (0.4,0)

      (0.1,0) =

      (0.6,0)

      = tr( )+tr( )

      Conversely,

      Hence ( )T = .

      tr( )+tr( ) =

      (max{max( ),max( )},min{min( ),min( )})

      1. The above said matrices may be considered for the

         = (max{max( , )},min{min( , )})

         ( + )T = T+ T .

         = tr( + )

         (0.2,0)

         T = (0.4,0)

         (0.5,0)

         (0.3,0)

         + = (0.5,0)

         (0.3,0)

         (0.1,0)

         (0.8,0)

         (0.6,0)

         (0.8,0)

         (0.6,0)

         (0.7,0) ;

         (0.4,0)

         (0.5,0)

         (0.8,0)

         Hence tr( + ) = tr( )+tr( ).

         Example 4.10:

         (0.6,0)

         Let = (0.4,0)

         (0.2,0)

         (0.5,0)

         (0.7,0)

         (0.8,0)

         (0.3,0)

         (0.9,0) and

         (0.1,0)

         (0.7,0)

         (0.3,0)

         ( + )T = (0.6,0)

         (0.5,0)

         (0.7,0)

         (0.5,0)

         (0.8,0)

         (0.8,0)

         (0.6,0)

         (0.7,0)

         (0.7,0) ;

         (0.6,0)

         (0.4,0)

         = (0.7,0)

         (0.6,0)

         (0.3,0)

         (0.9,0)

         (0.5,0)

         (0.1,0)

         (0.2,0)

         (0.3,0)

         The addition of two fuzzy soft matrices would be:

         forward that our work would enrich the scope of characterization of fuzzy soft matrices.

         (0.6,0)

         + = (0.7,0)

         (0.6,0)

         (0.5,0)

         (0.9,0)

         (0.8,0)

         (0.3,0)

         (0.9,0)

         (0.3,0)

         REFERENCES

         1. Ahmad.B and Kharal.A, On Fuzzy Soft Sets, Advances in Fuzzy systems, 2009, pp:1-6

         2. Cagman.N and Enginoglu.S, Soft matrix theory and its decision making, Journal computers and Mathematics with applications,

            Using the definition of trace of fuzzy soft matrices, we find the following results:

            tr = (max{0.6,0.7,0.1},min{0,0,0})= (0.7,0)

            tr = (max{0.4,0.9,0.3},min{0,0,0}) = (0.9,0)

            Thus we have tr( )+tr( ) = (max{0.7,0.9},min{0,0,0}) = (0.9,0)

            And tr( + ) = (max{0.6,0.9,0.3},min{0,0,0}) = (0.9,0) Hence tr( + ) = tr( )+tr( ).

            Proposition 4.11:

            Let = [( , )] be fuzzy soft square matrices each of order n. Then tr ( ) = tr( T), where T is the transpose of .

            Proof:

            Let = [( , )], then T = [( , )]

            Volume 59, Issue 10(2010), pp:3308-3314.

         3. Cagman.N and Enginoglu.S, Fuzzy Soft Matrix Theory and Its Application in Decision Making, Iranian Journal of Fuzzy systems, Volume 9, No.1,(2012),pp: 109-119.

         4. Jalilul Islam Mondal.Md, TapanKumar Roy, Theory ofFuzzy Soft Matrix and its Multi criteria in Decision Making Based on Three Basic t-Norm Operators, International Journal of innovative research in Science, engineering and Technology,Vol.2, Issue 10, October 2013,pp: 5715-5723.

         5. Maji.P.K, Biswas.R and Roy.A.R, Fuzzy Soft Sets, The Journal of fuzzy mathematics, Volume 9, No.3,(2001), pp:589-602.

         6. Manash Jyoti Borah, Tridiv Jyoti Neog, Dusmanta Kumar Sut, Fuzzy Soft Matrix Theory And its Decision Making,international Journal of Modern Engineering Research,Volume 2, Issue 2, Mar- Apr 2012, pp:121-127.

         7. Molodtsov.D, Soft set Theory First Results, Computer and Mathematics with applications, 37(1999),pp:19-31.

         8. Mamoni Dhar, Representation of Fuzzy Matrices Based on Reference Function, I.J. Intelligent systems and Applications, 2013, 02 ,pp: 84-90.

         9. Rajarajeswari .P, Dhanalakshmi .P, An Application of Similarity Measure of Fuzzy Soft Set Base on Distance, IOSR journal of Mathematics, Volume4, Issue 4(Nov-Dec 2012), pp:27-30.

         10. Said Brouni, Florentin smarandache, Mamoni Dhar, On Fuzzy Soft Matrix Based on Reference Function, I.J. Information Engineeriing and Electronic Business, 2013, 2, pp:52-59.

         11. Sarala .N, Rajkumari .S, Invention of Best Technology In

             tr( T) = (max( ),min( )) = tr( )

             Agriculture Using Intuitionistic Fuzzy Soft Matrices,

             Hence tr( ) = tr( T).

             Example 4.11:

             (0.5,0)

             Let = (0.6,0)

             (0.2,0)

             (0.3,0)

             (0.4,0)

             (0.7,0)

             (0.8,0)

             (0.9,0) ;

             (0.1,0)

             International Journal of Scientific and Technology Research, Volume3, Issue5, May 2014, pp:282-286.

         12. Sarala .N and Rajkumari .S, Application of Intuitionistic Fuzzy Soft Matrices in Decision Making Problem by Using Medical Diagnosis, IOSR Journal of Mathematics, Volume 10, Issue 3ver.VI (May- June 2014), pp:37-43.

         13. Sarala .N, Rajkumari .S, Role Model Service Rendered to Orphans by Using Fuzzy Soft Matrices, International Journal of Mathematics Trends and Technology, Volume 11, Number1,July 2014, pp:14-19.

         14. Tridiv Jyoti Neog, Dusmanta Kumar Sut,An Application of

             (0.5,0)

             T = (0.3,0)

             (0.8,0)

             (0.6,0)

             (0.4,0)

             (0.9,0)

             (0.2,0)

             (0.7,0)

             (0.1,0)

             Fuzzy Soft sets in Decision Making Problems Using Fuzzy Soft Matrices, International Journal of Mathematical Archive- 2(11),2011, pp:2258-2263.

         15. Triidiv Jyoti Neog, Manoj Bora, Dusmanta Kumar Sut, On Fuzzy Soft Matrix Theory, International Journal of Mathematical Archive-3(2), 2012, pp:491-500.

         16. Yong yang and Chenli Ji ,Fuzzy Soft Matrices and Their

             tr( ) = (max{0.5,0.4,0.1},min{0,0,0}) = (0.5,0)

             tr( T) = (max{0.5,0.4,0.1},min{0,0,0}) = (0.5,0)

             Hence tr( ) = tr( T).

   5. CONCLUSION

In this paper, we extend the concept with regard to the matrices of union, intersection, complement, addition, multiplication and trace of fuzzy soft matrices. For doing the said types of matrices with the help of reference function which are different from the existing definition by establishing some of its properties with example. We put

Applications, AICI 2011,Part I, LNAI7002,pp:618-627.

1. Zadeh .L .A, Fuzzy sets, Information and control, 8, 1965, pp:338-353.