Interval-valued Fuzzy Soft Matrix Theory

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1 Annals of Pure and Applied Mathematics Vol. 7, No. 2, 2014, ISSN: X (P, (online Published on 18 September 2014.researchmathsci.org Annals of Interval-valued Fuzzy Soft Matrix heory P.Rajarajesari 1 and P.Dhanalakshmi 2 1 Department of Mathematics, hikkanna Govt Arts ollege,irupur. p.rajarajesari29@gmail.com 2 Department of Mathematics, iruppur Kumaran ollege for Women,irupur. viga_dhanasekar@yahoo.co.in Received 4 August 2014; 20 August 2014 Abstract. oday is a orld of uncertainty ith its associated problems, hich can be ell handled by soft set theory. In this paper, e propose Interval valued fuzzy soft matrix,its types ith examples and some ne operators on the basis of eights. We also study their properties. Keyords Soft set, fuzzy soft set,, interval valued fuzzy soft set, interval valued fuzzy soft matrix. AMS Mathematics subject classification (2010: 08A72 1. Introduction In real life scenario, e face so many uncertainties, in all alks of life. Zadeh s classical concept of fuzzy set[20] is strong to deal ith such type of problems. Since the initiation of fuzzy set theory, there are suggestions for higher order fuzzy sets for different applications in many fields. Among higher fuzzy sets intuitionistic fuzzy set introduced by Atanassov [1,2,3] have been found to be very useful and applicable. Soft set theory has received much attention since its introduction by Molodtsov [11].he concept and basic properties of soft set theory are presented in [11,6]. Later on Maji et al. [7] have proposed the theory of fuzzy soft set. Majumdar et al. [9] have further generalised the concept of fuzzy soft sets. Maji et al. [8] extended fuzzy soft sets to intuitionistic fuzzy soft sets hich is based on the combination of the intuitionistic fuzzy set [1] and soft set. Yang et al. [18] presented the concept of the interval valued fuzzy soft sets by combining the interval valued fuzzy sets and soft set. hey have also given an algorithm to solve interval valued fuzzy soft set based decision making problems. Matrices play an important role in the broad area of science and engineering. Hoever,the classical matrix theory cannot solve the problems involving various types of uncertainties. Mondal et al. [12] and Shyamal et al. [17] initiated a matrix representation of a soft set and interval valued fuzzy set respectively. In [19], Yong et al initiated a matrix representation of a fuzzy soft set and applied it in certain decision making problems. Borah et al. [4] extended fuzzy soft matrix theory and its application. In [5], 61

2 P.Rajarajesari and P.Dhanalakshm hetia et al and in [13,15] Rajarajesari et al. defined intuitionistic fuzzy soft matrix and interval valued intuitionistic fuzzy soft matrix and its types. Also extended some operations and an algorithm for medical diagnosis in [14,16].In[10], Mitra Basu et al. described interval valued fuzzy soft matrix and its types. In this paper, e extended the concept of Interval valued fuzzy soft matrix, and defined different types of matrices along ith examples. We also studied some operators on the basis of eights and their properties. 2. Preliminaries 2.1. Soft set [6] Suppose that U is an initial Universe set and E is a set of parameters, let P(U denotes the poer set of U.A pair(f,e is called a soft set over U here F is a mapping given by F: E P(U.learly a soft set is a mapping from parameters to P(Uand it is not a set, but a parameterized family of subsets of the Universe Fuzzy soft set [7] Let U be an initial Universe set and E be the set of parameters. Let A E. A pair (F,A is called fuzzy soft set over U here F is a mapping given by F: A I U,here I U denotes the collection of all fuzzy subsets of U Fuzzy Soft Matrices [19] Let U={c 1,c 2,c 3 c m } be the Universal set and E be the set of parameters given by E={e 1,e 2,e 3 e n }. Let A E and (F,A be a fuzzy soft set in the fuzzy soft class (U,E. hen fuzzy soft set (F,A in a matrix form as A mxn =[a ] mxn or A=[a ] i=1,2, m, j=1,2,3,..n µ j ( ci if ej A here a = µ j ( ci represents the membership of c i in the 0 if e j A fuzzy set F(e j Interval valued fuzzy soft set [18] Let U be an initial Universe set and E be the set of parameters. Let A E. A pair (F,A is called interval valued fuzzy soft set over U here F is a mapping given by F: A I U, here I U denotes the collection of all interval valued fuzzy subsets of U. 3. Interval valued fuzzy soft matrix theory 3.1. Interval valued fuzzy soft matrix Let U={c 1,c 2,c 3 c m } be an Universal set and E be the set of parameters given by E={e 1,e 2,e 3 e n }.Let A E and (F,A be a interval valued fuzzy soft set over U, here F is a mapping given by F: A I U, here I U denotes the collection of all interval valued fuzzy subsets of U. hen the interval valued fuzzy soft set can be expressed in matrix form as A % mxn =[a ] mxn or A % =[a ] i=1,2, m, j=1,2,3,..n 62

3 here ( ( a µ µ = [ ] ( c, ( c Interval-valued Fuzzy Soft Matrix heory jl ci, ju ci if e j A 0,0 if e A µ jl i µ ju i represents the membership of c i in the interval valued fuzzy set F(e j. µ µ then the interval-valued fuzzy soft matrix (IVFSM Note that if ( c = ( c reduces to an FSM. ju i jl i Example 3.1. Suppose that there are four houses under consideration, namely the universes U={ h 1,h 2,h 3,h 4 },and the parameter set E={e 1,e 2,e 3, e 4 } here e i stands for beautiful, large, cheap, and in green surroundings respectively. onsider the mapping F from parameter set A = { e1, e2} E to the set of all interval valued fuzzy subsets of poer set U. onsider an interval valued fuzzy soft set (F,A hich describes the attractiveness of houses that is considering for purchase. hen interval valued fuzzy soft set (F,A is (F,A = { F(e 1 = {(h 1,[0.6, 0.8], (h 2,[0.8, 0.9], (h 3,[0.6, 0.7], (h 4,[0.5, 0.6]} F(e 2 = {(h 1,[0.7, 0.8], (h 2,[0.6, 0.7],(h 3,[0.5, 0.7], (h 4,[0.8, 0.9]}} We ould represent this interval valued fuzzy soft set in matrix form as [ 0.6, 0.8] [ 0.7, 0.8] [ 0.0,0.0] [ 0.0, 0.0] [ 0.8, 0.9] [ 0.6, 0.7] [ 0.0,0.0] [ 0.0, 0.0] [ 0.6, 0.7] [ 0.5, 0.7] [ 0.0, 0.0] [ 0.0, 0.0] [ 0.5, 0.6] [ 0.8, 0.9] [ 0.0,0.0] [ 0.0, 0.0] 3.2. Interval valued fuzzy soft sub matrix Let A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, hen A % is a Interval valued Fuzzy Soft sub matrix of B %, denoted by A% B% if µ µ, µ µ for all i and j. AL % BL % AU % BU % 3.3. Interval valued fuzzy soft null (zero matrix An interval valued fuzzy soft matrix of order mxn is called interval valued fuzzy soft null (zeromatrix if all its elements are [0,0].It is denoted by Φ% Interval valued fuzzy soft universal matrix An interval valued fuzzy soft matrix of order mxn is called interval valued fuzzy soft universal matrix if all its elements are [1,1].It is denoted by U %. Proposition 3.1. Let A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, % = [c ] IVFSM mxn then j 63

4 P.Rajarajesari and P.Dhanalakshm i Φ% A% ii A% U% iii A% A% iv A % B%, B% % A% % Proof: It follos from the definition Interval valued fuzzy soft equal matrix Let A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, hen A % is equal to B %,denoted by A % = B % if µ = µ, µ = µ for all i and j. AL BL AU BU 3.6. Interval valued fuzzy soft transpose matrix Let A % = [a ] IVFSM mxn here a = µ jl ( ci, µ. hen A % is interval valued fuzzy soft transpose matrix of A % if j= 1,2, n. A % = [a ji ] IVFSM nxm. A % = [a ji ] i=1,2,..m, 3.7. Interval valued fuzzy soft rectangular matrix Let A % = [a ] IVFSM mxn,here a = µ jl ( ci, µ. hen A % is called interval valued Fuzzy Soft rectangular Matrix if m n Interval valued fuzzy soft square matrix Let A % = [a ] IVFSM mxn, here a = µ jl ( ci, µ. hen A % is called interval valued Fuzzy Soft square Matrix if m= n Interval valued fuzzy soft ro matrix Let A % = [a ] IVFSM mxn, here a = µ jl ( ci, µ. hen A % is called interval valued Fuzzy Soft ro Matrix if m= 1.It means that the universal set contains only one element Interval valued fuzzy soft column matrix a = µ c, µ c. hen A % is called Let A % = [a ] IVFSM mxn, here ( ( jl i ju i interval valued Fuzzy Soft olumn Matrix if n=1. It means that the parameter set contains only one parameter Interval valued fuzzy soft diagonal matrix Let A % = [a ] IVFSM mxn, here a = µ jl ( ci, µ. 64

5 Interval-valued Fuzzy Soft Matrix heory hen A % is called interval valued Fuzzy Soft diagonal Matrix if m= n and a = [ 0,0] for all i j Interval valued fuzzy soft scalar matrix a = µ c, µ c. hen A % is called Let A % = [a ] IVFSM mxn, here jl ( i ju ( i interval valued Fuzzy Soft scalar Matrix if m= n and a = [ 0,0] a = [ α, α ] α, α [0,1], α α i =j for all i j and Interval valued fuzzy soft upper triangular matrix a = µ c, µ c. hen A % is called Let A % = [a ] IVFSM mxn, here jl ( i ju ( i interval valued Fuzzy Soft upper triangular Matrix if m= n and [ 0,0] a = for all i> j Interval valued fuzzy soft loer triangular matrix a = µ c, µ c. hen A % is called Let A % = [a ] IVFSM mxn, here jl ( i ju ( i interval valued Fuzzy Soft loer triangular Matrix if m= n and [ 0,0] a = for all i< j Interval valued fuzzy soft triangular matrix An interval valued Fuzzy Soft Matrix is said to be triangular if it is either interval valued Fuzzy Soft loer triangular Matrix or interval valued Fuzzy Soft upper triangular Matrix Addition of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % + B %,addition of A % and B % as A % + B % = [ c ] mxn = [max( µ AL, µ BL,max( µ AU, µ BU ] for all i and j. Example 3.2. onsider [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0.6, , , , 0.7 A% = and B= % 0.5, , , , X 2 2 X 2 are to interval valued fuzzy soft matrices then sum of these to is [0.8, 0.9] [0.7, 0.8] A% + B% = [0.6, 0.7] [0.8, 0.9] X 2 2 Proposition 3.2. Let A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, % = [c ] IVFSM mxn then 65

6 P.Rajarajesari and P.Dhanalakshm i A% + Φ % =A % ii A % + U% = U% iii A% +B % = B+A % % iv (A+ % B% + % = A+(B+ % % % v (A+ % B% =A % + B% vi (A% + B% + % = A% + B% + % vii(a % =A% Proof: It follos from the definition Subtraction of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % B %,subtraction of A % and B % as A % - B % = [ c ] mxn = [min( µ AL, µ BL,min( µ AU, µ BU ] for all i and j Example 3.2. onsider [ 0.6, 0.8] [ 0.7, 0.8] [ 0.8, 0.9] [ 0.6, 0.7] A% = and B= % [ 0.5, 0.6] [ 0.8, 0.9] [ 0.6, 0.7] [ 0.5, 0.7] 2 X 2 2 X 2 are to interval valued fuzzy soft matrices then subtraction of these to is [0.6, 0.8] [0.6, 0.7] A% B% = [0.5, 0.6] [0. 5, 0.7] X 2 2 Proposition 3.3. Let A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, % = [c ] IVFSM mxn then i A% Φ% = Φ % ii A % - U% = A% iii A% - B% = B% - A% iv ( A% - B% % = A% - ( B% - % v( A% + B % % = ( A% - % + ( B % - % vi( A% B % + % = ( A% + % ( B% + % vii A% ( B% + % = ( A% - B% + ( A% - % Proof: It follos from the definition Product of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b jk ] IVFSM nxp, then e define A % * B %, multiplication of A % and B % as A % * B % = [ c ik ] mxp = [max min( µ ALj, µ BLj, max min( µ AUj, µ BUj ], i,j,k 66

7 Interval-valued Fuzzy Soft Matrix heory Example 3.2. onsider [ 0.6, 0.8] [ 0.7, 0.8] A% = [ 0.5, 0.6] [ 0.8, 0.9] [ 0.8, 0.9] [ 0.6, 0.7] [ ] [ ] 2 X 2 and B= % 0.6, ,0.7 2 X 2 are to interval valued fuzzy soft matrices then product of these to matrices is [0.6, 0.8] [0.6, 0.7] A% * B% = [0.6, 0.7] [0.5, 0.7] X Remark: A % * B % B % * A % Interval valued fuzzy soft complement matrix Let A % = [a ] IVFSM mxn, here a = µ jl ( ci, µ. hen A % is called interval valued Fuzzy Soft omplement Matrix if here b = 1- µ,1 µ jl ( ci, i,j. A % =[ b ] mxn Example 3.3. [ 0.6, 0.8] [ 0.7, 0.8] Let A% = [ 0.5, 0.6] [ 0.8, 0.9] 2 X 2 be interval valued fuzzy soft matrix then complement of this matrix is [0.2, 0.4] [0.2, 0.3] A % = [0.4,0.5] [0.1,0.2] 2 X 2 Proposition 3.4. c c i ( A% = A % c ii Φ % = U% iii ( A% +U % = Φ % iv (A+ % B% = ( B% + A% c c v (A- % B% = ( B% A% c c c vi( A% + B% = A% B% vii( A% c c c - B% = A% + B% c c viii( A% = ( A% c c c c ix( A% - B% = ( A% + ( B% c c c x( A% - B% = A% + B% c c c xi( A% + B% = A% B% c c c Proof: It follos from the definition. 67

8 P.Rajarajesari and P.Dhanalakshm Scalar multiple of interval valued fuzzy soft matrix a = µ c, µ c. hen scalar multiple Let A % = [a ] IVFSM mxn, here ( ( 2 X 2 jl i ju i of interval valued Fuzzy Soft Matrix A by a scalar k is defined by k A % =[ ka ] mxn here 0 k 1. Example 3.4. [ 0.6, 0.8] [ 0.7, 0.8] Let A% = [ 0.5, 0.6] [ 0.8, 0.9] 2 X 2 be an interval valued fuzzy soft matrix then the scalar multiple of this matrix by k = 0.5 is [0.3, 0.4] [0.35, 0.4] ka% = [0.25, 0.3] [0.4, 0.45] Proposition 3.5. Let A % = [a ] IVFSM mxn, here a = µ jl ( ci, µ. If m,n are to scalars such that 0 m, n 1,then i m(n A% = (mna% ii m n ma% na% iii A% B% ma% mb% ivm(n A% = (mna% c c vm(n A% = (mna% Proof: It follos from the definition race of interval valued fuzzy soft matrix a = µ c, µ c. hen race Let A % = [a ] IVFSM mxn,here m=n and jl ( i ju ( i of interval valued Fuzzy Soft Matrix A % is % ( ( µ jl j ( µ ju ( j tra = max c, max c. Example 3.5. [ 0.6, 0.8] [ 0.7, 0.8] Let A% = [ 0.5, 0.6] [ 0.8, 0.9] 2 X 2 be an interval valued fuzzy soft matrix then trace of this matrix is tra% = [0.8, 0.9] 68

9 Interval-valued Fuzzy Soft Matrix heory Proposition 3.6. Let A % = [a ] IVFSM mxm, if k is a scalar such that 0 k 1,then i tr( ka% = k tra% ii( ka% = k A% iii tr( A% + B% = tra% + trb% c c iv tr( ka% = k tra% v( ka% = k ( A% vi tr( A% + B% = tra% + trb% c c c c c c Proof: It follos from the definition Interval valued fuzzy soft symmetric matrix a = µ c, µ c. hen an interval Let A % = [a ] IVFSM nxn,here ( ( jl i ju i valued fuzzy soft square matrix A % is called an Interval valued Fuzzy Soft Symmetric Matrix if A % = A % ie., if [ a ] = [ a ] i,j. ji Example 3.6. [0.8, 0.9] [0.4, 0.5] Let A% = [0.4, 0.5] [0.7, 0.8] 2 X 2 be an interval valued fuzzy soft matrix, then the transpose of this matrix [0.8, 0.9] [0.4, 0.5] is A% = [0.4, 0.5] [0.7, 0.8] 2 X 2 since A % = A %, A % is called an interval valued Fuzzy Soft Symmetric Matrix of order 2. Special operators of interval valued fuzzy soft matrices Arithmetic mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % & B %,arithmetic mean of A % and B % as A % & B % = [ c ] mxn µ AL % + µ BL % µ µ AU % + BU % =, for all i and j Weighted arithmetic mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % & B %, eighted arithmetic mean of A % and B % as A % & B % = [ c ] mxn 69

10 P.Rajarajesari and P.Dhanalakshm 1 Lµ AL % + 2 Lµ BL % 1 U µ AU % + 2U µ BU % =, 1 L + 2 L 1 U + 2 U,,, [0,1] 1L 1U 2L 2U + = 1, + = 1 1L 1U 2L 2U Geometric mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % $ B %,geometric mean of  and B% as A % $ B % = [ c ] mxn = ( µ. µ, µ. µ AL % BL % AU % BU % Weighted geometric mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A % $ B %, eighted geometric mean of A % and B % as A % $ B % = [ c ] mxn = (( µ W L.( µ W L W L W L W U W U W U W U,(( µ.( µ + + AL % BL % AU % BU %,,, [0,1] 1L 1U 2L 2U + = 1, + = 1 1L 1U 2L 2U Harmonic mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn,then e define A B % harmonic mean of A % and B % as A B % = [ c ] mxn µ. µ µ. µ AL % BL % AU % BU % = 2, 2 for all i and j. µ + µ µ + µ AL % BL % AU % BU % Weighted Harmonic mean of interval valued fuzzy soft matrices If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then e define A B %,eighted harmonic mean of A % and B % as A W B % = [ c ] mxn = 70

11 Interval-valued Fuzzy Soft Matrix heory ( µ. µ ( + 2 Lµ AL 1Lµ BL 2U µ AU 1U µ % + % % + BU % ( µ. ( 1 2 AL % µ BL % L + L, AU % BU % 1U 2U,,, [0,1] 1L 1U 2L 2U + = 1, + = 1 1L 1U 2L 2U Proposition 3.7. If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then i A& % B% = B% & A% and (A& % B% = ( B% & A% ii A$ % B% = B% $ A% and (A$ % B% = ( B% $ A% iiia@ % B% = A% and (A@ % B% = ( A% iv A& % B% = B% & A% and (A& % W B% = ( B% & W A% v A$ % B% = B% $ A% and (A$ % W B% = ( B% $ W A% via@ % B% = A% and (A@ % W B% = ( W A% Proof: It follos from the definition. Proposition 3.8. If A % = [a ] IVFSM mxn, B % = [b ] IVFSM mxn, then c c c c c c i (A% & B% = A% & B% ii (A% $ B% = A% $ B% c c c c c c iii(a B% = B% iv (A% & B% = A% & B% c c c c c c v (A% $ B% = A% $ B% vi(a B% = B% Proof: It follos from the definition. 4. onclusion In this paper, e proposed the concept of Interval valued fuzzy soft matrix, and defined different types of matrices in Interval valued fuzzy soft set theory along ith examples. hen e have studied some ne operations and properties on the basis of eights. As far as future directions are concerned, e hoped that our finding ould help enhancing this study on Interval valued fuzzy soft set theory. In future, e ill use special operators hich are proposed in this paper in decision making problem in medical diagnosis.we ill compare the results ith other types of disease diagnosis. REFERENES 1. K.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1 ( K. Atanassov, Intuitionistic fuzzy sets, Physica-Verlag, Heidelberg, Ne York, K.Atanassov and G.Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 3(1 (

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