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Advances in Applied Science Research, 2012, 3 (1):412-423
ISSN: 0976-8610
CODEN (USA): AASRFC
Some Results of Intuitionistic Fuzzy Soft Matrix Theory
B. Chetia* and P. K. Das
1Department of Mathematics, Brahmaputra Valley Academy, North Lakhimpur, Assam, India
2Department of Mathematics, NERIST, Nirjuli, Itanagar, Arunachal Pradesh, India
_____________________________________________________________________________________________
ABSTRACT
The concept of soft set is one of the recent topics developed for dealing with the uncertainties present in most of our
real life situations. The parametrization tool of soft set theory enhance the flexibility of its applications. In this
paper, we define intuitionistic fuzzy soft matrices and their operations which are more functional to make
theoretical studies in the intuitionistic fuzzy soft set theory. We also define five types of products and some results
are established.
Keywords: Soft sets,Intuitionistic fuzzy soft sets, Soft matrices, Intuitionistic fuzzy soft matrices and Products of
intuitionistic fuzzy soft matrices.
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INTRODUCTION
Most of our real life problems in medical sciences, engineering, management, environment and social sciences often
involve data which are not always all crisp, precise and deterministic in character because of various uncertainties
typical for these problems. Such uncertainties are usually being handled with the help of the topics like probability,
fuzzy sets, intuitionistic fuzzy sets, interval mathematics and rough sets etc. However, Molodtsov[8] has shown that
each of the above topics suffers from some inherent difficulties due to inadequacy of their parametrization tools and
introduced a concept called ‘Soft Set Theory’ having parametrization tools for successfully dealing with various
types of uncertainties. The absence of any restrictions on the approximate description in soft set theory makes this
theory very convenient and easily applicable in practice. Research on soft sets has been very wide spread and many
important results have been achieved in the theoretical aspect. Maji et al. introduced several algebraic operations in
soft set theory and published a detailed theoretical study on soft sets[7].The same authors also extended crisp soft
sets to fuzzy soft sets [4] and intuitionistic fuzzy soft sets[6]. At the same time, there has been some progress
concerning practical applications of soft set theory, especially the use of soft sets in decision making. Recently,
Çagman et al.[1] introduced soft matrix and applied it in decision making problems. In one of our earlier work [2],
we proposed the idea of ‘Fuzzy Soft Matrix Theory’ in sequel to [1] defining some operations.The present paper
aims to define intuitionistic fuzzy soft matrix and establish some results on them.This style of representation is
useful for storing an intutionistic fuzzy soft set in computer memory and which are very useful and applicable.
2.Preliminaries
Definition 2.1[8] ⊂
Let U be an initial universe, P (U) be the power set of U, E be the set of all parameters and A E . A soft set on the universe U is
defined by the set of ordered pairs (f , E ) = {(e, f (e)) |e∈ E , and f (e) ∈P (U )}, where f : E → P (U) such that f (e) = φ if e
∉ A A A A A
A.
Here, f is called an approximate function of the soft set (f , E ). The set f (e) is called
A A A
e-approximate value set or e-approximate set which consists of related objects of the parameter e∈ E .
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B. Chetia et al Adv. Appl. Sci. Res., 2012, 3(1):412-423
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Example 2.1
Let U={c1,c2,c3} be the set of three cars and E ={costly(e1), metallic colour (e2), cheap (e3)} be the set of
parameters,where A={e ,e }⊂E. Then f (e )={c ,c ,c }, f (e )={c ,c }, then we write a crisp soft set (f ,E)={( e
1 2 A 1 1 2 3 A 2 1 3 A 1,
{c1,c2,c3}),(e2,{ c1,c3})}over U which describes the
“ attractiveness of the cars” which Mr. S(say) is going to buy .
Definition 2.2[3,5]
Let U be a universal set, E a set of parameters and A E. Let F(U) denotes the set of all fuzzy subsets of U. A
⊂
set on the universe is defined as the set of F
fuzzy soft ordered pairs = {(e, f (e)) : e∈ E , f (e) ∈
(f , E ) U (f , E )
A F A A A
(U )},where fA: E → (U ) .
Here, f is called an approximate function of the fuzzy soft set (f , E ) . The set f (e) is called e-
A A A
approximate value set or e-approximate set which consists of related objects of the parameter e∈ E .
Example 2.2
Let U={c1,c2,c3} be the set of three cars and E ={costly(e1), metallic colour(e2),getup (e3)} be the set of
parameters,where A={e1,e2)⊂E. Then (G,A)={G(e1)={c1/.6,c2/.4,c3/.3},
G(e2)={c1/.5,c2/.7,c3/.8}} is the fuzzy soft set over U and describes the “ attractiveness of the cars” which Mr.
S(say) is going to buy .
Definition 2.3[1 ]
Let (f , E ) be a soft set over U . Then a subset of U × E is uniquely defined by
A
R = {(u, e) : e∈A, u∈f (e)} which is called a relation form of (f , E ). The characteristic function of R is
A A A A
1, (u,e)∈RA
written by χ :U×E→{0,1},χ (u,e)={
RA RA 0, (u,e)∉RA
If U = {u , u , . . . , u }, E = {e , e , . . . , e } and A ⊆ E , then the R can be presented by a table as given
1 2 m 1 2 n A
below:
R e e e
A 1 2 n
u χ (u ,e ) χ (u ,e ) ....... χ (u ,e )
1 RA 1 1 RA 1 2 RA 1 n
u χ (u ,e ) χ (u ,e ) ....... χ (u ,e )
2 RA 2 1 RA 2 2 RA 2 n
M M M M
u χ (u ,e ) χ (u ,e )....... χ (u ,e )
m RA m 1 RA m 2 RA m n
If aij = χR (ui,ej),we can define a matrix
A
a a ... a
11 12 1n
[a ]= a21 a22 .... a2n
ij M M M M
a a .... a
m1 m2 mn
which is called an m×n soft matrix of the soft set (fA,E) over U.
According to this definition, a soft set (f , E) is uniquely characterized by the matrix [a ] . It
A ij m×n
means that a soft set (fA, E) is formally equal to its soft matrix [aij]m×n.
3.Fuzzy soft matrices:
Definition 3.1[ 2 ]
Let (f , E ) be a fuzzy soft set over U . Then a subset of U × E is uniquely defined by
A
R = {(u, e) : e∈A, u∈f (e)} which is called a relation form of (f , E ). The characteristic function of R is
A A A A
written byµR :U ×E →[0,1], where µR (u,e)∈[0,1] is the membership value of u∈U for each e∈E.
A A
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If µij= µ (u ,e ),we can define a matrix
RA i j
µ µ ... µ
11 12 1n
[µ ] =µ21 µ22 .... µ2n
ij m×n M M M M
µ µ .... µ
m1 m2 mn
which is called an m×n fuzzy soft matrix of the fuzzy soft set (f , E ) over U.
A
Therefore, we can say that a fuzzy soft set (fA,E ) is uniquely characterized by the matrix [µ ] and both
ij m×n
concept are interchangeable.
The set of all m × n fuzzy soft matrices over U will be denoted by FSM .
m×n
Example 3.1
Assume that U={u , u , u , u , u } is a universal set and E={e ,e ,e ,e } is a set all parameters. If A E={e , e , e }
1 2 3 4 5 1 2 3 4 ⊂ 2 3 4
and f (e )={u /.4, u /.5, u /1, u /.3, u /.6}, f (e )={u /.3, u /.4, u /.6,u /.5,u /1}, f (e )={u /.5,u /.5,u /.4,u /.3,u /.9}.
A 2 1 2 3 4 5 A 3 1 2 3 4 5 A 4 1 2 3 4 5
Then the fuzzy soft set (f ,E) is a parametrized family { f (e ), f (e ), f (e )}of all fuzzy sets over U.Then the
A A 2 A 3 A 4
relation form of (f ,E) is written by
A
RA e1 e2 e3 e4
u µ (u ,e ) µ (u ,e ) µ (u ,e ) µ (u ,e )
1 RA 1 1 RA 1 2 RA 1 3 RA 1 4
u µ (u ,e ) µ (u ,e ) µ (u ,e ) µ (u ,e )
2 RA 2 1 RA 2 2 RA 2 3 RA 2 4
u µ (u ,e ) µ (u ,e ) µ (u ,e ) µ (u ,e )
3 RA 3 1 RA 3 2 RA 3 3 RA 3 4
u µ (u ,e ) µ (u ,e ) µ (u ,e ) µ (u ,e )
4 RA 4 1 RA 4 2 RA 4 3 RA 4 4
u µ (u ,e ) µ (u ,e ) µ (u ,e ) µ (u ,e )
5 RA 5 1 RA 5 2 RA 5 3 RA 5 4
R e e e e
A 1 2 3 4
u 0 .4 .3 .5
1
u2 0 .5 .4 .5
u
3 0 1 .6 .4
u4 0 .3 .5 .3
u 0 .6 1 .9
5
Hence the fuzzy soft matrix [µij] is written as
0 .4 .3 .4
0 .5 .4 .5
[µij] = 0 1 .6 .4
0 .3 .5 .3
0 .6 1 .4
Definition 3.2[2]
Let [µ ] ∈FSM Then [µ ] is called
ij m×n. ij
(a) a zero fuzzy soft matrix, denoted by [0], if µij=0 for all i and j.
(b) a universal fuzzy soft matrix, denoted by [1], if µij=1 for all i and j.
%
(c) [µ ] is a fuzzy soft submatrix of [λ ], denoted by [µ ] ⊆ [λ ], if µ ≤ λ
ij ij ij ij ij ij
for all i and j.
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(d) [µ ] and [λ ] are fuzzy soft equal matrices, denoted by [µ ] = [λ ], if µ = λ
ij ij ij ij ij ij
for all i and j.
Definition 3.3[2 ]
Let [µ ], [λ ] ∈FSM Then the fuzzy soft matrix [ν ] is called
ij ij m×n. ij
%
(a) union of [µ ] and [λ ], denoted by [µ ] ∪ [λ ] if ν = max{ µ , λ } for all i and j.
ij ij ij ij ij ij ij
%
(b) intersection of [µ ] and [λ ], denoted by [µ ] ∩ [λ ] if ν = min{ µ , λ }
for all i and j. ij ij ij ij ij ij ij
(c) complement of [µ ], denoted by [µ ]°, if ν =1- µ for all i and j.
ij ij ij ij
Example 3.2
.2 .4 .5 .6 .3 .4 .4 .6
.3 .5 .1 1 .3 .5 .1 1
Let [µ ]= and [λ ]= . Then
ij .2 .4 .4 .5 ij .4 .4 .4 .5
1 .9 .7 .5 .5 .1 .7 .5
.6 .5 .6 .3 .5 .7 .4 .8
.3 .4 .5 .6 .2 .4 .4 .6 .8 .6 .5 .4
.3 .5 .1 1 .3 .5 .1 1 .7 .5 .9 0
%
[µ ] ∪ [λ ]= .4 .4 .4 .5 ,[µ ]∩ [λ ] = .2 .4 .4 .5 and [µ ]° = .8 .6 .6 .5 .
ij ij ij ij ij
1 .9 .7 .5 .5 .1 .7 .5 0 .1 .3 .5
.6 .7 .6 .8 .5 .5 .4 .3 .4 .5 .4 .7
Proposition 3.1[2]
Let [µ ], [λ ] ∈FSM . Then
ij ij m×n
% %
(i) ([µij] ∪ [λij])°= [µij]° ∩ [λij]°
% %
(ii) ([µij] ∩ [λij])°= [µij]° ∪ [λij]°
Proof: (i )For all i and j,
%
([µij] ∪ [λij])°=[ max{ µij, λij}]°
=[1-max{ µij, λij}]
=[min{1- µij,1- λij}]
%
=[ µij]° ∩ [λij]°
(ii) similar to (i).
Proposition 3.2[2 ]
Let [µ ], [ν ], [λ ] ∈FSM , then
ij ij ij m×n
% % % % %
(i) [µ ] ∪ ([ν ] ∩ [λ ]) =([µ ] ∪ [ν ]) ∩ ([µ ] ∪ [λ ])
ij ij ij ij ij ij ij
% % % % %
(ii) [µ ] ∩ ([ν ] ∪ [λ ]) =([µ ] ∩ [ν ]) ∪ ([µ ] ∩ [λ ])
ij ij ij ij ij ij ij
4. Product of fuzzy soft matrices
In this section, four types of products of fuzzy soft matrices are defined in continuation to four special products of
soft matrices introduced by Çagman et al.[1].
Definition 4.1[2 ]
Let [µ ], [ν ] ∈FSM . Then And-product of [µ ] and [ν ] is defined by
ij ik m×n 2 ij ik
∧: FSM × FSM → FSM , [µ ] ∧[ν ]= [λ ]
m×n m×n m×n ij ik ip
where λ = min{µ , ν } such that p = n(j-1)+k.
ip ij ik
Definition 4.2[2 ]
Let [µ ], [ν ] ∈FSM . Then Or-product of [µ ] and [ν ] is defined by
ij ik m×n 2 ij ik
∨: FSM × FSM → FSM , [µ ] ∨ [ν ]= [λ ]
m×n m×n m×n ij ik ip
where λ = max{µ , ν } such that p = n(j-1)+k.
ip ij ik
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