International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 2261
ISSN 2229-5518
S. Ruban Raj1, M. Saradha2
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Most of our real life problems in medical sciences, engineering, management,
Maji, R. Biswas and A.R. Roy[6] studied the theory of soft sets initiated by Molodstov [2]
and developed several basic notions of Soft Set
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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, intuitionstic fuzzy sets, interval mathematics and rough sets etc. In this paper we introduced new concepts of soft intuitionstic fuzzy set. There has been incredible interest in the subject due to its diverse applications, ranging from engineering and computer science to social behavior studies. Fuzzy set was introduced by Lotfi A. Zadeh [1]. An Intuitionistic fuzzy set was provided by Atanassov [11]. However, Molodstov [2] has shown that each of the above topics suffers from some inherent difficulties due to inadequacy of their parameterization tools and introduced a concept called “Soft Set Theory” having parameterization 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. Fuzzy set theory proposed by Professor L.A. Zadeh[1] in 1965 is considered as a special case of the soft sets. In 2003, P.K.
Theory. At present, researchers are contributing a lot on the extension of soft set theory. In
2005, Pei and Miao[7] and Chen et al. [5] studied and improved the findings of Maji et al [6]. In 2011, T.J. Neog and D.K. Sut[8] studied the theory of soft sets initiated by Molodstov [2] and developed several basic notions of Soft Set Theory. Recently, Cagman et al. [9] introduced soft matrix and applied it in decision making problems. In one of our earlier work [10], we proposed the idea of ‘Properties of Soft Intuitionstic Fuzzy Set’ in sequel to [10] defining some operations. The present paper aims to define intuitionstic fuzzy soft matrices and establish some results on them. In this paper, we define a submatrixhood of intuitionstic soft fuzzy matrices along with several examples.
1. Associate Professor, Department of Mathematics, St.Joseph College, Trichirappalli, Tamilnadu
2. Research Scholar, Department of Mathematics, St.Joseph College, Trichirappalli, Tamilnadu
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Let X be an initial set and U be a set of parameters. Let P(X) denotes the power
set of X, and let A ⊂ U. A pair (F, A) is
called a soft set over X, where F is a mapping given by F: A → P(X).
In other words, a soft set over X is a parameterized family of subsets of the
universe X. For ξ ∈A, F (ξ) may be
considered as the set of ξ-elements of the
soft set (F, A) or as the ξ - approximate
elements of the soft set. Clearly, a soft set
is not a (crisp) set. To illustrate this idea, let we consider the following example.
set �ℎ2,ℎ5 �. Thus, we can view the soft set
(F, A) as a collection of approximations as
below
(F,A)=
⎧ 𝑒𝑥𝑝𝑒𝑛𝑠𝑖𝑣𝑒 ℎ𝑜𝑢𝑠𝑒𝑠 = �ℎ2, ℎ5 � ⎫
⎪ 𝑏𝑒𝑎𝑢𝑡𝑖𝑓𝑢𝑙 ℎ𝑜𝑢𝑠𝑒𝑠 = �ℎ1, ℎ3, ℎ5 � ⎪
⎨ 𝑤𝑜𝑜𝑑𝑒𝑛 ℎ𝑜𝑢𝑠𝑒𝑠 = �ℎ1, ℎ4, ℎ5 � ⎬
⎪ 𝑐ℎ𝑒𝑎𝑝 ℎ𝑜𝑢𝑠𝑒𝑠 = �ℎ1, ℎ2, ℎ5 � ⎪
⎩𝑖𝑛 𝑔𝑟𝑒𝑒𝑛 𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑𝑖𝑛𝑔 ℎ𝑜𝑢𝑠𝑒𝑠 = {ℎ5 }⎭
A pair (F, A) is called a fuzzy soft set over
X where F: A→ P�(X) is a mapping from A
intoP�(X).
Let X=�ℎ1, h2, h3, h4, h5, h6� be the set of six
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Let us consider a soft set (F, A) which describes the “attractiveness of houses” that Mr. P is considering to purchase.
Suppose that there are six houses in the universe X under consideration,
X=�ℎ1, h2,h3,h4, h5,h6�
and A=�𝑒1, e2,e3, e4,e5 �⊂ U
is a set of decision parameters, where 𝑒1
stands for the parameters “expensive”, 𝑒2
stands for the parameters “beautiful”, 𝑒3
stands for the parameters “wooden”, 𝑒4
stands for the parameters “cheap”, 𝑒5
stands for the parameters “in the green
surrounding”.
Suppose that
F(𝑒1)=�ℎ3,ℎ5 �,F(𝑒2)=�ℎ1, ℎ3, ℎ5 �,F(𝑒3)=
houses under consideration and
A=�𝑒1, e2,e3,e4, e5 �⊂ U is a set of decision
parameters,
Consider the mapping F: A→ P�(X).
(F, A)= {F (𝑒1) ={ℎ2 /0.9, ℎ5 /0.7}, F (𝑒2)
={ℎ1 /0.6, ℎ3 /0.9, ℎ5 /0.7},
F(𝑒3)={ℎ1 /0.5, ℎ4 /0.9, ℎ5 /0.6},
F(𝑒4)={ℎ1 /0.7, ℎ2 /0.8, ℎ5 /0.9},
F(𝑒5)={ℎ5 /0.8}}
is the fuzzy soft set representing the
“attractiveness of the houses” which Mr. X
is going to buy.
Let X be a universal set & U be the set of parameters. Let A ⊆ U & (F,A) be a fuzzy
soft set in the fuzzy soft class (X, U). Then
fuzzy soft set (F, A) in a matrix form as
�ℎ1, ℎ4, ℎ5 �,F(𝑒4)=�ℎ1, ℎ2 �,F(𝑒5)={ℎ5 }.
𝐴𝑚𝑥𝑛 = [𝑎𝑖𝑗 ]
Rmxn.
Or A= [𝑎𝑖𝑗 ] where
Therefore, F(𝑒1) means “houses
(expensive)”, whose functional value is the
i=1,2,…..m & j=1,2,…….n Where
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� �
𝜇𝑗 (𝑥𝑖 ) 𝑖𝑓 𝑒𝑗 ∈ 𝐴
If (𝜇𝑖𝑗 , 𝛾𝑖𝑗 ) = ( 𝐼𝜇𝛼 (x, y), 𝐼𝛾𝛼 (𝑥, 𝑦)), we
𝑎𝑖𝑗 = �0 𝑖𝑓 𝑒
∉ 𝐴
can define a matrix, [(𝜇𝑖𝑗 , 𝛾𝑖𝑗 )]𝑚𝑥𝑛 =
𝜇𝑗 (𝑥𝑖 ) represents the membership of 𝑥𝑖 in
the fuzzy set.
Let X=�ℎ1, h2, h3,h4� be the set of four houses under consideration and
A=�𝑒1, e2,e3,e4 �⊂ U is a set of decision
parameters,
The matrix representation is,
0.7 0 0.8 0
0 0.8 0.9 0.6
(𝜇11 , 𝛾11 ) (𝜇12 , 𝛾12 ) … (𝜇1𝑛 , 𝛾1𝑛 ) (𝜇21 , 𝛾21 ) (𝜇22 , 𝛾22 ) … (𝜇2𝑛 , 𝛾2𝑛 )
… … …
… … …
(𝜇𝑚1 , 𝛾𝑚1 ) (𝜇𝑚2 , 𝛾𝑚2 ) … (𝜇𝑚𝑛 , 𝛾𝑚𝑛 )
Which is called an m x n IFSM of the
IFSM (F, A) over X.
Therefore, we can say that an IFSS (F, A) is uniquely characterized by the matrix
[(𝜇𝑖𝑗 , 𝛾𝑖𝑗 )]𝑚𝑥𝑛 and both concepts are
interchangeable. The set of all m x n IFS
matrices over X will denoted
by IFSM𝑚 𝑥 𝑛 .
Let X be an initial universe, U be the set of parameters and A ⊆ U. Let (F, A) be an
intuitionistic fuzzy soft matrix (IFSM) over U. Let F(X×Y) denotes the family of an
intuitionstic fuzzy relationship on X to Y.
Suppose that X=�ℎ1, h2,h3, h4,h5, h6� is the set of houses and U = {expensive (𝑒1),
beautiful (𝑒2), wooden (𝑒3), cheap (𝑒4)}.
The set
If A ⊆ U = {𝑒 , 𝑒 , 𝑒
} and
1 2 3
� 𝐼� =(0.6, 0.4)⁄(ℎ
, 𝑒 )+(0.7, 0.3)⁄(ℎ
, 𝑒 )
I�∝ = {(x, y)∈X×Y: 𝐼𝜇𝛼 (x, y) ≥ α and ∝ 1 1 1 2
𝐼�𝛾 (𝑥, 𝑦) ≤ α} ⊂ X×Y is defined as α-cut set if Ĩ ∈ F(X×Y) for α ∈ [0, 1]. Then a
subset of X x U is uniquely defined by,
F: A → F� (X) and F(x) = {y ∈X: (x, y)
∈ 𝐼�∝ , x ∈ A, y ∈ X, α ∈ [0, 1]}.
The membership function and non
membership function are written by
+(0, 1)⁄(ℎ1, 𝑒3)+(0.9, 0.1)⁄(ℎ2, 𝑒1)+
(0, 1)⁄(ℎ2, 𝑒2)+(0.7, 0.3)⁄(ℎ2, 𝑒3)+
(0.5, 0.5)⁄(ℎ3, 𝑒1)+(0.7, 0.3)⁄(ℎ3, 𝑒2)+
(0.9, 0.1)⁄(ℎ3, 𝑒3)+(0.3, 0.5)⁄(ℎ4, 𝑒1)+
(0.9, 0.1)⁄(ℎ4, 𝑒2)+(0.6, 0.4)⁄(ℎ4, 𝑒3)+
(0.7, 0.3)⁄(ℎ5, 𝑒1)+(0.6, 0.4)⁄(ℎ5, 𝑒2)+
(0.5, 0.5)⁄(ℎ5, 𝑒3)+(0.8, 0.1)⁄(ℎ6, 𝑒1)+
(0.5, 0.2)⁄(ℎ6, 𝑒2)+(0.7, 0.2)⁄(ℎ6, 𝑒3).
Then, a soft intuitionistic fuzzy matrix
� �
𝐼𝜇𝛼 : X x U → [0, 1] and 𝐼𝛾𝛼 : X x U → [0,
1] where 𝐼�𝜇 : (x, y) ∈ [0, 1] and
𝐼�𝛾 : (x, y) ∈ [0, 1] are the membership
value respectively of x ∈ X for each y ∈ Y.
(F, A) is a parameterized family
{F(e1 ), F(e2), F (𝑒3)} of all IFS matrices over X.
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Hence the IFSM [(𝜇𝑖𝑗 , 𝛾𝑖𝑗 )] is written by
′ ′
′′ ′′
[(𝜇64 , 𝛾64 )]6𝑥4 =
(0.6, 0.4) (0.7, 0.3) (0, 1) 0 (0.9, 0.1) (0, 1) (0.7, 0.3) 0
Let (F, A) = [(𝜇𝑖𝑗, 𝛾𝑖𝑗 )], (G, B) = [(𝜇𝑖𝑗, 𝛾𝑖𝑗 )] ∈
IFSM𝑚 𝑥 𝑛 . Then the IFSM
(H, C) = [(𝜇𝑖𝑗 , 𝛾𝑖𝑗 )] is called
(a)Union of (F, A) and (G, B), denoted by
(F, A) ⋃� (G, B) if 𝜇𝑖𝑗 = max {(𝜇𝑖𝑗 , 𝜇𝑖𝑗 )}
′ ′′
′ ′′
(0.5, 0.5) (0.7, 0.3) (0.9, 0.1) 0
(0.3, 0.5) (0.9, 0.1) (0.6, 0.4) 0
and 𝛾𝑖𝑗 = min {(𝛾𝑖𝑗, 𝛾𝑖𝑗 )} for all i and j.
(b)Intersection of (F, A) and (G, B),
denoted by (F, A) ⋂� (G, B) if
(0.7, 0.3) (0.6, 0.4) (0.5, 0.5) 0
𝜇𝑖𝑗 = min {(𝜇𝑖𝑗
𝑖𝑗
𝑖𝑗
′ , 𝜇′′ )} and 𝛾
′ 𝛾′′ )} for all i and j.
= max
(0.8, 0.1) (0.5, 0.2) (0.7, 0.2) 0
{(𝛾𝑖𝑗,
𝑖𝑗
Let (F, A) = [(𝜇𝑖𝑗 , 𝛾𝑖𝑗 )] ∈ IFSM𝑚 𝑥 𝑛 . Then
(F, A) is called
(c)Complement of (F, A) is denoted by
(F, A)c=[(𝛾𝑖𝑗 , 𝜇𝑖𝑗 )] for all i and j.
Let (F, A) =
(a)A zero IFSM, denoted by 0� = [(0, 0)], if
𝜇𝑖𝑗 = 0 and 𝛾𝑖𝑗 = 0 for all i and j.
(b)A 𝜇 - universal IFSM, denoted by
𝐼̌ = [(1, 0)], if 𝜇𝑖𝑗 = 1 and 𝛾𝑖𝑗 = 0 for all i
and j.
(c)A 𝛾- universal IFSM, denoted by
𝐼̌ = [(0, 1)], if 𝜇𝑖𝑗 = 0 and 𝛾𝑖𝑗 = 1 for all i
and j.
(0.1, 0.2) (0.5, 0.4) (0.3, 0.6)
(0.4, 0.4) (0.2, 0.3) (0.5, 0.1) (0.5, 0.2) (0.3, 0.4) (0.6, 0.2) (0.7, 0.2) (0.6, 0.1) (0.5, 0.3)
and (G, B) =
′ ′
(d)A = [(𝜇𝑖𝑗, 𝛾𝑖𝑗 )] is a intuitionistic fuzzy soft
′′ ′′
sub matrix of B = [(𝜇𝑖𝑗, 𝛾𝑖𝑗 )], denoted by (F, A)
(0.5, 0.3) (0.1, 0.6) (0.7, 0.1)
′ ′ ′′ ′′
⊂ (G, B) if 𝜇𝑖𝑗
and j.
≤ 𝜇𝑖𝑗 and 𝛾𝑖𝑗
≤ 𝛾𝑖𝑗 for all i
(0.8, 0.1) (0.4, 0.3) (0.5, 0.2)
(e)(F, A) = [(𝜇′ 𝛾′ )] and(G, B) = [(𝜇′′ 𝛾′′ )]
𝑖𝑗, 𝑖𝑗 𝑖𝑗, 𝑖𝑗
are intutionistic fuzzy soft equal matrices,
(0.2, 0.5) (0.3, 0.6) (0.4, 0.5)
(0.1, 0.7) (0.2, 0.5) (0.5, 0.1)
denoted by(F, A) = (G, B), if 𝜇′
′ = 𝛾′′ for all i and j.
= 𝜇′′ and
𝛾𝑖𝑗
𝑖𝑗
Then (F, A) ⋃� (G, B) =
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(0.5, 0.2) (0.5, 0.4) (0.7, 0.1) (0.8, 0.1) (0.4, 0.3) (0.5, 0.1) (0.5, 0.2) (0.3, 0.4) (0.6, 0.2)
(0.7, 0.2) (0.6, 0.1) (0.5, 0.1)
(F, A) ⋂� (G, B) =
,
(0.1, 0.3) (0.1, 0.6) (0.3, 0.6) (0.4, 0.4) (0.2, 0.3) (0.5, 0.2) (0.2, 0.5) (0.3, 0.6) (0.4, 0.5) (0.1, 0.7) (0.2, 0.5) (0.5, 0.3)
We define for a finite fuzzy soft matrix (F,
A), the Set Cardinality |(F, A)| is defined
as,
|(F, A)| = � exp 𝐹(𝑒𝑗 )
𝑒𝑗 ∈𝐴
Let X=�ℎ1, h2,h3,h4 � be the set of houses under consideration and U = {expensive
(𝑒1), near by city (𝑒2), cheap (𝑒3), beautiful (𝑒4)} be the set of parameters.
Consider the fuzzy soft matrices (F, A) and
(G, B) where A = {𝑒1, e2, e3,} ⊆ U. Then
(F, A) =
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and
(F, A)c =
(0.6, 0.3) (0.6,0.2) (0.9,0.1) (0.8, 0.1) (0.7, 0.2) (0.9, 0.1) (0.5, 0.4) (0.7, 0.3)
(0.5, 0.4) (0, 0) (0.6, 0.3) (0, 0) (0.5,0.4) (0, 0) (0.9, 0.1) (0, 0)
(0.2, 0.1) (0.4, 0.5) (0.6, 0.3) (0.4, 0.4) (0.3, 0.2) (0.1, 0.5) (0.2, 0.5) (0.4, 0.3) (0.2, 0.6) (0.2, 0.7) (0.1, 0.6) (0.3, 0.5)
For a finite fuzzy soft matrix (F, A), the
|(F, A)| = ∑𝑒𝑖𝑗 ∈𝐴 exp 𝐹(𝑒𝑖𝑗 ) = 3.3.
Let X be a Universal, U be a set of
parameters and let (F, A)Ĩ and (G, B)Ĩ are
two soft intuitionistic fuzzy matrix of X.
Then the degree of subset hood denoted by
S (A, B)Ĩ is defined as,
Cardinality |(F, A)| is defined as,
|(F, A)| = ∑xϵU µij (x) = ∑(F , A).
S (A, B)
I𝝁
= 1
|(F,A)|I�µ
�|(F, A)|
Ĩµ −
And its relative cardinality ‖(𝐹, 𝐴)‖ is,
|(F,A)|
‖(F, A)‖ =
∑ max{0, (F, A)Ĩ µ − (G, B)Ĩµ �
where
|F, A|𝐼̃𝜇
=∑ 𝜇𝐴(𝑥)
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|F, A|𝐼̃𝛾 =∑ 𝛾𝐴 (𝑥)
(F, A) – (G, B) = (-0.1, 0, -0.1, -0.1)
|G, B|𝐼̃𝜇 =∑ 𝜇𝐵 (𝑥)
S(A, B) = 1
3.2
{3.2 − ∑ max{0, (−0.1, 0,
|G, B|𝐼̃𝛾 =∑ 𝛾𝐵 (𝑥)
−0.1, −0.1)}} = 1 and
S(B,A)=
and
1
3.5
{3.5 ∑ max{0, (0.1, 0, 0.1, 0.1)}}
S (B, A)
I 𝝁
= 1
|(G,B)|I�µ
�|(G, B)|
Ĩµ −
= 0.9143 ≅ 0.9.
4. Conclusion:
∑ max{0, (G, B)Ĩµ − (F, A)Ĩ µ �
Let X=�ℎ1, h2,h3,h4 � be the set of houses under consideration and U = {expensive
(𝑒1), near by city (𝑒2), cheap (𝑒3), beautiful (𝑒4)} be the set of parameters.
Consider the fuzzy soft matrices (F, A) and
In this article, we have explained a very important consistency principle to Degree of Sub matrix hood of the fuzzy soft matrix. In this paper the basic concept of a vague soft matrix is recalled. We have introduced the concept of sub matrix hood of the soft fuzzy matrix as an extension to the fuzzy soft matrix. The basic properties
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(G, B) where A, B ⊆ U.
Then (F, A) =
(0.8, 0.1) (0.8, 0.2)
(0, 0) (0.6, 0.4)
on soft fuzzy matrix are also presented.
The null, union, intersection, sub matrix and sub matrix hood as far as future direction are concerned. It is hoped that
(0.9, 0.1) (0.7,0.2)
(0.8, 0.1) (0.5, 0.4)
(0.7, 0.2) (0.6, 0.2)
(0.4, 0.5) (0.5, 0.4)
(0.2,0.7) (0.6, 0.3)
(0.1, 0.7) (0.4, 0.5)
our findings will help enhancing this study
on fuzzy soft matrices for the researchers.
and (G, B) =
(0.8, 0.1) (0.9, 0.1) (0.9, 0.1) (0.8, 0.2)
(0.9, 0.1) (0.8, 0.2) (0.8, 0.1) (0.6, 0.3)
(0.6, 0.3) (0.7, 0.2) (0.8, 0.2) (0.6, 0.3)
(0.4, 0.6) (0.9, 0.1) (0.8, 0.1) (0.7, 0.2)
[1] Zadeh, L. A., “Fuzzy Sets”, Information and Control, 8, pp.
338-353, 1965.
[2] Molodstov. D.A., “Soft Set Theory – First Result”, Computers and Mathematics with Applications, Vol. 37, pp. 19-31,
1999.
|(F, A)| = ∑𝑒𝑖𝑗 ∈𝐴 exp 𝐹(𝑒𝑖𝑗 ) = 3.2
|(G, B)| = ∑𝑒𝑖𝑗 ∈𝐴 exp 𝐺(𝑒𝑖𝑗 ) = 3.5
[3] Ahmad, B. and Kharal, A., “Mappings on Soft Classes” (Originally Submitted on 17th oct,
2008 to Information Sciences with the title of “Mappings on Soft Sets” and was given MS#INS-D-
08-1231 by EES.
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 2267
ISSN 2229-5518
[4] Ali, M. Irfan, Feng F, Liu, X., Min, W.K. Shabir, M. “On some new opereations in soft set theory”, Computers and Mathematics with Applications 57 (2009) 1547-1553.
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