International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 5, July-August 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
Some New Operators on Multi
Intuitionistic Fuzzy Soft Matrix Theory
P. Rajarajeswari1, J. Vanitha2
1Assistant
2Assistant
Professor of Mathematics, Chikkanna Government Arts College, Tirupur, Tamil Nadu, India
Professor of Mathematics, RVS College of Education, Sulur, Coimbatore, Tamil Nadu, India
ABSTRACT
In this paper, we have defined four new operators namely □, ◊,
, ⊗ on a
new type of matrix namely Multi Intuitionistic Fuzzy soft Matrix were
defined and some of their properties are studied. The concepts are
illustrated with suitable numerical examples.
How to cite this paper: P. Rajarajeswari
| J. Vanitha "Some New Operators on
Multi Intuitionistic Fuzzy Soft Matrix
Theory" Published in International
Journal of Trend in
Scientific Research
and
Development
(ijtsrd), ISSN: 24566470, Volume-4 |
Issue-5,
August
IJTSRD33009
2020, pp.843-848,
URL:
www.ijtsrd.com/papers/ijtsrd33009.pdf
KEYWORDS: Soft Set, Fuzzy Soft Set, Intuitionistic Fuzzy soft set, Multi-Fuzzy
Soft Set
Copyright © 2020 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
by/4.0)
1. INTRODUCTION
In real world problems we have uncertainties. Zadeh [11]
in 1965, has introduced the concept namely Fuzzy sets to
deal uncertainties which consists of degree of
membership. Intuitionistic Fuzzy Sets are introduced by
Atanasov[1,2] which are extension of Fuzzy Sets and
consists of both membership value and non-membership
value associated with every element. The concept Soft set
theory have been introduced by Molodtsor [8] in 1999 and
he also studied various properties of soft set.
Representation of Soft sets in matrix form was given by
Cagman et.al [5]. Maji et. Al. [7] have introduced the
concept of Intuitionistic fuzzy soft set. Multi sets and Multi
Fuzzy Sets were studied in [3,4] and [10]. Intuitionistic
Multi fuzzy soft sets were introduced by Sujit Das and
Samarjit Kar [9].
Definition 2.2
An Intuitionistic Fuzzy Set (IFS) A in E is defined as an
object of the following form A= {( x, A (x), A (x))/ x  E}
where the functions, A : E [ 0, 1 ] and A : E [ 0, 1 ]
define the degree of membership and the degree of nonmembership of the element x  E respectively and for
every x  E: 0 ≤ A (x) + A (x) ≤ 1.
AMS Mathematics subject classification (2010): 08A72
Definition 2.4
Let IP(U) denotes the set of all intuitionistic fuzzy set of U.
A pair (F,A) is called a intuitionistic fuzzy soft set (IFSS) of
over U, where F is a mapping given by F:A IP (U ) . For
any parameter e A, F(e) is an intuitionistic fuzzy subset
of U and is called Intuitionistic fuzzy value set of
parameter e. Clearly F(e) can be written as an
Intuitionistic fuzzy set such that F(e) = { x, F(e) (x), F(e) (x)
 x U }. Here F(e) (x) , F(e) (x) are membership amd nonmembership functions respectively and x U, F(e) (x) +
F(e) (x) ≤ 1,
2. PRELEMINARIES
In this section we have given some basic definitions and
properties which are required for this paper.
Definition 2.1
Let X denotes a Universal set. Then the membership
function A by which a fuzzy set (FS) A is usually defined
has the form A: X
[0, 1], where [0, 1] denotes the
interval of real numbers from 0 to 1 inclusive.
@ IJTSRD
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Unique Paper ID – IJTSRD33009
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Definition 2.3
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
U
over U where F is a mapping given by F:A I , where
I U denotes the collection of all fuzzy subsets of U. An
fuzzy soft set is a parameterized family of fuzzy subsets of
Universe U.
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International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Definition 2.5
Let IMFS(U) denotes the set of all intuitionistic multi fuzzy
set of U. A pair (F,A) is called a intuitionistic multi-fuzzy
soft set (IMFSS) of dimension k over U, where F is a
K
mapping given by F:A IMFS (U ) . An intuitionistic
multi fuzzy soft set is a mapping from parameters A to
IMFS K (U ) . It is a parameterized family of intuitionistic
multi fuzzy subsets of U. For e A, F(e) may be considered
as the set of e- approximate elements of the intuitionistic
multi fuzzy soft set( F,A).
3. Some New Operators on Multi Intuitionistic Fuzzy
Soft Matrices
In this section, some new Operators on Multi Intuitionistic
Fuzzy Soft Matrices are defined and based on these
operators some properties are studied.
Definition 3.1
Let U = {u1 ,u2 ,…, um } be the universal set and E = {e1 ,e2 ,…,
en } be the set of parameters. Let A⊆E and be a Multi
Intuitionistic Fuzzy Soft Set on U. Then the matrix
associated with this set namely Multi Intuitionistic Fuzzy
̃ (K) = [𝑎ij (K) ]
Soft Matrix A
, i=1,2,…,m j=1,2,…,n
m×n
(K)
(K)
(μj (ui ), νj (ui ))
(or) (μ(K)
̌ ij
A
(K)
, νǍ )
ij
(K)
ij
ij
(K)
represents
(K)
(K)
the
membership
and
non-
The set of all m × n Multi Intuitionistic Fuzzy Soft Matrices
are denoted by M(K) IVFSM.
Example 3.2
Suppose that U = {u1 ,u2 ,u3 } is the universal set of students
and E = {e1 ,e2 } is the set of parameters where e1 =
Academic performance and e2 = Sports performance. Let A
= E. Then the Multi Intuitionistic Fuzzy Soft Set (F,A),
where F : n → M(K) IFS (Multi Intuitionistic Fuzzy Sets on U)
and is given by
F(A) = { F(e1 ) = {( u1 , (0.8,0.1),(0.7,0.2))}, {( u2 ,
(0.5,0.3),(0.4,0.2))}, {( u3 , (0.6,0.2),(0.8,0.1))}} , F(e2 ) = {(
u1 , (0.7,0.2),(0.6,0.3))}, {( u2 , (0.4,0.2),(0.5,0.1))}, {( u3 ,
(0.7,0.1),(0.6,0.2))}}}
We can represent the above Multi Intuitionistic Fuzzy Soft
Set in Matrix as follows.
e1 e2
u1 ((0.8,0.1), (0.7,0.2)) ((0.7,0.2), (0.6,0.3))
̃(K)
u
A
3×2 = 2 (((0.5,0.3), (0.4,0.2)) ((0.4,0.2), (0.5,0.1)))
u3 ((0.6,0.2), (0.8,0.1)) ((0.7,0.1), (0.6,0.2))
3×2
Definition 3.3
̃(K) = [𝑎ij (K) ]
Let A
m×n
(K)
ij
Definition 3.4
A Multi Intuitionistic Fuzzy Soft Matrix of order m × n
with cardinality K is called Multi Intuitionistic Fuzzy Soft
Null (Zero) Matrix if all of its elements are (0(K) ,1(K) ). It is
̃ (K).
denoted by ɸ
Definition 3.5
A Multi Intuitionistic Fuzzy Soft Matrix of order m × n
with cardinality K is called Multi Intuitionistic Fuzzy Soft
Absolute Matrix if all of its elements are (1(K) ,0(K) ). It is
(K)
denoted by Ĩ .
Definition 3.6
̃ (K) = [𝑎ij (K) ]
If A
(K)
m×n
̃(K) = [bij (K) ]
∈ M(K) IFSM and B
m×n
∈
M IFSM then Addition, Subtraction and Multiplication of
̃ (K) and B
̃(K)
two Multi Intuitionistic Fuzzy Soft Matrices A
are defined as
̃ (K) + B
̃(K)
A
=
ij
(K) (K)
μǍ , μB̌
ij
ij
̃ (K) = [bij (K) ]
∈ M(K) IFSM and B
m×n
∈
is a Multi Intuitionistic Fuzzy Soft Sub
Unique Paper ID – IJTSRD33009
|
ij
ij
̃ (K) − B
̃(K)
A
) , max
(K)
=
(K) (K)
( νǍ , νB̌
ij
ij
(K)
) )] for all i,j and K.
̃
∗B
=
max min (K) (K) min max (K) (K)
[(
( μǍ , μB̌ ) ,
( νǍ , νB̌ ) )].
l ij
ij
ij
jk
jl
l ij
[Cil ]m×p=
Example 3.7
̃ 2×2 (2)
Consider
A
((0.7,0.2)(0.8,0.1)) ((0.6,0.2)(0.7,0.1))
(
) and
((0.5,0.4)(0.4,0.5)) ((0.2,0.6)(0.4,0.5))
̃2×2 (2) = (((0.6,0.2)(0.5,0.4)) ((0.6,0.1)(0.7,0.2)))
B
((0.7,0.1)(0.6,0.2)) ((0.5,0.3)(0.2,0.7))
=
are two Multi Intuitionistic Fuzzy Soft Matrices then
̃(K) + B
̃(K) = (((0.6,0.2)(0.5,0.4))
A
((0.7,0.1)(0.6,0.2))
(K)
(K)
̃ −B
̃ = (((0.6,0.2)(0.5,0.4))
A
((0.5,0.4)(0.4,0.5))
(K)
(K)
((0.6,0.2)(0.6,0.2))
̃ ∗B
̃ =(
A
((0.5,0.4)(0.4,0.5))
Definition 3.8
̃(K) = [𝑎ij (K) ]
Let A
m×n
((0.6,0.1)(0.7,0.1))
)
((0.5,0.3)(0.4,0.5))
((0.6,0.2)(0.7,0.2))
)
((0.2,0.6)(0.2,0.7))
((0.6,0.2)(0.7,0.2))
)
((0.5,0.4)(0.4,0.5))
̃T (K) is the Multi
∈ M(K) IFSM then A
̃(K) and is
Intuitionistic Fuzzy Soft Transpose Matrix of A
̃T (K) =[𝑎 (K) ]
given by A
∈M(K) IVFSM.
𝑗𝑖
n×m
Definition 3.9
̃(K) = [𝑎𝑖𝑗 (K) ]
Let A
(K)
(K)
(μj (ui ),νj (ui )).
̃
M(K) IFSM. Then A
|
ij
ij
̃
A
if ej ∉A
membership of the Multi Intuitionistic Fuzzy Soft Set.
@ IJTSRD
ij
[(min (
if ej ∈A
(0(K) ,1(K) )
(K)
(K)
(K)
ij
Also 0 ≤ μj (ui )+ νj (ui ) ≤ 1 and μj (ui ) , νj (ui ) and
μǍ , νǍ
(K)
νǍ ≥ νB̌ for all i,j and K.
(K)
(K) (K)
[(max ( μ(K)
̌ ) )] for all i,j and K.
̌ , μB
̌ , νB
̌ ) , min ( νA
A
where,
𝑎ij (K) = {
̃(K) , denoted by A
̃(K) ⊆ B
̃(K) if μ(K) ≤ μ(K) and
Matrix of B
̌
̌
A
B
m×n
∈ M(K) IFSM
̃C
Then A
(K)
where
𝑎𝑖𝑗 (K)
=
, the Multi Intuitionistic Fuzzy
(K)
̃ is defined as
Soft Complement Matrix A
(K)
(K)
(K)
(K)
̃C = [𝑏
]
A
= (ν (u ),μ (u )) , For all i,j and K.
𝑖𝑗
Volume – 4 | Issue – 5
m×n
|
j
i
j
i
July-August 2020
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International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Definition 3.10
̃(K) = [a ij (K) ]
Let
A
(K)
(K)
(μj (Ci ), νj (Ci )).
(K)
I.
m×n
∈
(K)
M IFSM
a ij (K) =
where
Then
(K)
is called a Multi Intuitionistic Fuzzy Soft
̃(K) and is defined as
Necessity Matrix of A
(K)
̃(K) = [bij (K) ]
□A
, where bij (K) = (μ (Ci ), 1 −
j
m×n
νj (Ci )) for all i,j and K.
̃(K) is called a Multi Intuitionistic Fuzzy Soft
II. ◊A
̃ (K) and is defined as
Possibility Matrix of A
̃(K)
[bij (K) ]
(1 −
◊A
=
,
where
bij (K) =
(K)
(K)
(K)
(K)
(K)
[(μj (Ci ), 1-μj (Ci ))] ⊆ [(μj (Ci ), νj (Ci ))] ⊆ [(1(K)
νj (Ci ), νj (Ci ))] for all all i,j and K.
̃
□A
(K)
(K)
(K)
(iii). We have 0 ≤ μj (Ci ) + νj (Ci ) ≤ 1 for all all i,j and K.
̃(K) ⊆ A
̃(K) ⊆ ◊A
̃(K) .
Hence □A
(K)
̃
(iv). □A
(K)
(K)
= [(μj (Ci ), 1-μj (Ci ))] for all all i,j and K.
̃ (K) = [(μ(K)
(Ci ), 1-μ(K)
(Ci ))] for all all i,j and K.
□□A
j
j
(K)
̃
= □A
̃ (K) ) = ◊A
̃(K)
(v).Similarly we can prove ◊ (◊A
m×n
(K)
μj (Ci ), νj (Ci )) for all i,j and K.
̃(K) =
◊A
(vi).
Example 3.11
̃2×2 (2) be a Multi Intuitionistic Fuzzy Soft Matrix given
Let A
by
̃2×2 (2) = (([0.6,0.3], [0.7,0.3]) ([0.6,0.2], [0.5,0.3]))
A
([0.5,0.4], [0.6,0.2]) ([0.7,0.2], [0.8,0.1]) 2×2
(K)
(K)
([(1-νj (Ci ), νj (Ci ))]) for all i,j and K.
(K)
(K)
= [(1-νj (Ci ), νj (Ci ))] for all i,j and K.
̃ (K)
= ◊A
(K)
̃ ) = ◊ ([(μ(K)
(Ci ), 1-μ(K)
(Ci ))]) for all i,j and K.
(vii). ◊ (□A
j
j
(K)
(K)
= [(μj (Ci ), 1-μj (Ci ))] for all i,j and K.
̃(K)
= □A
Then Multi Intuitionistic Fuzzy Soft Necessity Matrix of
̃(2) is given by
A
̃2×2 (2) = (([0.6,0.4], [0.7,0.3]) ([0.6,0.4], [0.5,0.5]))
□A
([0.5,0.5], [0.6,0.4]) ([0.7,0.3], [0.8,0.2]) 2×2
(viii). Proof follows from (iv)
Also Multi Intuitionistic Fuzzy Soft Possibility Matrix of
̃(2) is given by
A
̃2×2 (2)= (([0.7,0.3], [0.7,0.3]) ([0.8,0.2], [0.7,0.3]))
◊A
([0.6,0.4], [0.8,0.2]) ([0.8,0.2], [0.9,0.1]) 2×2
Remark 3.13
̃ (K) be a Multi Intuitionistic Fuzzy Soft Matrix. Then in
Let A
̃(K) ≠ ◊ □A
̃(K).
general □ ◊ A
Proposition 3.12
̃(K) = [a ij (K) ]
Let
A
m×n
∈
M(K) IFSM
a ij (K) =
where
(K)
(K)
(μj (Ci ), νj (Ci )). Then
C
I.
̃C (K) ) = ◊A
̃(K)
(□A
II.
̃C
(◊ A
III.
IV.
V.
VI.
(K) C
(K)
̃
) = □A
(K)
(K)
̃C
(□A
) = (1 −
(K)
(K)
νj (Ci ), νj (Ci )) for all i,j
∈
and K.
Unique Paper ID – IJTSRD33009
m×n
=
m×n
(K)
(K)
ij
ij
= [(μǍ , νǍ )] ∈ M(K) IFSM and
(K)
[(μB̌
ij
(K)
, νB̃ )]
ij
∈ M(K) IFSM then
(K)
̃(K) + B
̃(K) = [(max(μ(K) , μ(K) ), min(ν(K)
A
̃ ))] for all i,j and
̌ , νB
̌
̌
A
A
B
ij
ij
ij
ij
K
̃(K) + B
̃(K) ) = [(max(μ(K) , μ(K) ) , 1 − max(μ(K) , μ(K) ))] for
□(A
̌
̌
̌
̌
A
B
A
B
ij
ij
ij
ij
all i,j and K --------(1)
̃(K) = [(μ(K) , 1 − μ(K) )] for all i,j and K
□A
(K)
=
̌ ij
A
[(μ(K)
̌ ij , 1
B
−
(K)
μB̌ )]
ij
for all i,j and K
̃(K) + □B
̃(K) = [(max(μ(K) , μ(K) ) , min (1 − μ(K) , 1 − μ(K) ))]
□A
̌
̌
̌
̌
A
B
A
B
ij
ij
ij
ij
for all i,j and K
(K)
(K)
(K)
(K)
ij
ij
ij
ij
= [(max(μǍ , μB̌ ) , 1 − max(μǍ , μB̌ ))] for all i,j and K ----
̃
=◊A
Similarly (ii) can be proved.
|
̃(K) = [bij (K) ]
B
̃
□B
(K)
@ IJTSRD
= [a ij (K) ]
̌ ij
A
i
̃C (K) = (ν(K) (C ), 1 − ν(K) (C )) for all i,j and K.
Now, □A
i
i
j
j
(K) C
m×n
C
̃C (K) ) = (□A
̃(K) )
◊(A
(K)
Proof
̃C (K) = (ν(K) (C ), μ(K) (C )) for all i,j and K.
(i) A
j
M IFSM then
̃(K) + B
̃ (K) ) = □A
̃(K) + □B
̃(K)
(i)
□(A
(K)
(K)
(K)
̃ +B
̃ )=◊A
̃ +◊ B
̃(K)
(ii)
◊ (A
C
̃C (K) ) = (◊A
̃(K) )
(iii)
□(A
̃
(i). Let A
̃ ) = □A
̃
VII. ◊ (□A
n ̃ (K)
̃(K) for all integer n >
VIII. □ (A ) = (□(□( … (A))) = □A
0.
̃(K) ) = ◊(◊ (◊ ( … (A))) = ◊ A
̃(K) for all integer n >
IX.
◊n (A
0.
i
m×n
(K)
Proof
(K)
j
Proposition 3.14
̃ (K) = [a ij (K) ]
̃(K) = [bij (K) ]
If A
∈ M(K) IFSM and B
(iv)
(K)
̃ ⊆A
̃ ⊆ ◊A
̃
□A
(K)
(K)
̃
̃
(□A ) = □A
̃(K) ) = ◊A
̃(K)
◊ (◊A
̃ (K) ) = ◊A
̃(K)
(◊A
(K)
(ix). Proof follows from (v)
----(2)
̃(K) + B
̃(K) ) = □A
̃(K) + □B
̃(K).
From (1) and (2) □(A
Similarly (ii) can be proved.
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International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
̃C (K) = [(ν(K) , μ(K) )] for all i,j and K
(iii). Consider A
̌
̌
A
A
ij
Corollary 3.17
(K)
̃ (K)
Let A
m×n ∈ M IFSM then
̃(K) ⊆ A
̃(K) ⊕ A
̃(K)
(i)
A
(K)
(K)
̃ ⊗A
̃ ⊆A
̃(K)
(ii)
A
̃(K) ⊗ A
̃(K) ⊆ A
̃(K) ⊆ A
̃(K) ⊕ A
̃(K)
(iii)
A
̃(K) ⊗ A
̃(K) ⊆ A
̃(K) ⊕ A
̃(K)
(iv)
A
Proof is Obvious.
ij
̃C (K) = [(ν(K) ,1 − ν(K) )] for all i,j and K ---(3)
Now, □A
̌
̌
A
A
ij
(K)
̃
◊A
= [(1 −
ij
(K)
(K)
νǍ ,
ij
ν
)] for all i,j and K
̌ ij
A
C
(K)
̃(K) ) = [(ν(K)
(◊A
̌ ,1 − νA
̌ )] for all i,j and K ---(4)
A
ij
ij
C
̃C (K) ) = (◊A
̃(K) )
From (3) and (4), □(A
Similarly, (iv) can be proved.
Definition 3.15
̃(K) = [a ij (K) ]
If A
(K)
m×n
̃ (K) = [bij (K) ]
∈ M(K) IFSM and B
m×n
∈
M IFSM then the operators ⊕ and ⊗ are defined on
these matrices as follows.
(K)
̃
(i). A
̃
⊕B
(K)
(K)
(K)
(K)
(K)
(K)
(K)
ij
ij
ij
ij
ij
̃ij
B
(K)
(K)
ij
̃ij
B
= [(μǍ + μB̌ − μǍ . μB̌ , νǍ . ν
i,j and K
(K)
̃
(ii). A
̃
⊗B
(K)
(K)
(K)
(K)
ij
ij
ij
(K)
= [(μǍ . μB̌ , νǍ + ν
i,j and K
̃ij
B
)] for all
− νǍ . ν
)] for all
̃(K) ⊕ B
̃(K) ∈ M(K) IFSM and A
̃(K) ⊗ B
̃(K) ∈
A
M(K) IFSM . Here +, -, . represents usual addition,
subtraction and multiplication.
Proposition 3.18
̃ (K) , B
̃(K) and C̃ (K) ∈ M(K) IVFSM then
Let A
̃(K) ⊕ B
̃(K) = B
̃(K) ⊕ A
̃(K) ( Commutative Law)
(i)
A
̃(K) ⊕ B
̃ (K) ) ⊕ C̃ (K) =A
̃(K) ⊕ ( ̃B(K) ⊕ C̃ (K) )
(ii)
(A
(Associative Law)
̃(K) ⊗ B
̃(K) = B
̃(K) ⊗ A
̃(K) ( Commutative Law)
(iii)
A
(K)
(K)
(K)
(K)
̃ ⊗B
̃ ) ⊗ C̃ =A
̃(K) ⊗ ( ̃B(K) ⊗
(iv)
(A
C̃ )
(Associative Law)
Proof
(K) ]
̃ (K)
Let A
m×n =[a ij
(K)
̃(K)
]
B
m×n =[bij
(K)
C̃ m×n =[cij (K) ]
Proposition 3.16
(K)
(K)
̃(K)
̃ (K)
Let A
m×n ∈ M IFSM and Bm×n ∈ M IFSM then
̃(K) + B
̃(K) ⊆ A
̃(K) ⊕ B
̃(K)
(i)
A
Now,
(ii)
(iii)
= [(μB̌ +μ
(K)
ij
ij
̃
+B
(K)
(K) (K)
(K) (K)
= [(max(μǍ , μB̌ ), min(νǍ , νB̃ ))] for all
ij
ij
ij
ij
i,j and K
(K)
̃
A
̃
⊕B
and K
Since
(K)
=
(K)
[(μǍ
ij
+
(K)
(K)
ij
ij
(K)
μB̌
ij
max(μǍ , μB̌ )
(K)
(K)
(K)
ij
ij
ij
(K) (K)
− μǍ . μB̌
ij
ij
≤
,
(K)
νǍ .
ij
(K)
ν
̃ij
B
)] for all i,j
(K)
(K)
(K)
(K)
ij
ij
ij
ij
μǍ + μB̌ − μǍ . μB̌
(K)
and
min(νǍ , νB̃ ) ≥ νǍ . ν
(K)
̃
Hence A
(ii).
(K)
(K)
ij
̃ij
B
(K)
νǍ + ν
̃ij
B
̃ij
B
(K)
̃
⊆A
(K)
̃
A
m×n
(K)
(μǍ
ij
̃
⊕B
̃
⊗B
(K)
(K)
(K)
(K)
(K)
[(μ(K)
̌ +ν
̌ . μB
̌ , νA
A
=
ij
(K)
(K)
(K)
ij
ij
ij
, μB̌ ) ≥ μǍ . μB̌
(K)
(K)
ij
̃ij
B
(K)
ij
ij
̃ij
B
−
(K)
(K)
ij
̃ij
B
and min ( νǍ , ν ) ≤
ij
(K)
(K)
ij
ij
∈ M(K) IFSM and
m×n
∈ M(K) IFSM and
m×n
̌ ij
A
(K)
ij
ij
ij
ij
(K)
(K)
(K)
(K)
ij
ij
ij
ij
̃ij
B
Unique Paper ID – IJTSRD33009
|
)]
− μB̌ . μǍ , νB̃ . νǍ )] for all i,j and K
̃
⊕A
Similarly (ii) can be proved.
(K)
̃(K) ⊗ B
̃(K) = [(μ(K) . μ(K) , ν(K)
(ii). For all i,j and K. A
̌ +ν
̌
̌
A
A
B
(K)
(K)
ij
̃ij
B
νǍ . ν
ij
ij
̃ij
B
−
)]
(K)
(K)
(K)
(K)
(K)
(K)
ij
ij
ij
ij
ij
ij
= [(μB̌ . μǍ , νB̃ + νǍ − νB̃ . νǍ )]
̃(K) ⊗ A
̃(K)
=B
Similarly (iv) can be proved.
Proposition 3.19
(K) ]
̃ (K)
Let A
m×n =[a ij
m×n
(K)
(K)
ij
ij
= [(μǍ , νǍ )]
(K)
(K)
ij
ij
= [(μB̌ , νB̃ )]
∈ M(K) IFSM and
m×n
̃(K) and
∈ M(K) IFSM . If A
m×n
̃(K) are Symmetric, then A
̃(K) ⊕ B
̃(K) and A
̃(K) ⊗ B
̃(K) are
B
also Symmetric.
Proof
(K) ]
̃ (K)
Let A
m×n =[a ij
(K)
̃(K)
]
B
m×n =[bij
− νǍ . ν
|
̃
=B
(K)
(K)
(K)
̃(K)
]
B
m×n =[bij
̃(K) ⊗ B
̃ ⊆A
̃(K) + B
̃(K)
Therefore A
(iii). From above results (i) and (ii) combining both we
have
̃(K) ⊗ B
̃(K) ⊆ A
̃(K) + B
̃(K) ⊆ A
̃(K) ⊕ B
̃(K)
A
@ IJTSRD
(K)
ij
= [(μČ , νC̃ )]
m×n
for all i,j and K
)]
Since max
ij
(K)
Now,
νǍ . ν
(K)
̃
+B
(K)
∈ M(K) IFSM ,
m×n
ij
ij
(K)
ij
(K)
(K)
(K)
[(μ(K)
̃ )] ∈ M IFSM
̌ , νB
B
̃
Now, A
m×n
(K)
ij
= [(μB̌ , νB̃ )]
ij
Proof
(K)
(K)
(K)
̃(K)
̃(K)
Let A
m×n = [(μA
̌ )] ∈ M IFSM and Bm×n =
̌ , νA
ij
m×n
(K)
= [(μǍ , νǍ )]
̃(K) ⊕ B
̃(K) = [(μ(K) + μ(K) − μ(K) . μ(K) , ν(K)
A
̌ .ν
̌
̌
̌
̌
A
A
B
A
B
̃(K) ⊗ B
̃(K) ⊆ A
̃(K) + B
̃(K)
A
̃(K) ⊗ B
̃(K) ⊆ A
̃(K) + B
̃(K) ⊆ A
̃(K) ⊕ B
̃(K)
A
ij
m×n
m×n
m×n
(K)
(K)
ij
ij
= [(μǍ , νǍ )]
(K)
(K)
ij
ij
= [(μB̌ , νB̃ )]
(K)
(K)
ij
ij
∈ M(K) IFSM and
m×n
∈ M(K) IFSM .
m×n
(K)
(K)
ij
ij
Then we have 0 ≤ μǍ + νǍ ≤ 1 and 0 ≤ μB̌ + νB̃ ≤ 1 for
all i,j and K.
̃T (K) =
̃(K) and B
̃(K) are symmetric. We have A
Given that A
̃T (K) = B
̃(K) and B
̃(K) .
A
Volume – 4 | Issue – 5
|
July-August 2020
Page 846
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
(K)
(K)
(K)
(K)
ji
ij
ji
ij
Therefore, μǍ = μǍ and νǍ = νǍ .
(K)
(K)
ji
ij
(K)
(K)
ji
ij
̃ ⊕B
̃
For all i,j and K, (A
(K)
ij
ij
(K)
(K)
ij
̃ij
B
=
[(μ(K)
̌ ji
A
(K)
+ μB̌
ji
=
(K)
[(μǍ
ij
(K)
+ μB̌
ij
(K)
(K)
(K) T
=
)
(K)
[(μǍ
ij
(K)
+ μB̌
ij
−
(K)
(K)
ji
ji
(K)
(K)
̃ ⊕B
̃
(A
(i)
(K)
̃
(A
(ii)
ji
ji
(K)
νǍ .
ij
,
(K)
ν
)]
̃ij
B
(K) T
̃T
) ≠A
(K)
̃T
⊗B
(K)
C̃ =[cij (K) ]
m×n
m×n
ij
ij
(K)
(K)
ij
ij
= [(μB̌ , νB̃ )]
(K)
(K)
= [(μČ , νC̃ )]
ij
ij
T
T
̃T
̃(K) ⊕B
̃(K) ) =A
(A
(K)
̃ ⊗B
̃
(A
T
(K)
̃T
) =A
(K)
∈ M IFSM ,
m×n
m×n
∈ M(K) IFSM,
m×n
(K)
(K)
ij
ij
̃ij
B
and
(K)
and
(K)
and
(K)
(K)
(K)
(K)
(K)
ji
ji
ji
ji
ji
ji
(K)
(K)
= [(μǍ + μB̌ −
ij
ij
C
̃(K) ⊗B
̃(K) ) =
(ii) (A
̃C (K)
̃C (K) ⊗B
A
⊆
(K)
(K)
ij
ij
ij
̃ij
B
(K)
(K)
ij
̃ij
B
(K)
(K)
= [(μǍ + μB̌ −
ij
ij
C
)]
(K)
(K)
(K)
(K)
ij
ij
ij
ij
, μǍ + μB̌ − μǍ . μB̌ )] ----(1)
̃C
For all i,j and K, A
[(ν(K)
̃ij
B
C
(K)
(K) (K)
̃C (K) =
= [( νǍ , μǍ )] and B
ij
ij
(K)
, μB̌ )]
ij
̃C (K) = [( ν(K) . ν(K) , μ(K) + μ(K) − μ(K) . μ(K) )] ----(2)
̃C (K) ⊗B
A
̌
̌
̌
̌
̌
A
A
B
A
B
ij
̃ij
B
(K)
̃ ⊕B
̃
From (1) and (2) , (A
ij
(K)
C
̃C
) =A
ij
(K)
ij
̃C
⊗B
(K)
(K)
ji
(2)
ji
ji
T
(K)
̃T
̃(K) ⊕B
̃ (K) ) = A
From (1) and (2), (A
Similarly we can prove (ii).
ij
ij
≥
(K)
(K)
ij
̃ij
B
≤ νǍ + ν
̃ij
B
(K) (K)
μǍ . μB̌
ij
ij
(K)
(K)
ij
̃ij
B
− νǍ . ν
̃ij
B
ij
ij
(K)
(K)
ij
ij
and μǍ + μB̌ −
In a Similar way result (iv) can be proved.
ji
ji
ji
̃T (K) .
⊕B
̃(K) ⊆B
̃(K) . Therefore for all i,j and K we have
(iii). Let A
(K)
(K)
(K)
(K)
μǍ ≤ μB̌ and νǍ ≤ νB̃
(K)
(K)
Since, νǍ . ν
ij
̃ij
B
C
̃C (K)
̃C (K) ⊕B
̃(K) ⊕B
̃(K) ) ⊆A
From (1) and (3) we have, (A
̃T (K) ⊕B
̃T (K)= [(μ(K) + μ(K) − μ(K) . ν(K) , ν(K) . ν(K) )] ----Now, A
̃
̃
̌
̌
̌
̌
B
B
A
A
B
A
ji
∈ M(K) IFSM . Then
m×n
̃C (K)
ji
ji
ij
(K)
(3)
T
̃T (K) = [(μ(K) , ν(K) )], B
̃T (K) = [(μ(K) , ν(K) )]
A
̃
̌
̌
̌
B
A
A
B
ji
∈ M(K) IFSM and
m×n
̃(K) ⊕B
̃(K) )
and K, (A
(K)
(K) (K)
μǍ . μB̌
ij
ij
ji
ij
⊕C̃ .
C
ij
(K)
(K)
ij
ij
̃C (K) = [ ν(K) + ν(K) − ν(K) . ν(K) , μ(K) . μ(K) ] ---̃C (K) ⊕B
Now, A
̌
̌
̌
̌
A
A
A
B
)]
(K)
ij
(K)
Similarly we can prove (ii).
T
(K)
ij
ij
ij
= [(μǍ + μB̌ − μǍ . νB̃ , νǍ . νB̃ )] -----(1)
ji
ij
̃C (K)
̃C (K) ⊕B
A
= [( νǍ . ν
T
̃(K) ⊕B
̃(K) )
For all i,j and K, (A
(K)
ij
ij
(K)
(K)
= [(μB̌ , νB̃ )]
μǍ . μB̌ , νǍ . ν
∈ M(K) IFSM and
̃(K) ⊕B
̃(K) ) = (B
̃(K) ⊕A
̃ )
(A
ij
̃
⊆B
(K)
C
̃C (K) ⊗B
̃(K) ⊕B
̃(K) ) =A
(A
For all i,j
(K) T
(K)
m×n
̃(K) ⊗B
̃(K) )
(A
̃(K) ⊕C̃ (K) ⊆B
̃(K) ⊕C̃ (K)
then A
T
μǍ . μB̌ , νǍ . ν
(K)
C
̃C (K) (iv)
̃C (K) ⊕B
̃(K) ⊕B
̃(K) ) ⊆A
(iii) (A
(K)
̃(K) ⊗A
̃(K)
B
̃T
⊗B
̃(K) ⊆B
̃(K)
(iii)
If A
̃(K) ⊗C̃ (K) ⊆B
̃ (K) ⊗C̃ (K)
A
Proof
∈
(K)
T
(K)
m×n
̃(K) ⊕A
̃(K) )
(B
̃T
⊕B
ij
ij
ij
Proof
̃(K) ⊗B
̃(K) ) =
(A
(ii)
̃(K) =[bij (K) ]
B
̃T (K) ⊕B
̃T (K)
) ≠A
̃(K) ⊕B
̃(K) ) =
(A
(i)
ij
m×n
(K) T
m×n
ij
Proposition 3.22
(K)
(K)
̃ (K) =[a ij (K) ]
Let A
= [(μǍ , νǍ )]
(K)
̃ (K)
]
∈ M(K) IFSM and B
m×n =[bij
m×n
Proposition 3.21
(K)
(K)
̃(K) =[a ij (K) ]
Let A
= [(μǍ , νǍ )]
̃(K) =[bij (K) ]
B
ij
ij
(i)
̃
⊗B
ij
(K)
̃(K) ⊕C̃ (K) = [(μ(K) + μ(K) − μ(K) . μ(K) , ν(K)
Also, B
̃ . νC̃ )]
̌
̌
B
B
Č
B
Č
In the following Proposition, we prove that the
Complement based on the operators
and follows DeMorgan’s Laws.
(K)
̃ ⊕B
̃
=A
̃(K) ⊕B
̃(K) is Symmetric.
Therefore A
̃(K) ⊗ B
̃(K) is Symmetric.
Similarly we can prove A
M(K) IFSM . Then
ij
(K)
(K)
(K)
(K) (K)
(K)
(K)
⊕C̃ = [(μǍ + μČ − μǍ . μČ , νǍ . νC̃ )]
̃ ⊕C̃
From (3) we have, A
)]
(K) (K)
μǍ . μB̌
ij
ij
Remark 3.20
(K) ]
̃(K)
Let A
m×n =[a ij
(K)
ij
(K)
T
− μǍ . νB̃ , νǍ . νB̃ )]
−
(K)
ij
̃
Now, A
(K)
(K)
(K)
ij
(K)
Also μB̌ = μB̌ and νB̃ = νB̃ .
μǍ . μB̌ , νǍ . ν
(K)
and νB̃ . νC̃ ≥ νB̃ . νC̃ .
ij
ij
(K)
(K)
(K)
(K)
ij
ij
ij
ij
Conclusion:
In this paper, we have defined some new operators
namely □, ◊ on a new type of matrix namely Multi
Intuitionistic Fuzzy soft Matrix were defined and some of
their properties are studied. Also we have defined another
two new operators namely
, ⊗ on Multi Intuitionistic
Fuzzy soft Matrix and we have studied some of their
properties. The concepts are illustrated with suitable
numerical examples.
⇒ μǍ + μČ − μǍ . μČ ≤ μB̌ + μČ − μB̌ . μČ ----(3)
@ IJTSRD
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ij
Unique Paper ID – IJTSRD33009
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Volume – 4 | Issue – 5
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Reference
[1] Atanassov, K., Intuitionistc fuzzy sets, Fuzzy Sets and
Systems, 20(1) (1986), 87-96.
[6] Gorzalczany, M. B., A method of inference in
approximate reasoning based on interval valued
fuzzy sets, Fuzzy sets and systems 21(1987), 1-7.
[2] Atanassov, K., Intuitionistic fuzzy sets, PhysicaVerlag, Heidelberg, Newyork, 1999.
[7] Maji, P. K., Biswas, R. and Roy, A. R., Fuzzy Soft sets,
Journal of Fuzzy Mathematics, 9(3)(2001), 677-691.
[3] Baruah H. K., Towards Forming a Field of Fuzzy Sets.
International Journal of Energy, Information and
Communications, Vol.2, Issue 1(2011) 16-20.
[8] Molodtsov, D., Soft set theory-first result, Computers
and Mathematics with Applications, 37(1999), 19-31.
[4] Baruah H. K., The Theory of Fuzzy Sets. Beliefs and
Realities.
International Journal of
Energy,
Information and Communications, Vol.2, Issue
2(2011) 1-22.
[5]


Cagman , N. Enginoglu , S ., Soft matrix theory
and its decision making, Computers and Mathematics
with Applications, 59(2010), 3308-3314.
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[9] Sujit Das and Samarjit Kar Intuitionistic Multi Fuzzy
Soft Set and its Application in Decision Making
International Conference on Pattern Recognition and
Machine Intelligence (2013), LNCS 8251, 587-592.
[10] Yager R. R., On the theory of bags (Multi Sets).
International Journal of General System, 13(1986)
23-37.
[11] Zadeh, L. A., Fuzzy sets, Information and Control,
8(1965), 338-353.
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