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International Journal of Scientific Research and Modern Education (IJSRME)
Impact Factor: 6.225, ISSN (Online): 2455 – 5630
(www.rdmodernresearch.com) Volume 2, Issue 1, 2017
122
OPERATIONS ON P-ORDER AND R-ORDER OF INTERNAL
CUBIC SOFT MATRICES
V. Chinnadurai* & S. Barkavi** Department of Mathematics, Annamalai University, Annamalai Nagar,
Chidambaram, Tamilnadu
Cite This Article: V. Chinnadurai & S. Barkavi, “Operations on P-Order and R-Order of
Internal Cubic Soft Matrices”, International Journal of Scientific Research and Modern Education, Volume 2,
Issue 1, Page Number 122-145, 2017.
Copy Right: © IJSRME, 2017 (All Rights Reserved). This is an Open Access Article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract: In this paper, we introduce the notion of P-order (R-order) of internal cubic soft matrices. Further, we
develop union and intersection of P-order(R-order) of internal cubic soft matrices and their related properties have
been investigated.
Key Words: Cubic Soft Sets, Cubic Soft Matrices, P-Order Internal Cubic Soft Matrices & R-Order Internal
Cubic Soft Matrices.
1. Introduction:
Fuzzy set theory was proposed by Lotifi A. Zadeh [12] and it has extensive applications in various fields.
In 1999, Molodstov[8] introduced the novel concept of soft sets and established the fundamental results of the new
theory. In 2003, Maji et al.[6] studied some properties of soft sets. Pei and Miao [10] and Chen [1]et al.,
improved the work of Maji et al. [5, 6].In [3], Jun et al., introduced a new notion of cubic set which is a
combination of fuzzy set and interval-valued fuzzy set and also investigated several properties of cubic sets.
Muhiuddin and Al-roqi [9], introduced the concepts of internal, external cubic soft sets, P-cubic
(R-cubic) soft subsets, R-union(R-intersection, P-union and P-intersection) of cubic soft sets and the complement
of a cubic soft sets. They also investigated several related properties and applied the notion of cubic soft sets to
BCK/BCI-algebras.
Fuzzy matrix was introduced by Thomason [11] and the concept of uncertainty was discussed by using
fuzzy matrices. Different concepts and ideas of fuzzy matrices have been given earlier mainly by Kim,
Meenakshi and Thomson [4, 7, 11]. Fuzzy matrix plays a vital role in fuzzy modeling, fuzzy diagnosis and fuzzy
controls. It also has applications in fields like psychology, medicine, economics and sociology.
In the early investigation, Chinnadurai and Barkavi introduced a new concepts of cubic soft matrix,
internal cubic soft matrix, external cubic soft matrix and discussed various result in[2].
In this paper, the new definition of P-order internal cubic soft matrices and R-order internal cubic soft
matrices have been introduced, along with some algebraic properties are discussed with suitable illustrations.
2. Preliminiaries:
In this section we recall some basic definitions and results which will be needed in the sequel.
Definition 2.1 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set over U is
defined to be a pair ),( A where is a mapping from A to )(UP and .EA Then the pair ),( A
can be represented as, AeeA )/(=),( where AeUuuuAue ee ,/)(),(~
,=)( is a
cubic soft set in which )(~
uAe is the interval valued fuzzy set and )(ue is a fuzzy set.
Definition 2.2 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set ),( A over
U is said to be an internal cubic soft set if )()()( uAuuA eee
for all Ae and for all .Uu
Definition 2.3 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set ),( A over
U is said to be an external cubic soft set if ))(),(()( uAuAu eee
for all Ae and for all .Uu
Definition 2.4 [9] Let U be an initial universal set and E be a set of parameters. For any subsets A and B of E,
),( A and ),( B be cubic soft sets over .U
1. The R-union of ),( A and ),( B is a cubic soft set ),( C where BAC = and
BAeee
ABee
BAee
e
R if)()(
,/ if)(
,/ if)(
=)
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for all Ce and it is denoted by ).,(),(=),( BAC R
2. The R-intersection of ),( A and ),( B is a cubic soft set ),( C where BAC = and
BAeee
ABee
BAee
e
R if)()(
,/ if)(
,/ if)(
=)
for all Ce and it is denoted by ).,(),(=),( BAC R
Definition 2.5 [9] Let U be an initial universal set and E be a set of parameters. For any subsets A and B of E,
),( A and ),( B be cubic soft sets over .U
1. The P-union of ),( A and ),( B is a cubic soft set ),( C where BAC = and
BAeee
ABee
BAee
e
P if)()(
,/ if)(
,/ if)(
=)
for all Ce . This is denoted by ).,(),(=),( BAC P
2. The P-intersection of ),( A and ),( B is a cubic soft set ),( C where BAC = and
BAeee
ABee
BAee
e
P if)()(
,/ if)(
,/ if)(
=)
for all Ce . This is denoted by ).,(),(=),( BAC P
Definition 2.6 [9] Let U be an initial universal set and E be a set of parameters. The complement of a cubic soft
set ),( A over U is denoted by cA),( and defined by
),(=),( AA c where )(: UPAc and AeeA cc )/(=),( where
.,/)(),(~
,=)( AeUuuuAue c
e
c
e
c
Definition 2.7 [7] A matrix nmijaA ][= is said to be fuzzy matrix if miaij [0,1],1 and nj 1 .
Definition 2.8 [7] For any two fuzzy matrices ][=],[= ijij bBaA and a scalar Fk . Then,
(i) ],[=],[= bijabasupBA ijijij .
(ii) .],[=],[= ijijijij babainfsupAB
(iii) ].,[=],[= ijij akakinfkA
Definition 2.9 [2] Let muuuU ,...,,= 21 be an initial universal set and neeeE ,...,,= 21 be a set of
parameters. Let .EA Then cubic soft set ),( A can be expressed in matrix form as
][= ijaA=
mnmm
n
n
aaa
aaa
aaa
21
22221
11211
such that a
ij
a
ijij
eij
eij AuuAaA ,~
=)(),(~
=][= which is called an nm cubic soft matrix (shortly
CS-matrix or CSM) of the cubic soft set ).,( A
According to this definition, a cubic soft set ),( A is uniquely characterized by matrix nmija ][ where
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mi 1,2,3,...,= and .1,2,3,...,= nj
Example 2.10 Let 4321 ,,,= uuuuU is a set of cars and 321 ,,= eeeA is a set of parameters, which
stands for mileage, engine and prize respectively. Then cubic soft set is defined as
}.)0.4[0.3,0.7],,(),0.5[0.2,0.9],,(),0.9[0.6,0.7],,(),0.4[0.1,0.8],,(,
,)0.1[0.3,0.5],,(),0.2[0.2,0.7],,(),0.7[0.3,0.6],,(),0.3[0.2,0.5],,(,
,)0.4[0.3,0.9],,(),0.9[0.2,0.6],,(),0.5[0.1,0.7],,(),0.6[0.5,0.8],,(,{=),(
43213
43212
43211
uuuue
uuuue
uuuueA
Then the CS-matrix A is written as,
A =
0.4[0.3,0.7],0.1[0.3,0.5],0.4[0.3,0.9],
0.5[0.2,0.9],0.2[0.2,0.7],0.7[0.2,0.6],
0.9[0.6,0.7],0.7[0.3,0.6],0.5[0.1,0.7],
0.4[0.1,0.8],0.3[0.2,0.5],0.6[0.5,0.8],
.
Definition 2.11 [2] Let nmij CSMaA ][=. Then
A is an internal cubic soft matrix (ICSM), if
a
ij
a
ij
a
ij AA~~
for all i, j.
Definition 2.12 [2] Let nmijaA ][=, nmijbB ][=
be the two cubic soft matrix of order .nm Then
P-order matrix is denoted and defined as ][][ ijPij ba , if b
ij
a
ij BA~~
and b
ij
a
ij for all i, j.
Definition 2.13 [2] Let nmijaA ][=, nmijbB ][=
be the two cubic soft matrix of order .nm Then
R-order matrix is denoted and defined as ][][ ijRij ba , if b
ij
a
ij BA~~
and b
ij
a
ij for all i, j.
Definition 2.14 [2] Let .],~
,~
[=,],~
,~
[= nm
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij CSMBBBAAA
Then
1. P-union of A and
B is denoted by BA P and defined as
CBA P = , if
c
ij
c
ijij CcC ,~
=][=, where b
ij
a
ij
c
ij BAC~
,~
max=~
and b
ij
a
ij
c
ij ,max= for all i, j.
2. P-intersection of A and
B is denoted by BA P and defined as
CBA P = , if
c
ij
c
ijij CcC ,~
=][=, where b
ij
a
ij
c
ij BAC~
,~
min=~
and b
ij
a
ij
c
ij ,min= for all i, j.
3. R-union of A and
B is denoted by BA R and defined as
CBA R = , if
c
ij
c
ijij CcC ,~
=][=, where b
ij
a
ij
c
ij BAC~
,~
max=~
and b
ij
a
ij
c
ij ,min= for all i, j.
4. R-intersection of A and
B is denoted by BA R and defined as
CBA R = , if
c
ij
c
ijij CcC ,~
=][=, where b
ij
a
ij
c
ij BAC~
,~
min=~
and b
ij
a
ij
c
ij ,max= for all i, j.
3. P-order and R-order of Internal Cubic Soft Matrices:
In this section we define P-order and R-order of internal cubic soft matrix and discussed some algebraic
properties.
Definition 3.1 Let a
ij
a
ijAA ,~
= and
b
ij
b
ijBB ,~
= be internal cubic soft matrices, if
b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j. Then it is defined as P-order internal cubic soft matrices and
it is denoted by .PICSM
Example 3.2 Let 21,= uuU be the universal set and 4321 ,,,= eeeeE be the set of parameters. Let
., EBA Then
})0.35[0.3,0.8],,(),0.6[0.4,0.9],,(,
,)0.4[0.3,0.7],,(),0.5[0.2,0.8],,(,{=),(
212
211
uue
uueA
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and
}.)0.75[0.7,0.9],,(),8[0.6,1],0.,(,
,)0.6[0.5,0.8],,(),0.7[0.4,0.9],,(,{=),(
212
211
uue
uueB
Then P-order internal cubic soft matrices is defined as follows,
A = BP
0.5[0.3,0.8],0.4[0.3,0.7],
0.6[0.4,0.9],0.5[0.2,0.8],=
0.75[0.7,0.9],0.6[0.5,0.8],
8[0.6,1],0.0.7[0.4,0.9],.
(i.e) . BA P
Definition 3.3 Let a
ij
a
ijAA ,~
= and
b
ij
b
ijBB ,~
= be internal cubic soft matrices, if
b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j. Then it is defined as R-order internal cubic soft matrices and
it is denoted by .RICSM
Example 3.4 Let 21,= uuU be the universal set and 4321 ,,,= eeeeE be the set of parameters. Let
., EBA Then
})0.4[0.2,0.4],,(),0.7[0.4,0.8],,(,
,)0.5[0.1,0.6],,(),0.3[0.2,0.4],,(,{=),(
212
211
uue
uueA
and
}.)0.35[0.3,0.8],,(),6[0.5,1],0.,(,
,)0.4[0.3,0.9],,(),0.3[0.3,0.7],,(,{=),(
212
211
uue
uueB
Then R-order internal cubic soft matrices is defined as follows,
A = BR
0.4[0.2,0.4],0.5[0.1,0.6],
0.7[0.4,0.8],0.3[0.2,0.4],= .
0.35[0.3,0.8],0.4[0.3,0.9],
6[0.5,1],0.0.3[0.3,0.7],
(i.e) . BA R
Definition 3.5 Let ., P
nmICSMBA Then
a) Union of BA , is defined by ],[== ijP cCBA where
},{max}],~
,~
{max},~
,~
{max[= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABAC
for all i,j.
b) Intersection of BA , is defined by ],[== ijP cCBA where
},{min}],~
,~
{min},~
,~
{min[= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABAC
for all i,j.
Example 3.6 Consider Example 3.2. Then union and intersection of BA , are given by,
a) BA P = .
0.75[0.7,0.9],0.6[0.5,0.8],
8[0.6,1],0.0.7[0.4,0.9],
b) BA P = .
0.35[0.3,0.8],0.4[0.3,0.7],
0.6[0.4,0.9],0.5[0.2,0.8],
Definition 3.7 Let ., R
nmICSMBA Then
a) Union of BA , is defined by ],[== ijR cCBA where
},{min}],~
,~
{max},~
,~
{max[= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABAC
for all i,j.
b) Intersection of BA , is defined by ],[== ijR cCBA where
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},{max}],~
,~
{min},~
,~
{min[= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABAC
for all i,j.
Example 3.8 Consider Example 3.4. Then union and intersection of BA , are given by,
a) BA R = .
0.35[0.3,0.8],0.4[0.3,0.9],
6[0.5,1],0.0.3[0.3,0.7],
b) BA R = .
0.4[0.2,0.4],0.5[0.1,0.6],
0.7[0.4,0.8],0.3[0.2,0.4],
Proposition 3.9 Let .,, P
nmICSMCBA Then
(i) .= AAA P
(ii) .= BBA P
(iii) .== CCBACBA PPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then (i) a
ij
a
ij
a
ijP
a
ij
a
ij
a
ijP AAAAAA ],~
,~
[],~
,~
[=
for all i, j.
Since a
ij
a
ijP AAAA~~
and
a
ij
a
ij for all i, j.
By Definition 3.5(a),
(ii) b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijP BBAABA ],~
,~
[],~
,~
[=
for all i, j.
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
B
B
BB
BABABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijP
(iii) .],~
,~
[],~
,~
[= CBBAACBA P
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijPP
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
},{max}],~
,~
{max},~
,~
{max[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ij
P
b
ij
b
ij
b
ij
P
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijPP
CCBB
CBB
CBABACBA
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.5(a),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
C
C
CC
CBCBCBA
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijPP
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Similarly, .= CCBA PP
Hence, .== CCBACBA PPPP
Proposition 3.10 Let .,, R
nmICSMCBA Then
(i) .= AAA R
(ii) .= BBA R
(iii) .== CCBACBA RRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then (i) a
ij
a
ij
a
ijR
a
ij
a
ij
a
ijR AAAAAA ],~
,~
[],~
,~
[=
for all i, j.
Since a
ij
a
ijR AAAA~~
and
a
ij
a
ij for all i, j.
By Definition 3.7(a),
.=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
A
AA
AAAAAA
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ijR
(ii) b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijR BBAABA ],~
,~
[],~
,~
[=
for all i, j.
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
B
B
BB
BABABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijR
(iii) .],~
,~
[],~
,~
[= CBBAACBA R
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijRR
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
},{min}],~
,~
{max},~
,~
{max[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ij
R
b
ij
b
ij
b
ij
R
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijRR
CCBB
CBB
CBABACBA
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.7(a),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
C
C
CC
CBCBCBA
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijRR
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Similarly, .= CCBA RR
Hence, .== CCBACBA RRRR
Proposition 3.11 Let .,, P
nmICSMCBA Then
(i) .= AAA P
(ii) .= ABA P
(iii) .== ACBACBA PPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then (i) a
ij
a
ij
a
ijP
a
ij
a
ij
a
ijP AAAAAA ],~
,~
[],~
,~
[=
for all i, j.
Since a
ij
a
ijP AAAA~~
and
a
ij
a
ij for all i, j.
By Definition 3.5(b),
.=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
A
AA
AAAAAA
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ijP
(ii) b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijP BBAABA ],~
,~
[],~
,~
[=
for all i, j.
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
A
A
AA
BABABA
a
ij
a
ij
a
ij
a
ij
a
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijP
(iii) .],~
,~
[],~
,~
[= CBBAACBA P
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijPP
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ij
P
a
ij
a
ij
a
ij
P
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijPP
CCAA
CAA
CBABACBA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
Again by Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
A
A
AA
CACACBA
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijPP
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129
Similarly, .= ACBA PP
Hence, .== ACBACBA PPPP
Proposition 3.12 Let .,, R
nmICSMCBA Then
(i) .= AAA R
(ii) .= ABA R
(iii) .== ACBACBA RRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then (i) a
ij
a
ij
a
ijR
a
ij
a
ij
a
ijR AAAAAA ],~
,~
[],~
,~
[=
for all i, j.
Since a
ij
a
ijR AAAA~~
and
a
ij
a
ij for all i, j.
By Definition 3.7(b),
.=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
A
AA
AAAAAA
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ijR
(ii) b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijR BBAABA ],~
,~
[],~
,~
[=
for all i, j.
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(b),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
A
A
AA
BABABA
a
ij
a
ij
a
ij
a
ij
a
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijR
(iii) .],~
,~
[],~
,~
[= CBBAACBA R
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijRR
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(max)],~
,~
(min),~
,~
(min[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ij
R
a
ij
a
ij
a
ij
R
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijRR
CCAA
CAA
CBABACBA
Since, c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
Again by Definition 3.7(b),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
A
A
AA
CACACBA
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijRR
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130
Similarly, .= ACBA RR
Hence, .== ACBACBA RRRR
Theorem 3.13 Let .,...,,, 321
P
nmk ICSMAAAA Since ....321
kPPPP AAAA
Then .=...321
kkPPPP AAAAA
Proof. Let
.],~
,~
[=
,...,],~
,~
[= ,],~
,~
[=,],~
,~
[= 333322221111
P
nm
a
ijk
a
ijk
a
ijkk
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
ICSMAAA
AAAAAAAAA
Then, ....=... 321321
kPPPPkPPPP AAAAAAAA
Since, a
ij
a
ijP AAAA 2121
~~
and a
ij
a
ij21 for all i, j.
By Definition 3.5(a),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{max}],~
,~
{max},~
,~
{max[=
4333222
3222
3212121
kPPP
a
ij
a
ij
a
ijP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Since, a
ij
a
ijP AAAA 3232
~~
and a
ij
a
ij32 for all i, j.
Again by Definition 3.5(a),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{max}],~
,~
{max},~
,~
{max[=
5444333
4333
3323232
kPPP
a
ij
a
ij
a
ijP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Continuing the same process we get,
.],~
,~
[],~
,~
[= 1)(1)(1)(
a
ijk
a
ijk
a
ijkP
a
ijk
a
ijk
a
ijk AAAA
Since, a
ijk
a
ijkkPk AAAA
~~1)(1)(
and
a
ijk
a
ijk 1)( for all i, j.
Again by the Definition 3.5(a),
.=
.,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[= 1)(1)(1)(
k
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
A
A
AA
AAAA
Theorem 3.14 Let .,...,,, 321
R
nmk ICSMAAAA Since ....321
kRRRR AAAA
Then .=...321
kkRRRR AAAAA
Proof. Let
.],~
,~
[=
,...,],~
,~
[= ,],~
,~
[=,],~
,~
[= 333322221111
R
nm
a
ijk
a
ijk
a
ijkk
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
ICSMAAA
AAAAAAAAA
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Then, ....=... 321321
kRRRRkRRRR AAAAAAAA
Since, a
ij
a
ijR AAAA 2121
~~
and a
ij
a
ij21 for all i, j.
By Definition 3.7(a),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{min}],~
,~
{max},~
,~
{max[=
4333222
3222
3212121
kRRR
a
ij
a
ij
a
ijR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Since, a
ij
a
ijR AAAA 3232
~~
and a
ij
a
ij32 for all i, j.
Again by Definition 3.7(a),
....],~
,~
[],~
,~
[=
...],~
,~
[=
jforalli,...},{min}],~
,~
{max},~
,~
{max[=
5444333
4333
3323232
kRRR
a
ij
a
ij
a
ijR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Continuing the same process we get,
.],~
,~
[],~
,~
[= 1)(1)(1)(
a
ijk
a
ijk
a
ijkR
a
ijk
a
ijk
a
ijk AAAA
Since, a
ijk
a
ijkkRk AAAA
~~1)(1)(
and
a
ijk
a
ijk 1)( for all i, j.
Again by Definition 3.7(a),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[= 1)(1)(1)(
k
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
a
ijk
A
A
AA
AAAA
Theorem 3.15 Let .,...,,, 321
P
nmk ICSMAAAA Since ....321
kPPPP AAAA
Then .=... 1321
AAAAA kPPPP
Proof. Let
.],~
,~
[=
,...,],~
,~
[= ,],~
,~
[=,],~
,~
[= 333322221111
P
nm
a
ijk
a
ijk
a
ijkk
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
ICSMAAA
AAAAAAAAA
Then, ....=... 321321
kPPPPkPPPP AAAAAAAA
Since, a
ij
a
ijP AAAA 2121
~~
and a
ij
a
ij21 for all i, j.
By Definition 3.5(b),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{min}],~
,~
{min},~
,~
{min[=
4333111
3111
3212121
kPPP
a
ij
a
ij
a
ijP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
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132
Since, a
ij
a
ijP AAAA 3131
~~
and a
ij
a
ij31 for all i, j.
Again by Definition 3.5(b),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{min}],~
,~
{min},~
,~
{min[=
5444111
4111
3313131
kPPP
a
ij
a
ij
a
ijP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
kPPP
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Continuing the same process we get,
.],~
,~
[],~
,~
[= 111
a
ijk
a
ijk
a
ijkP
a
ij
a
ij
a
ijAAAA
Since, a
ijk
a
ijkP AAAA
~~11
and a
ijk
a
ij 1 for all i, j.
Again by Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
1
11
111
111
A
A
AA
AAAA
a
ij
a
ij
a
ij
a
ij
a
ij
a
ijk
a
ij
a
ijk
a
ij
a
ijk
a
ij
Theorem 3.16 Let .,...,,, 321
R
nmk ICSMAAAA Since ....321
kRRRR AAAA
Then .=... 1321
AAAAA kRRRR
Proof. Let
.],~
,~
[=
,...,],~
,~
[= ,],~
,~
[=,],~
,~
[= 333322221111
R
nm
a
ijk
a
ijk
a
ijkk
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
ICSMAAA
AAAAAAAAA
Then ....=... 321321
kRRRRkRRRR AAAAAAAA
Since, a
ij
a
ijR AAAA 2121
~~
and a
ij
a
ij21 for all i, j.
By Definition 3.7(b),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{max}],~
,~
{min},~
,~
{min[=
4333111
3111
3212121
kRRR
a
ij
a
ij
a
ijR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Since, a
ij
a
ijR AAAA 3131
~~
and a
ij
a
ij31 for all i, j.
Again by Definition 3.7(b),
....],~
,~
[],~
,~
[=
...],~
,~
[=
ji, allfor ...},{max}],~
,~
{min},~
,~
{min[=
5444111
4111
3313131
kRRR
a
ij
a
ij
a
ijR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
kRRR
a
ij
a
ij
a
ij
a
ij
a
ij
a
ij
AAAAAA
AAAA
AAAAAA
Continuing the same process we get,
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133
.],~
,~
[],~
,~
[= 111
a
ijk
a
ijk
a
ijkR
a
ij
a
ij
a
ijAAAA
Since, a
ijk
a
ijkR AAAA
~~11
and a
ijk
a
ij 1 for all i, j.
Again by Definition 3.7(b),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
1
11
111
111
A
A
AA
AAAA
a
ij
a
ij
a
ij
a
ij
a
ij
a
ijk
a
ij
a
ijk
a
ij
a
ijk
a
ij
Theorem 3.17 Let ,,, P
nmICSMCBA then
.== BCBCACBA PPPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[= CBBAACBA P
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijPP
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(max)],~
,~
(max),~
,~
(max[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ij
P
b
ij
b
ij
b
ij
P
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijPP
CCBB
CBB
CBABACBA
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
B
B
BB
CBCB
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ij
R.H.S= . CBCA PPP
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ijP CCAACA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.5(b),
.,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijP
A
AA
CACACA
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134
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ijP CCBBCB
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
Again by Definition 3.5(b),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
BA
BACBCA
B
BB
CBCBCB
P
b
ij
b
ijP
a
ij
a
ijPPP
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijP
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.==Hence,
.=
],~
,~
[=
],~
,~
[=
},{max}],~
,~
{max},~
,~
{max[=
BCBCACBA
B
BB
BB
BABACBCA
PPPPP
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijPPP
Theorem 3.18 Let ,,, R
nmICSMCBA then
.== BCBCACBA RRRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[= CBBAACBA R
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijRR
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
},{min}],~
,~
{max},~
,~
{max[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ij
R
b
ij
b
ij
b
ij
R
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijRR
CCBB
CBB
CBABACBA
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.7(b),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
B
B
BB
CBCB
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ij
R.H.S= . CBCA RRR
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135
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ijR CCAACA
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.7(b),
.,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijR
A
AA
CACACA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ijR CCBBCB
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
Again by Definition 3.7(b),
.=
,~
,~
=
.,~
=
],~
,~
[=
},{max}],~
,~
{min},~
,~
{min[=
BA
BACBCA
B
BB
CBCBCB
R
b
ij
b
ijR
a
ij
a
ijRRR
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijR
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.==Hence,
.=
],~
,~
[=
],~
,~
[=
},{min}],~
,~
{max},~
,~
{max[=
BCBCACBA
B
BB
BB
BABACBCA
RRRRR
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijRRR
Theorem 3.19 Let ,,, P
nmICSMCBA then
.== CCBCACBA PPPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[= CBBAACBA P
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijPP
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ij
P
a
ij
a
ij
a
ij
P
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijPP
CCAA
CAA
CBABACBA
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136
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor ),(min)],~
,~
(min),~
,~
(max[=
C
C
CC
CACA
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ij
R.H.S= . CBCA PPP
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ijP CCAACA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.5(a),
.,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijP
C
CC
CACACA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ijP CCBBCB
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
Again by Definition 3.5(a),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
CC
CCCBCA
C
CC
CBCBCB
P
c
ij
c
ijP
c
ij
c
ijPPP
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijP
By Proposition 3.11(i),
.==Hence,.= CCBCACBACCBCA PPPPPPPP
Theorem 3.20 Let ,,, R
nmICSMCBA then
.== CCBCACBA RRRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[= CBBAACBA R
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijRR
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ij
R
a
ij
a
ij
a
ij
R
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijRR
CCAA
CAA
CBABACBA
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Impact Factor: 6.225, ISSN (Online): 2455 – 5630
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137
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.7(a),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
C
C
CC
CACA
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ij
R.H.S= . CBCA RRR
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ijR CCAACA
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.7(a),
.,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijR
C
CC
CACACA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ijR CCBBCB
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
Again by Definition 3.7(a),
.=
,~
,~
=
.,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
CC
CCCBCA
C
CC
CBCBCB
R
c
ij
c
ijR
c
ij
c
ijRRR
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijR
By Proposition 3.12(i), .= CCBCA RRR
.==Hence, CCBCACBA RRRRR
Theorem 3.21 Let ,,, P
nmICSMCBA then
.== BCABACBA PPPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ijPPP CCBBACBA
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.5(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(min)],~
,~
(min),~
,~
(min[=
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ij
b
ij
b
ij
b
ijP
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijPPP
BBAA
BBA
CBCBACBA
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138
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
B
B
BB
BABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij
R.H.S= . CABA PPP
Then, .],~
,~
[],~
,~
[=
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijP BBAABA
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(a),
.,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijP
B
BB
BABABA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ijP CCAACA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
Again by Definition 3.5(a),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{max},~
,~
{max[=
CB
CBCABA
C
CC
CACACA
P
c
ij
c
ijP
b
ij
b
ijPPP
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijP
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.5(b),
.=
],~
,~
[=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
B
BB
BB
CBCBCABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijPPP
Hence, .== BCABACBA PPPPP
Theorem 3.22 Let ,,, R
nmICSMCBA then
.== BCABACBA RRRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
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139
Then, L.H.S= .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ijRRR CCBBACBA
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.7(b),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(max)],~
,~
(min),~
,~
(min[=
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ij
b
ij
b
ij
b
ijR
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijRRR
BBAA
BBA
CBCBACBA
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
B
B
BB
BABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij
R.H.S= . CABA RRR
Then, .],~
,~
[],~
,~
[=
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijR BBAABA
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a),
.,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijR
B
BB
BABABA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ijR CCAACA
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j. Again by Definition 3.7(a),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{max},~
,~
{max[=
CB
CBCABA
C
CC
CACACA
R
c
ij
c
ijR
b
ij
b
ijRRR
c
ij
c
ij
c
ij
c
ij
c
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijR
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.7(b),
.=
],~
,~
[=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
B
BB
BB
CBCBCABA
b
ij
b
ij
b
ij
b
ij
b
ij
b
ij
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijRRR
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140
Hence, .== BCABACBA RRRRR
Theorem 3.23 Let ,,, P
nmICSMCBA then
.== ACABACBA PPPPP
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= P
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
b
ij
b
ij
b
ijPPP CCBBACBA
Since c
ij
b
ijP CBCB~~
and
c
ij
b
ij for all i, j. By Definition 3.5(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(max)],~
,~
(max),~
,~
(max[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ij
c
ij
c
ij
c
ijP
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijPPP
CCAA
CCA
CBCBACBA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j. By Definition 3.5(b),
.=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
A
A
AA
CACA
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ij
R.H.S= . CABA PPP
Then, .],~
,~
[],~
,~
[=
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ijP BBAABA
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j. By Definition 3.5(b),
.,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
a
ij
a
ij
a
ij
a
ij
a
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijP
A
AA
BABABA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijP
a
ij
a
ij
a
ijP CCAACA
Since c
ij
a
ijP CACA~~
and
c
ij
a
ij for all i, j.
Again by Definition 3.5(b),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{min}],~
,~
{min},~
,~
{min[=
AA
AACABA
A
AA
CACACA
P
a
ij
a
ijP
a
ij
a
ijPPP
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijP
By Proposition 3.9(i), .= ACBCA PPP
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Hence, .== ACABACBA PPPPP
Theorem 3.24 Let ,,, R
nmICSMCBA then
.== ACABACBA RRRRR
Proof. Let .],~
,~
[=,],~
,~
[=,],~
,~
[= R
nm
c
ij
c
ij
c
ij
b
ij
b
ij
b
ij
a
ij
a
ij
a
ij ICSMCCCBBBAAA
Then, L.H.S= .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
b
ij
b
ij
b
ijRRR CCBBACBA
Since c
ij
b
ijR CBCB~~
and
c
ij
b
ij for all i, j.
By Definition 3.7(a),
.],~
,~
[],~
,~
[=
],~
,~
[=
ji, allfor ),(min)],~
,~
(max),~
,~
(max[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ij
c
ij
c
ij
c
ijR
c
ij
b
ij
c
ij
b
ij
c
ij
b
ijRRR
CCAA
CCA
CBCBACBA
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
By Definition 3.7(b),
.=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
A
A
AA
CACA
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ij
R.H.S= . CABA RRR
Then, .],~
,~
[],~
,~
[=
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ijR BBAABA
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(b),
.,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
a
ij
a
ij
a
ij
a
ij
a
ij
b
ij
a
ij
b
ij
a
ij
b
ij
a
ijR
A
AA
BABABA
Then, .],~
,~
[],~
,~
[=
c
ij
c
ij
c
ijR
a
ij
a
ij
a
ijR CCAACA
Since c
ij
a
ijR CACA~~
and
c
ij
a
ij for all i, j.
Again by Definition 3.7(b),
.=
,~
,~
=
,~
=
],~
,~
[=
ji, allfor },{max}],~
,~
{min},~
,~
{min[=
AA
AACABA
A
AA
CACACA
R
a
ij
a
ijR
a
ij
a
ijRRR
a
ij
a
ij
a
ij
a
ij
a
ij
c
ij
a
ij
c
ij
a
ij
c
ij
a
ijR
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142
By Proposition3.10(i), .= ACBCA RRR
.==,Hence ACABACBA RRRRR
Theorem 3.25 Let ., P
nmICSMBA Then
(i) .=c
P
cc
P ABBA
(ii) .=c
P
cc
P ABBA
Proof. Let a
ij
a
ij
a
ij AAA ],~
,~
[=
and .],~
,~
[= b
ij
b
ij
b
ij BBB
(i) L.H.S .],~
,~
[],~
,~
[= .=c
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ij
c
P BBAABA
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By the Definition 3.5(a), c
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{max}],~
,~
{max},~
,~
{max[= for all i, j
},{max}],1~
,~
{max},1~
,~
{max[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{max}],1~
,~
{max},1~
,~
{max[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
R.H.S =c
P
cAB
ca
ij
a
ij
a
ijP
cb
ij
b
ij
b
ij AABB
],~
,~
[],~
,~
[=
a
ij
a
ij
a
ijP
b
ij
b
ij
b
ij AABB
],1~
,1~
[1],1~
,1~
[1=
.],1~
,1~
[1],1~
,1~
[1= a
ij
a
ij
a
ijP
b
ij
b
ij
b
ij AABB
Since ,c
P
cAB by Definition 3.5(b),
},1{1min}],~
,1~
{1min},~
,1~
{1min[= a
ij
b
ij
a
ij
b
ij
a
ij
b
ij ABAB
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
L.H.S = R.H.S
So .=c
P
cc
P ABBA
(ii) L.H.S .],~
,~
[],~
,~
[= =c
b
ij
b
ij
b
ijP
a
ij
a
ij
a
ij
c
P BBAABA
Since b
ij
a
ijP BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.5(b), c
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{min}],~
,~
{min},~
,~
{min[= for all i, j
},{min}],1~
,~
{min},1~
,~
{min[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
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},{min}],1~
,~
{min},1~
,~
{min[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
R.H.S =c
P
cAB
ca
ij
a
ij
a
ijP
cb
ij
b
ij
b
ij AABB
],~
,~
[],~
,~
[=
a
ij
a
ij
a
ijP
b
ij
b
ij
b
ij AABB
],1~
,1~
[1],1~
,1~
[1=
.],1~
,1~
[1],1~
,1~
[1= a
ij
a
ij
a
ijP
b
ij
b
ij
b
ij AABB
Since ,c
P
cAB by Definition 3.5(a),
},1{1max}],~
,1~
{1max},~
,1~
{1max[= a
ij
b
ij
a
ij
b
ij
a
ij
b
ij ABAB
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
L.H.S = R.H.S
So .=c
P
cc
P ABBA
Theorem 3.26 Let ., R
nmICSMBA Then
(i) .=c
R
cc
R ABBA
(ii) .=c
R
cc
R ABBA
Proof. Let a
ij
a
ij
a
ij AAA ],~
,~
[=
and .],~
,~
[= b
ij
b
ij
b
ij BBB
(i) L.H.S .],~
,~
[],~
,~
[= =c
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ij
c
R BBAABA
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(a), c
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{min}],~
,~
{max},~
,~
{max[= for all i, j
},{min}],1~
,~
{max},1~
,~
{max[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{min}],1~
,~
{max},1~
,~
{max[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
R.H.S = .c
R
cAB
ca
ij
a
ij
a
ijR
cb
ij
b
ij
b
ij AABB
],~
,~
[],~
,~
[=
a
ij
a
ij
a
ijR
b
ij
b
ij
b
ij AABB
],1~
,1~
[1],1~
,1~
[1=
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.],1~
,1~
[1],1~
,1~
[1= a
ij
a
ij
a
ijR
b
ij
b
ij
b
ij AABB
Since ,c
R
cAB by Definition 3.7(b),
},1{1max}],~
,1~
{1min},~
,1~
{1min[= a
ij
b
ij
a
ij
b
ij
a
ij
b
ij ABAB
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
L.H.S = R.H.S
So .=c
R
cc
R ABBA
(ii) L.H.S .],~
,~
[],~
,~
[= =c
b
ij
b
ij
b
ijR
a
ij
a
ij
a
ij
c
R BBAABA
Since b
ij
a
ijR BABA~~
and
b
ij
a
ij for all i, j.
By Definition 3.7(b), c
b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
},{max}],~
,~
{min},~
,~
{min[= for all i, j
),(max)],1~
,~
(min),1~
,~
(min[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
),(max)],1~
,~
(min),1~
,~
(min[1= b
ij
a
ij
b
ij
a
ij
b
ij
a
ij BABA
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
R.H.S =c
R
cAB
ca
ij
a
ij
a
ijR
cb
ij
b
ij
b
ij AABB
],~
,~
[],~
,~
[=
a
ij
a
ij
a
ijR
b
ij
b
ij
b
ij AABB
],1~
,1~
[1],1~
,1~
[1=
.],1~
,1~
[1],1~
,1~
[1= a
ij
a
ij
a
ijR
b
ij
b
ij
b
ij AABB
Since ,c
R
cAB by Definition 3.7(a),
},1{1min}],~
,1~
{1max},~
,1~
{1max[= a
ij
b
ij
a
ij
b
ij
a
ij
b
ij ABAB
))],~
,~
[= c
ij
c
ij
c
ij CC
.= C
L.H.S = R.H.S
So .=c
R
cc
R ABBA
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International Journal of Scientific Research and Modern Education (IJSRME)
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(www.rdmodernresearch.com) Volume 2, Issue 1, 2017
145
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