International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
Database Oriented Validation of Theorems on
Approximations of Classifications in Optimistic
and Pessimistic Multigranulation Based Rough
Sets
R. Raghavan
SITE, VIT University, Vellore-632014, Tamil Nadu, India
Abstract— The concept granular computing was
in ability to classify objects. The concept of
introduced by Zadeh which deals with granules and
approximations of classifications was introduced and
includes all methods and tools which provide flexibility
studied by Busse [1]. In his work [1] he revealed some
and adaptability in the decision at which knowledge or
important and interesting results. After analysis it has
information is extracted and represented. According to
granular computing an equivalence relation on the
universe can be considered as a granulation, and a
partition on the universe can be considered as a
been found that not all the properties of basic rough sets
can be extended to the category of classifications. In [1]
Busse established four theorems as properties of
granulation space. Two types of multigranulation of rough
classifications and the obtained results could be used in
sets were defined from the view of granular computing.
rule generation. The unification process of the four
One is called to be the optimistic multigranular rough set
theorems of Busse was done by Tripathy et al [7, 8] in
was introduced by Qian et al [4]. During evolution there
the idea that they established two theorems of the
was only one type of multigranulation and it was not
necessary and sufficient type, from which several
named as optimistic multigranulation rough sets. After the
results including the four theorems of Busse can be
improvement of a second type of multigranulation, the
first one was called optimistic multigranulation rough set
and
the
second
one
was
called
as
pessimistic
multigranulation rough sets. For granulations of higher
derived. Also, their results confirm to the prediction of
Pawlak. From the view of information systems, the role
of these four theorems was found to be useful in
order (more than two), the definitions and properties are
deriving rules from such systems. It was observed that
similar. Using multigranular rough sets several theorems
only five cases were considered by Busse out of eleven
with respect to approximations of classifications was
cases as far as the types of classifications, on the other
introduced in [11]. In this paper the validation for the
hand, the other types of classifications reduce either
theorems on approximations of classification in [11] is
directly or indirectly to the five cases considered by
provided with a database support for both optimistic and
Busse. Another interesting aspect of the results in [7, 8]
pessimistic multigranular rough sets.
is the enumeration of possible types [2, 3, 9] of
elements in a classification, which is based upon the
Keywords— validation, approximations of classification,
types of rough sets introduced by Pawlak [3] and
rough
carried out further by Tripathy et al [7, 8]. From the
sets,
optimistic
multi-granular
rough
sets,
pessimistic multi-granular rough sets
view of granular computing the above results on
approximation of classifications were based upon single
I.
INTRODUCTION
granulation. Two types of multigranulations using
In rough set theory role of classification of universes
rough sets were recently introduced in the literature.
were considered to be more important due to its usage
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
These are termed as optimistic [4] and pessimistic [5]
R, S  R . The pessimistic multigranular lower
multigranulation.
approximation and pessimistic multigranular upper
The result of Tripathy et al [7, 8] is extended to
approximation of X with respect to R and S in U as
the multigranulation context. In [6] the study about the
types of basic rough sets was done from the view of
optimistic multigranulation rough set and from the
pessimistic multigranulation point of view was done in
R  S X  { x | [x]R  X and [x]S  X}
and R  S X  ~ ( R  S (~ X )).
III. PROPERTIES OF MULTIGRANULATIONS
[10].
A classification F = { X1 , X 2 , ..., X n } of a universe U is
n
such that X i  X j =  for i  j and  X k  U .
k 1
Some properties of multigranulations which
shall be used in this work to establish the results is
given below.
3.1
classification F was defined by Busse as
PROPERTIES
OF
OPTIMISTIC
MULTIGRANULAR ROUGH SETS
The following properties of the optimistic
multigranular rough sets were established in [4].
RF  {RX1 , RX 2 , ..., RX n }
(R + S)(X)  X  (R + S)(X)
The
approximations
(lower
and
upper)
of
and RF  {RX1, RX 2 ,...RX n }
a
.
(3.1)
(R +S)( ) =  = (R +S)( ) , 


(3.2)
For any two equivalence relations R and S
(R +S)(U) = U = (R +S)(U) 

over U, the lower and upper optimistic multigranular
(R + S)(~ X) =~ (R + S)(X)
(3.3)
rough approximations and pessimistic multigranular
(R + S)(X) = RX  SX
(3.4)
( R  S )( X )  R X  S X
(3.5)
approximations are defined in a natural manner as
R +SF={R +SX1 ,R +S X 2 ,...,R +SX n }

and
 …..1.1

R +SF={R +SX1 ,R +S X 2 ,...R +SX n } 
R*SF={R*SX1,R*SX 2 ,...,R*SX n }

and
 ……1.2

R*SF={R*SX1,R*SX 2 ,...R*SX n } 
II. DEFINITIONS
(R + S)(X) = (S + R)(X), 


(3.6)
(R + S)(X) = (S + R)(X) 

(R + S)(X  Y)  (R + S)X  (R + S)Y
(3.7)
(R + S) (XUY)  (R + S)X U (R + S)Y
(3.8)
(R + S)(XUY)  (R + S)XU (R + S)Y
(3.9)
(R + S) (X  Y)  (R + S)X  (R + S)Y
(3.10)
3.2
2.1 DEFINITION: Let K= (U, R) be a knowledge base,
R be a family of equivalence relations, X  U and
R , S  R . The optimistic multigranular lower
PROPERTIES
OF
PESSIMISTIC
MULTIGRANULAR ROUGH SETS
The following properties of the pessimistic
multigranular rough sets which are parallel to the above
said
properties
were
established
in
approximation and optimistic multigranular upper
approximation of X with respect to R and S in U as
R  S X  { x | [x]R  X or [x]S  X}
and
R  S X  ~ ( R  S (~ X )).
2.2 DEFINITION: Let K= (U, R) be a knowledge base,
R be a family of equivalence relations, X  U and
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[5].
International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
(R*S)(X)  X  (R *S)(X)
(3.11)
(i) R  S (  X )
i
  

(R*S)() =  = (R*S)(), (R *S)(U) = U = (R*S)(U)
(3.12)
(R*S)(~ X) =~ (R *S)(X)
(3.13)
(R*S)(X) = RX  SX
(3.14)
(R*S)(X) = RXUSX
(3.15)
(R*S)(X) = (S*R)(X), (R*S)(X) = (S*R)(X)
(3.16)
(R*S)(X  Y)  (R*S)X  (R*S)Y
(3.17)
(R*S)(X  Y)  (R*S)X  (R*S)Y
(3.18)
(R*S)(X  Y)  (R*S)X  (R*S)Y
(3.19)
(R*S)(X  Y)  (R*S)X  (R*S)Y
(3.20)
R  S Xj
 U.
j I C
i I
(ii) R * S(  X j )     R * SX j  U.
jI
jIC
PROOF: R + S(  X j )   
jI
x  Usuch that [x]R   X j or[x]S   X j .
jI
jI
Thus [x]R  (  X j ) =  or [x]S  (  X j ) =  .
jIC
jIC
C
C
So, [x]R  (  X j ) or [x]S  (  X j ) .
jIC
jIC
C
Hence, x  R + S(  X j ) .
C
jI
IV.THEOREMS
ON
APPROXIMATIONS
OF
CLASSIFICATIONS IN MULTIGRANULATIONS
C C
This implies that x  ( R + S (  X j ) )
j I C
.
The notation used in this section is denoted as
N n  {1, 2, 3,....n } .
So, x  R + S(  X j ).
jI C
For any I  N n , I C denotes the complement
of I in N n .
This proves that R + S ( 
THEOREM 4.1: For any I  N n
The proof of (ii) is similar.
R  S (  X i ) = U, if and only if
iI
R  S(  Xi )  U
iI
if and only if
 Xj )= 
jI C

R  S ( U \  Xi ) = 
iI

~ R  S (  Xi ) = 
iI

R  S (  Xi ) = U
iI
The proof for pessimistic case is similar.
ISSN: 2231-5381
ON
THEOREMS
OF
APPROXIMATIONS OF CLASSIFICATIONS IN
Facult
y
Name
Sam
PROOF:
THEOREM 4.2: For any I  N n
VALIDATION
The following faculty details database of an university
with assumed data is used to validate the theorems of
approximations of classifications in multigranulation
rough sets.
R  S (  X j )  .
jI C
RS (
V.
X j) U.
MULTIGRANULATIONS ROUGH SET THEORY
 Xj )=.
jI C
RS(
j I C
Ram
Div
isio
n
NW
Grade
AP
Highes
t
Degree
M.C.A
Pr
Ph.D
APJ
M.sc.,
IS
Shyam SE
Peter
AI
AS P
Ph.D
Roger
ES
P
Ph.D
Native
State
Tamil
Nadu
Andhra
Pradesh
Tamil
Nadu
Tamil
Nadu
Tamil
Nadu
,
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Page 127
International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
AI
APJ
M.Sc.,
Mishra ES
APJ
M.Sc.,
Albert
Andhra
Pradesh
Tamil
Nadu
Hari
IS
SP
Ph.D
Orissa
John
SE
AP
M.C.A
Smith
NW
ASP
Ph.D
West
Bengal
Orissa
Linz
AI
SP
Ph.D
Orissa
Keny
SE
P
Ph.D
Karnataka
ASP
Ph.D
APJ
M.Sc.,
Tamil
Nadu
Karnataka
Willia ES
ms
Martin IS
APJ
M.Sc.,
Karnataka
Lakma SE
n
Sita
AI
ASP
Ph.D
AP
M.Tech
West
Bengal
Karnataka
Fatima ES
AP
M.Tech
Muker
jee
Preeth
a
IS
SP
Ph.D
SE
SP
Ph.D
Jacob
NW
West
Bengal
West
Bengal
Karnataka
  {shyam, albert, mishra
jacob,sam, john,sita, fatima, ram, peter,
roger, hari,smith, keny, linz, williams, mukherjee,
preetha}
 / Highest Degree  {{shyam, albert, mishra
jacob},{sam, john},{sita, fatima},{ram, peter,
roger, hari, smith, keny, linz, williams, mukherjee,
preeth}}
 / Native State  {{sam,shyam, roger, mishra,
williams},{ram, peter},{hari, smith, linz},{peter,
john, lakman, fatima, mukherjee},{keny, martin,
jacob,sita, pretha}}
 / Division  {{sam,smith, jacob},{shyam, john,
keny, lakman, pretha},{peter, albert, linz, sita},
{roger, mishra, williams, fatima},{ram, hari, martin,
mukherjee}}
Let us define a classification F namely grade
from the faculty details table as given below
F   / Grade  {{shyam, albert, mishra, martin, jacob},
{sam, john,sita, fatima},{peter,smith, williams,
lakman},{linz, mukherjee, preetha, hari},
{ram, roger, keny}}
F  {X1, X 2 , X 3 , X 4 , X 5}
N n  {1, 2, 3, 4, 5}
The database which is stated above is used for the
I  N
validation. Here NW indicates networks, SE indicates
I  {1, 2, 3, 4}
Software
I C  {5}
engineering,
AI
indicates
Artificial
Intelligence, ES indicated Embedded Systems, IS
indicated Information Systems. Similarly AP indicates
Assistant Professor, APJ indicates Assistant Professor
(Junior), ASP indicates Associate Professor, SP
n
Let
Let U/Grade be the classification named F
U/ R be U / Native state
U/S be U / Division
For validating theorem 4.1
indicates Senior Professor, Pr indicates Professor.
The universe and other equivalence relations are
defined below.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
5.1 VALIDATION OF THEOREM 4.1
Validation of Theorem 4.2
OPTIMISTIC CASE
For any I  N n
For any I  N n
R  S( iI Xi )  ,if and only if
,
R  S( Xi )     R  SX j  
R  S( jIC X j )  
jI C
iI
(ii) R * S(  X j )     R * SX j  U.
jI
C
jI
Take
R  S( iI Xi )  R  S(X1  X 2  X3  X 4 )
F  {X1, X 2 , X 3 , X 4 , X 5}
~ (R  S(~ (X1  X 2  X 3  X 4 )))
N n  {1, 2, 3, 4, 5}
~ (R  S(X 5 ))  ~ ([x]R  X5 or [x]S  X 5 )
I  N
n
I  {1, 2}
Here fifth classification X 5 is {ram, roger, keny}
I C  {3, 4, 5}
Let
Let U/Grade be the classification named F
R  S( jIC X j ) R  S(X 5 )  [x]R  X 5 or [x]S  X 5 U/R be U / Highest Degree
U/ S be U / Native state
for validating theorem 4.2
 {}or{}
~ ( or )  

 {}
So R  S( iI X i )  ,if and only if
OPTIMISTIC CASE
R  S( jIC X j )  
R  S( iI Xi ) R  S(X1  X 2 )
Hence Validated
 [x]R  {X1  X 2 }or[x]S  {X1  X 2 }
 {sam, john}or{}
 {sam, john}

PESSIMISTIC CASE
To Pr ove
 R  SX
R *S( iI X i )  , if and only if
j
R  S(X3  X 4  X 5 )
jIC
R *S( jIC X j )  
~ R  S(~ (X 3  X 4  X 5 ))
Take
~ R  S(X1  X 2 )
R *S( iI X i )  R *S(X1  X 2  X 3  X 4 )
~ ({sam, john}or }
 ~ (R *S(~ (X1  X 2  X 3  X 4 )))
~ ({sam, john})

 ~ (R *S(X 5 ))  ~ ([x]R  X 5 and [x]S  X 5 )
Here fifth classification X 5 is {ram, roger, keny}
 ~ ( and )  
So R  S( X i )   
iI
R *S( jIC X j )  R *S(X 5 )  [x]R  X 5 and [x]S  X 5
 R  SX
j

jIC
Hence Validated
 {}and{}  {}
So R *S( iI X i )  ,if and only if
R *S( jIC X j )  
Hence Validated
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
PESSIMISTIC CASE
To Prove R *S( Xi )     R *SX j  
jIC
iI
Take I  {3, 4,5}
IC  {1, 2}
R *S( iI X i )  R *S(X3  X 4  X 5 )
 [x]R  {X3  X 4  X 5}and[x]S  {X 3  X 4  X 5}
 {ram, peter, roger, hari,smith, linz, keny, williams,
lakman, mukherjee, pretha}and{hari,smith, linz}
 {hari,smith, linz}  
 R *SX
j
R *S(X1  X 2 )
jIC
~ R *S(~ (X1  X 2 ))
~ R *S(X3  X 4  X 5 ) ~ ({hari, smith, linz})  
So R *S( X j )   
jI
 R *SX
j

jIC
Hence Validated
V. CONCLUSION
In this paper the validation for the theorems on
approximations of classification was successfully
validated with a database support for both optimistic
and pessimistic multigranular rough sets. This
visualizes how the internal segments of theorems had
the flow with the database used. Such practical
implementation with a database gives more
understandability to the analytical work carried in the
previous paper.
Studies in Computational Intelligence, vol.174, Rough
Set Theory: A True Landmark in Data Analysis,
Springer Verlag, (2009), pp.85 - 136.
[8] Tripathy, B.K., Ojha, J., Mohanty, D. and Prakash
Kumar, Ch. M.S: On rough definability and types of
approximation of classifications, vol.35, no.3, (2010),
pp.197-215.
[9] Tripathy, B.K. and Mitra, A.: Topological properties
of rough sets and their applications, International
Journal of Granular Computing, Rough Sets and
Intelligent Systems (IJGCRSIS), (Switzerland),vol.1,
no.4, (2010),pp.355-369 .
[10] Tripathy, B.K. and Nagaraju, M.: On Some
Topological Properties of Pessimistic Multigranular
Rough Sets, International Journal of Intelligent Systems
and Applications, Vol.4, No.8, (2012), pp.10-17.
[11] Tripathy, B.K. and Raghavan, R.: Some Algebraic
properties of Multigranulations and an Analysis of
Multigranular Approximations of Classifications,
International Journal of Information Technology and
Computer Science, Volume 7, 2013, pp. 63-70.
VII.BIOGRAPHY
R. Raghavan is an Assistant
Professor (Senior) in the School
of Information Technology and
Engineering
(SITE),
VIT
University at Vellore in India. He
obtained his masters in computer
applications from University of
Madras. He completed his M.S., (By Research) in
information technology from school of information
technology and engineering, VIT University. He is a
life member of CSI. His current research interest
includes Rough Sets and Systems, Knowledge
Engineering, Granular Computing, Intelligent
Systems, image processing.
VI. REFERENCES
[1] Grzymala Busse, J.: Knowledge acquisition under
uncertainty- a rough set approach, Journal of Intelligent
and Robotics systems, 1, (1988), pp. 3 -1 6.
[2] Pawlak, Z.: Rough Classifications, international
Journal of Man Machine Studies, 20, (1983), pp.469 –
483.
[3] Pawlak, Z.: Theoretical aspects of reasoning about
data, Kluwer academic publishers (London), (1991).
[4] Qian, Y.H and Liang, J.Y.: Rough set method based
on Multi-granulations, Proceedings of
the 5th IEEE Conference on Cognitive Informatics,
vol.1, (2006),pp.297 – 304.
[5] Qian, Y.H., Liang, J.Y and Dang, C.Y.: Pessimistic
rough decision, in: Proceedings of RST 2010,
Zhoushan, China, (2010), pp. 440-449.
[6] Raghavan, R. and Tripathy, B.K.: On Some
Topological Properties of Multigranular Rough Sets,
Journal of Advances in Applied science Research,
Volume 2(3), 2011, pp.536-543.
[7] Tripathy, B.K.: On Approximation of
classifications, rough equalities and rough equivalences,
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