International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
Authentication of Algebraic Properties of
Optimistic and Pessimistic
Multi-Granular Rough Sets
R. Raghavan
SITE, VIT University, Vellore-632014, Tamil Nadu, India
Abstract— Rough set theory was introduced by
sets lead to the concept called algebraic rough sets [7].
Pawlak [1, 2] mainly to handle uncertainty and
Hence the morphological operators seem to be a good
impreciseness. The rough set theory got extended to multi-
tool for defining lower and upper approximation of
granular based rough sets by Qian, Y.H and Liang, J.Y [3,
rough sets. Moreover these operators lead to a
4]. In multi-granular rough sets, the granules were
represented through multi-equivalence relations. Since it
uses multi-equivalence relations it was highly enriched to
handle practical applications. In general, the algebraic
generalization of rough sets. In this context the concept
of algebraic properties of general rough sets got
extended in view with multigranulation rough sets.
property used to deal with elements and studies the
In general rough set theory some of the interesting
relationship among the elements. In this paper, the
properties were studied and obtained on intersection and
database based validation for algebraic property of
union of rough sets of different types [5]. This study has
multigranular
associative,
been further extended to multigranulation rough sets to
commutative properties are provided. Also this paper
rough
sets
particularly
find out the validity of other algebraic properties like
contains the database based validation for the conditions
associativity and commutativity [6].
under which the two types of multigranulations reduced to
single granulation.
II.
ALGEBRAIC PROPERTIES OF
MULTIGRANULATIONS
Index Terms— algebraic property, multi-granular rough
With the same notations as in previous Theorem, the
sets, equivalence relations, database
following algebraic associative
and commutative
properties are satisfied by ‘+’ and ‘*’ type of
I.
INTRODUCTION
multigranulation rough sets.
Algebraic properties generally deal with elements and
study
of
the
relationship
of
elements.
Lower
approximations of rough sets and upper approximations
R  S X  S  RX and R  S X  S  RX
( R  S )  T X  R  ( S  T )X
and
of rough sets are having their similarity with
mathematical morphology in the operators used. Lower
approximation involve subset operation as like erosion
( R  S )  T X  R  ( S  T )X
R  S X  S  RX and R  SX  S  RX
and opening, while upper approximations involve set
intersection as do dilation and closing [7]. Applying
( R  S )  T X  R  ( S  T )X
and
algebraic erosion, dilation, opening and closing for
defining the corresponding approximations of rough
ISSN: 2231-5381
( R  S )  T X  R  ( S  T )X
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
III.VALIDATION ON ASSOCIATIVE AND
COMMUTATIVE PROPERTY ON OPTIMISTIC
MULTI-GRANULAR ROUGH SETS
The following database is used to validate the
associative and commutative property of
optimistic multigranulation rough sets.
Facult
y
Name
Sam
Ram
Div
isio
n
NW
IS
Shyam SE
Grade
AP
Highes
t
Degree
M.C.A
Pr
Ph.D
APJ
M.sc.,
Fatima ES
AP
M.Tech
Muker
jee
Preeth
a
IS
SP
Ph.D
SE
SP
Ph.D
West
Bengal
West
Bengal
Karnataka
The database which is stated above is used for the
Native
State
validation. Here NW indicates networks, SE indicates
Software
engineering,
AI
indicates
Artificial
Intelligence, ES indicated Embedded Systems, IS
Tamil
Nadu
Andhra
Pradesh
Tamil
Nadu
indicated Information Systems. Similarly AP indicates
Assistant Professor, APJ indicates Assistant Professor
(Junior), ASP indicates Associate Professor, SP
indicates Senior Professor, Pr indicates Professor.
The universe and other equivalence relations are
defined below.
Peter
AI
AS P
Ph.D
Roger
ES
P
Ph.D
Albert
AI
APJ
M.Sc.,
Mishra ES
APJ
M.Sc.,
Tamil
Nadu
Tamil
Nadu
Andhra
Pradesh
Tamil
Nadu
  {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,
Hari
IS
SP
Ph.D
Orissa
roger, hari, smith, keny, linz, williams, mukherjee,
preeth}}
John
SE
AP
M.C.A
Smith
NW
ASP
Ph.D
West
Bengal
Orissa
 / Native State  {{sam,shyam, roger, mishra,
williams},{ram, peter},{hari, smith, linz},{peter,
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
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
Jacob
NW
Lakma SE
n
Sita
AI
APJ
M.Sc.,
Karnataka
ASP
Ph.D
AP
M.Tech
West
Bengal
Karnataka
ISSN: 2231-5381
 / P be highset degree
 / Q be Native state
 / R be Division
X  {sam, john}
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
IV. AUTHENTICATION ON ASSOCIATIVE AND
COMMUTATIVE PROPERTY ON PESSIMISTIC
MULTI-GRANULATION BASED ROUGH SETS
P  QX  {sam, john}
Q  PX  {sam, john} Hence validated.
To Prove P  QX  Q  PX
Let X  {sam, john}
To Prove P *QX  Q * PX
P  QX ~ (P  Q(~ X))  ~ ([x]p  ~ X or
P * QX  ~ (P * Q(~ X)) ~ ([x]p  ~ X and
[x]Q  ~ X) ~ ({shyam, albert, mishra
[x]Q  ~ X)  ~ ({sam, john, sita, fatima}and {hari,
jacob,sita, fatima, ram, peter,
smith, linz, peter, john, lakman, fatima, mukherjee}
Let X  {ram, albert, shyam, mishra, martin, jacob}
roger, hari,smith, keny, linz, williams, mukherjee,
preetha})  {sam, john}
Q  PX  ~ (Q  P(~ X)) ~ ([x]Q ~ X or
~ ({john, fatima})
 {sam, ram,shyam, peter, roger, albert, mishra, hari,
smith, linz, keny, williams, martin, jacob, lakman,sita,
mukherjee, preetha}
[x]P  ~ X)  ~ ({shyam, albert, mishra
jacob,sita, fatima, ram, peter,
roger, hari, smith, keny, linz, williams, mukherjee,
preetha})  {sam, john}
Hence validated
To Pr ove (P  Q)  RX  P  (Q  R)X
X  {sam, john, ram}
(P  Q)  RX  ([x]p  X or[x]Q  X) or [x]R  X
 ({sam, john}or ) or   {sam, john}
P  (Q  R)X  [x]p  X or([x]Q  X or [x]R  X)
 {sam, john} or { or }  {sam, john}
so (P  Q)  RX  P  (Q  R)X Hence Validated
Q* PX  ~ (Q * P(~ X)) ~ ([x]Q ~ X and
[x]P  ~ X) ~ ({john, lakman, fatima, mukherjee}
and {sam, john, sita, fatima}  ~ ({john, fatima})
 {sam, ram,shyam, peter, roger, albert, mishra, hari,
smith, linz, keny, williams, martin, jacob, lakman,sita,
mukherjee, preetha}
so Q* PX  P * QX Hence validated
To Pr ove P *QX  Q * PX
X  {ram, albert,shyam, mishra, martin, jacob}
P *QX  [x]P  X and [x]Q  X  {shyam, albert,
To prove (P  Q)  RX  P  (Q R)X
X  {sam, ram, albert,smith, fatima, keny}
mishra, martin, jacob} and {ram, albert}  {albert}
Q* PX  [x]Q  X and [x]P  X  {ram, albert}and
(P  Q)  RX  ~ ((P  Q)  R(~ X))
~ (([x]P  ~ X or [x]Q  ~ X) or [x]R  ~ X))
~ ( or ( or )) ~ ()  
So (P  Q)  RX  P  (Q R)X Hence validated
{shyam, albert, mishra, martin, jacob}  {albert}
So P *QX  Q *PX Hence validated.
To Pr ove (P *Q) *RX  P *(Q *R)X
X  {shyam, albert, mishra, martin, jacob, ram, linz,
peter,sita}
(P *Q) *RX  ([x]P  X and [x]Q  X) and [x]R  X
 ({shyam, albert, mishra, martin, jacob}and{ram,
albert}) and{peter,albert, linz,sita}
 {albert}and{peter, albert, linz, sita}  {albert}
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International Journal of Engineering Trends and Technology (IJETT) – Volume 6 Number 3- Dec 2013
P * (Q* R)X  [x]P  X and ([x]Q  X and [x]R  X)
 {shyam, albert, mishra, martin, jacob}and({ram,
albert}) and{peter, albert, linz, sita})
 {shyam, albert, mishra, martin, jacob}and{albert}
 {albert}
So P *(Q* R)X  (P *Q)* RX Hence validated
To Pr ove (P *Q) *RX  P *(Q *R)X
X  {ram, albert,shyam, mishra, martin, jacob}
International Journal of Granular Computing, Rough
Sets
and
Intelligent
Systems
(IJGCRSIS),
(Switzerland),vol.1, no.4, (2010),pp.355-369 .
[6] 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.
[7] Isabelle Bloch, On links between mathematical
morphology and rough sets, The Journal of the Pattern
Recognition Society, 2000. pp. 1487 – 1496.
VII.BIOGRAPHY
(P *Q) *RX  ~ ((P *Q) *R(~ X))
~ (([x]P  ~ X and [x]Q ~ X) and [x]R  ~ X)
~ (({sam, john,sita, fatima}and{hari,smith,linz,
peter, john, lakman, fatima, mukherjee}) and {})
~ ({john, fatima}and{}) ~ ()  
P *(Q * R)X ~ (P *(Q * R)(~ X))
~ ([x]P  ~ X and ([x]Q ~ X and [x]R  ~ X))
~ ({sam, john,sita, fatima}and({hari,smith, linz,
peter, john, lakman, fatima, mukherjee} and {}))
~ ({sam, john,sita, fatima}and{}) ~ ( )  
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
(SITE), 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.
so P * (Q * R)X  (P * Q) * RX Hence validated.
V. CONCLUSION
This paper provided database based validation and
authentication on associative and commutative property
on optimistic and pessimistic multigranulation rough
sets. More particularly here the associative property is
used to prove how even different grouping of
equivalence relations with respect to similar
approximations gives the same result. In future work,
this would be extended to suitable industrial
applications where such grouping of different phases of
product manufacturing gives no change in performance
evaluation of a developed product.
VI. REFERENCES
[1] Pawlak, Z., Rough Sets, Int. jour. of Computer and
Information Sciences, 11, (1982), pp.341-356.
[2] Pawlak, Z.: Theoretical aspects of reasoning about
data, Kluwer academic publishers (London), (1991).
[3] 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.
[4] Qian, Y.H, Liang, J.Y. and Dang, C.Y.: MGRS in
Incomplete Information Systems, IEEE
Conference on Granular Computing,(2007),pp.163 168.
[5] Tripathy, B.K. and Mitra, A.: Topological
properties of rough sets and their applications,
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