1. Novel Set Approximations in
Generalized Multi-valued
Decision Information Systems
(GMDIS)
2006

2. By
A. M. Kozea1 & S. Abd El-Badie2
1Mathematics Department, Faculty of Science,
Tanta University, Egypt
2Mathematics Department, Faculty of
Engineering, Kafr Elsheikh University, Egypt.
1Email: akozae55@yahoo.com
2Email: savvymore@yahoo.com
2Homepage: www.savvymore.mysite.com

3. Outlines
• The Aim of The Work
• Rough Set Theory: Basic Concept
• Information Systems
• Single Valued Information System (SIS)
• Single Valued Decision Information System (SDIS)
• Multi-valued Information System (MIS)
• Multi-valued Decision Information System (MDIS)
• New Thinking
• Generalized Multi-Valued Decision Information System (GMDIS)
• New Set Approximations in GMDIS
• Exactness Determination
• Linkage of knowledge
• Linkage Grade of Knowledge
• Conclusion

4. The Aim of The Work
• In this paper a partial order relations are defined on
the objects of the Multi-valued Information System
(MIS) or Multi-valued Decision Information System
(MDIS) which we called a Generalized Multi-valued
Information System (GMIS) or Generalized Multi-
valued Decision Information System (GMDIS).
• The resulted classes from this relation are used as a
general knowledge base for defining basic rough set
concepts on GMIS .
• Some examples are given and a comparison with
classical rough set concepts is obtained.

5. Rough Set Theory:
Basic Concepts
• Information/Decision Systems (Tables)
• Indiscernibility
• Set Approximation
• Reducts and Core
• Rough Membership
• Dependency of Attributes

6. Information Systems

7. Information Systems (IS)
The first concept of IS was developed by
Grzymala-Busse (1988); there are many
types of IS as follows:
Single valued Information System (SIS) ,
The data in this system are presented
when it takes a single value for each
element,
Single Valued Decision Information System
(SDIS) ,where D is the decision attribute

8. A Multi-valued Information System (MIS)
is an ordinary information, it can be
defined as
MIS = (U , At ,{Va : a  At}, f a )
A Multi-valued Decision Information
System (MDIS) can be defined as,
MDIS = (U , At U D, {Va : a  At}, f a )
where D is the decision attribute.

9. New Thinking

10. New Thinking (1)
Following the new thinking which we present
in the 1st workshop of ERS group, August 2006,
Faculty of Science, Alexandria University,
Egypt.
“New Approaches for Data Reduction in
Generalized Multi-valued Decision Information
System (GMDIS)
Case Study: Rheumatic Fever Patients”

11. Generalized Multi-
Valued Decision
Information System
(GMDIS)

12. The structure of the Generalized
Multi-valued Information System can
be defined as follows.
(1) GMIS = (U , At , {y a : a  At}, fa , {h B : B  At})
A Generalized Multi-valued Decision
Information System (GMDIS) is
denoted by,
(2) GMDIS = (U , At U D , {y : a  At}, f , {h : B  At})
a a B

13. New Set Approximations in GMDIS
(3) hB = {(x, y) : fa ( x)c  fa (y) , "a  B , B  At}
(4) hB = {(x, y) : fa ( y)  fa ( x), "a  B , B  At}
Define the set of all intersections of
members of as the Meeting Point Relation
(MPR) can be written as:
(5) α = {α = A I A , α  U A , A , A , A  A , i  j }
a l i j l k i j k ha
k

14. Moreover,
D
(6) POS B (D ) = U X hB , B  At
X  Ah
D
Where, for any subset X  U the lower and
upper approximations are defined by,
X h
= U {h : h Bx  X }, B  At
Bx
(7) B
X h = U {h : h Bx I X  F }, B  At
B Bx

15. For the relation defined in (3)

16. For the relation defined in (4)

17. Important Fact
It can be noticed that the properties of
approximations of the relation defined
in (3) & (4) are just eight properties
but in Pawalk 1991, there were twelve
properties. This means that there are
four properties that are not satisfied
for the above suggested relations. This
fact is illustrated by the following
examples.

18. Illustrative Example
U S1 S2 S3
x1 {a1, a2} {b3} {c2}
x2 {a1} {b2, b3} {c1,c2}
x3 {a3} {b4} {c2}
x4 {a2, a3} {b1, b3, b4} {c1,c2}
x5 {a1, a2. a3} {b1, b4} {c1 }
x6 {a2, a3} {b1, b2, b4} {c1,c2}
x7 {a1, a3} {b3, b4} {c2}
MIS DATA

19. (3) hB = {(x, y) : fa ( x)c  fa (y) , "a  B , B  At}

20. The four non satisfied properties for
relation (3)

21. Important Note
Relation (3) does not satisfied the
property that ,
Because the relation is not reflexive

22. (4) hB = {(x, y) : fa ( y)  fa ( x), "a  B , B  At}

23. The four non satisfied properties for
relation (4)

24. Exactness Determination
X h
(X ) =
B
E hB
X h B
Where X  f , Y is the cardinality of Y.
The exactness factor Eh ( X ) is intended to
B
capture the degree of completeness of our
,
knowledge about the set X
Y
.We note that
0  Eh ( X )  1for every h Band X  U .If Eh ( X ) =1
, B
h B -border line region of x is empty and
B
,the
the set X is h -definable. If E ( X ) <1 , the
B h
set X has some non-empty h -borderline region
B
and consequently it is h -indefinable.
B
B

25. Linkage of knowledge
 One can get that, if P , Q  At and αP ,
αQ are the MPR of knowledge ηP and ηQ
then it can be obtained the following.
 Knowledge ηQ linked with knowledge ηP,
denoted by
h P  h Q iff α P  α Q

26. Grade of Linkage of knowledge
Knowledge ηQ linked in a grade G ( 0  G  1 ) with
knowledge ηP iff
G = gh (h Q ) = POS h (h Q )
P
P
U
 If G =1  ηQ is totally linked with ηP.
 If 0 < G < 1  ηQ is roughly linked with ηP.
 If G= 0  ηQ is not totally linked with ηP.

27. Conclusion
The suggested method for obtaining new set
approximation on GMIS and GMDIS is a
general case. this approach tends to Pawlak
approach if the system is single valued and
the relation is equivalence.
Pawlak Properties for the equivalence
relations are not satisfied for the partial
order relations .

28. Conclusion
The New approximations can be applied for
all types of general relation and find which is the
best one, this is a future work of this work
The method opens the way for other
approximation if we use the general topological
recent concepts.. Other concepts of RST such as
reduction and decision rule can be investigated
using this approach.. Some of the above items
will be presented in a future work.