1. AN APPLICATION OF INTERVAL-
VALUED FUZZY SOFT SETS IN
MEDICAL DIAGNOSIS
Guide:Dr. Sunil Jacob John Jobish VD
M090054MA

2. Contents.
1. Preliminaries.
2. Application of interval valued fuzzy
soft set in medical diagnosis.
3. Algorithm.
4. Case Study.

3. 1. Preliminaries.

4.  Definition 1.1[3]:
Let U - initial universe set
E - set of parameters.
P (U) - power set of U. and,
A - non-empty subset of E.
A pair (F, A) is called a soft set over U,
where F is a mapping given by F: A P (U).

5.  Example 1.1;
Let U={c1,c2,c3} - set of three cars.
E ={costly(e1), metallic color (e2), cheap (e3)}
- set of parameters.
A={e1,e2} ⊂ E. Then;
(F,A)={F(e1)={c1,c2,c3},F(e2)={c1,c3}}
“ attractiveness of the cars” which Mr. X is going
to buy .

6.  Definition 1.2[3]:
Let U - universal set,
E - set of parameters and A ⊂ E.
Let F (U) - set of all fuzzy subsets of U.
Then a pair (F,A) is called fuzzy soft set over U,
where F :A F (U).

7.  Example 1.2;
Let U = {c1,c2,c3} - set of three cars.
E ={costly(e1),metallic color(e2) , getup (e3)}
- set of parameters.
A={e1,e2 } ⊂ E.
Then;
(G,A) = { G(e1)={c1/.6, c2/.4, c3/.3},
G(e2)={c1/.5, c2/.7, c3/.8} }.
- fuzzy soft set over U.
Describes the “ attractiveness of the cars” which
Mr. S want.

8. .
 Definition1.3[3]: An interval-valued fuzzy
ˆ
sets X on the universe U is a mapping
such that;
ˆ
X : U → Int ([0,1]).
where, Int ([0,1]) - all closed subintervals of [0,1].
The set of all interval-valued fuzzy sets on
~
U is denoted by F (U )

9.  If, ~
ˆ
X F (U ), x U
L U
ˆ ( x)
X
[ ˆ
X
( x), ˆ
X
( x)]
ˆ
T hedegree of membershipof an elementx to X
L
ˆ
ˆ
( x) Lower degree of membership x to X
X
U
ˆ
ˆ
( x) Upperdegree of membership x to X
X
L U
0 ˆ
X
( x) ˆ
X
( x) 1

10. ˆ ˆ ~
 Let X , Y F ( U ). T hen,
ˆ ˆ ˆ ˆ
Unionof X and Y , denotedby, X  Y ,
is given by -
ˆ ˆ
XY
( x ) sup [ ˆ
X
( x), Yˆ ( x)]
L L
[ sup ( ( x ) ,
ˆ
X Yˆ ( x)),
U U
sup ( ( x ) ,
ˆ
X Yˆ ( x ) ) ].

11.  ˆ ˆ ~
Let X , Y F ( U ).
ˆ ˆ
Then the intersection of X and Y , denoted by
ˆ ˆ
X  Y and is given by,
ˆ ˆ
X Y
( x ) inf [ ˆ
X
( x), Yˆ ( x)]
L L
[ inf ( ˆ
X
( x), Yˆ ( x)),
U U
inf ( ˆ
X
( x), Yˆ ( x ) ) ].

12. 
ˆ ˆ ~
Let X , Y F ( U ). T hen,
complement X
of ˆ denoted by X c ,
ˆ
and is given by -
ˆ
Xc
(x) 1- ˆ
X
(x) .
[1 - U
Xˆ ( x), 1 -
L
Yˆ
( x ) ].

13.  Definition 1.7 [4]:
Let U universal set.
E set of parameters.
and A ⊂E.
~ set of all interval-valued fuzzy sets on U.
F (U )
Then a pair (F, A) is called interval-valued fuzzy
soft set over U.
~
where F : A F (U ).

14.  Definition 1.8[5]: The complement of a
interval valued fuzzy soft set (F,A) is,
(F,A)C = (FC,¬A),
~
FC: ¬A F ( U ).
FC(β ) = (F ( ¬β ))C , ∀β ∈ ¬A

15.  Example2.3:
Let U={c1,c2,c3} set of three cars.
E ={costly(e1), grey color(e2),mileage (e3)},
set of parameters.
A={e1,e2} ⊂ E. Then,
(G,A) = {G(e1), G(e2)}
G(e1)={〈c1,[.6,.9]〉, 〈c2,[.4,.6]〉, 〈c3,[.3,.5]〉}
G(e2)= {〈c1,[.5,.7]〉, 〈c2,[.7,.9]〉, 〈c3,[.6,.9]〉}
“ attractiveness of the cars” which Mr. X wants.

16.  Example 2.4:
In example 2.3,
(G,A)C = {G(¬e1), G(¬e2)}.
G(¬e1)= { 〈c1,[0.1,0.4]〉, 〈c2,[0.4,0.6]〉,
〈c3,[0.5,0.7]〉}
G(¬e2)= { 〈c1,[0.3,0.5]〉, 〈c2,[0.1,0.3]〉
〈c3,[0.1,0.4]〉}

17. 2. Application –
in
medical diagnosis

18. S - Symptoms, D – Diseases, and P - Patients.
 Construct an I-V fuzzy soft set (F,D) over S
~
F:D→ F ( S ).
A relation matrix say, R1 - symptom-disease
matrix- constructed from (F,D).
 Its complement (F,D)c gives R2 - non
symptom-disease matrix.

19.  We construct another I-V fuzzy soft set (F1,S)
~
over P, F1:S→ F ( P).
 Relationmatrix Q - patient-symptom matrix-
from (F1,S).
Then matrices,
 T1=Q R1 - symptom-patient matrix, and
 T2= Q R2 - non symptom-patient matrix.

20. The membership values are calculated by,
T1 ( pi , d k ) [ a, b]
L L
a infj
{ Q ( pi , e j ) R1 (e j , d k , )},
b sup { U
Q ( pi , e j ) U
R1 (e j , d k , )}
j
T2 ( pi , dk ) [ x, y ]
L L
x infj
{ Q ( pi , e j ) R2 (e j , d k , )},
y sup { U
Q ( pi , e j ) U
R2 (e j , d k , )}
j

21. The membership values are calculated by,
S T1 ( pi , d j ) x y
L L
x { T1 ( pi , d j ) T1 ( p j , d i )}
j
U U
y { T1 ( pi , d j ) T1 ( p j , d i )}
j
S T 2 ( pi , d j ) s t
L L
s { T2 ( pi , d j ) T2 ( p j , d i )}
j
U U
t { T2 ( pi , d j ) T2 ( p j , d i )}
j

22. 3. Algorithm.

23. 1. Input the interval valued fuzzy soft sets (F,D)
and (F,D)c over the sets S of symptoms, where
D -set of diseases.
2. Write the soft medical knowledge R1 and R2
representing the relation matrices of the
IVFSS (F,D) and (F,D)c respectively.

24. 3. Input the IVFSS (F1,S) over the set P of
patients and write its relation matrix Q.
4. Compute the relation matrices T1=Q R1 and
T2=Q R2.
5. Compute the diagnosis scores ST1 and ST2

25. 6. Find Sk= maxj { ST1 (pi , dj) ─ ST2 (pi,┐dj)}.
Then we conclude that the patient pi is
suffering from the disease dk.

26. 4. Case Study.

27.  Patients - p1, p2 and p3.
 Symptoms (S) - Temperature, Headache, Cough
and Stomach problem
 S={ e1,e2,e3,e4} as universal set.
D ={d1,d2}.
d1 - viral fever, and
d2 - malaria.

28. Suppose that,
F(d1) ={ 〈e1, [0.7,1]〉, 〈e2, [0.1,0.4]〉,
〈e3, [0.5,0.6]〉, 〈e4,[0.2,0.4]〉) }.
F(d2) ={ 〈e1,[0.6,0.9] 〉, 〈e2,[0.4,0.6] 〉,
〈e3,[0.3,0.6] 〉, 〈e4,[0.8, 1] 〉 }.
 IVFSS - (F,D) is a parameterized family
={ F(d1), F(d2) }.

29.  IVFSS - (F,D) can be represented by a relation
matrix R1 - symptom-disease matrix- given by,
R1 d1 d2
e1 [0.7, 1.0 ] [ 0.6, 0.9 ]
e2 [0.1, 0.4 ] [0.4, 0.6 ]
e3 [0.5, 0.6 ] [0.3, 0.6 ]
e4 [0.2, 0.4 ] [0.8, 1.0 ]

30.  TheIVFSS - (F, D)c also can be represented by
a relation matrix R2, - non symptom-disease
matrix, given by-
R2 d1 d2
e1 [0 , 0.3 ] [ 0.1, 0.4 ]
e2 [0.6, 0.9 ] [0.4, 0.6 ]
e3 [0.4, 0.5 ] [0.4, 0.7 ]
e4 [0.6, 0.8 ] [0 , 0.2 ]

31.  We take P = { p1, p2, p3} - universal set .
S = { e1, e2, e3, e4} - parameters.
Suppose that,
F1(e1)={〈p1, [.6, .9]〉, 〈p2, [.3,.5]〉,〈p3, [.6,.8]〉},
F1(e2)={ 〈p1, [.3,.5] 〉, 〈p2, [.3,.7] 〉, 〈p3, [.2,.6] 〉},
F1(e3)={〈p1, [.8, 1]〉, 〈p2, [.2,.4]〉,〈p3, [.5,.7]〉} and
F1(e4)={〈p1, [.6,.9] 〉,〈p2, [.3,.5] 〉, 〈p3, [.2,.5] 〉},

32.  IVFSS - (F1,S) is a parameterized family
={ F1(e1), F1(e2), F1(e3), F1(e4) }.
which gives a collection of approximate
description of the patient-symptoms in the
hospital.

33.  (F1,S) - represents a relation a relation matrix
Q - patient-symptom matrix - given by;
Q e1 e2 e3 e4
p1 [0.6, 0.9] [0.3, 0.5] [0.8, 1] [0.6, 0.9]
p2 [0.3, 0.5] [0.3, 0.7] [0.2, 0.4] [0.3, 0.5]
p3 [0.6, 0.8] [0.2, 0.6] [0.5, 0.7] [0.2, 0.5]

34.  Combining the relation matrices R1 and R2
separately with Q, we get,
T1=Q o R1 - patient-disease matrix.
T2=Q o R2 - patient-non disease -
matrix.

35. Here we want to calculate,- membership
values using the equations ,
For the matrix T1
a = inf {min(0.6, 0.7 ) , min(0.3, 0.1 ) ,
min(0.8, 0.5 ), min(0.6, 0.2 ) }
= inf {0.6, 0.1, 0.5, 0.2} = 0.1.
b = sup {min(0.9, 1.0 ) , min(0.5, 0.4 ) ,
min(1.0, 0.6 ), min(0.9, 0.4 ) }
= sup {0.9, 0.4, 0.6, 0.4 } = 0.9.

36. T1 d1 d2
p1 [0.1 ,0.9] [0.3 ,0.9]
p2 [0.1 ,0.5] [0.2 ,0.6]
p3 [0.1 ,0.8] [0.2 ,0.8]
T2 d1 d2
p1 [0 , 0.8] [0 , 0.7]
p2 [0 , 0.7] [0 , 0.6]
p3 [0 , 0.6] [0 , 0.7]

37. Now we calculate,
ST1-ST2 d1 d2
p1 0.2 0.6
p2 -0.7 -0.4
p3 0.5 -0.1
The patient p3 is suffering from the disease d1.
 Patients p1 and p2 are both suffering from
disease d2.

38. References
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9(3)(2001), 677-692.

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