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AN APPLICATION OF INTERVAL-
VALUED FUZZY SOFT SETS IN

MEDICAL DIAGNOSIS

Guide:Dr. Sunil Jacob John Jobish VD
M090054MA

Contents.

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

 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).

 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 .

 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).

 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.

.

 Definition 1.3[3]: An interval-valued fuzzy
sets X on the universe U is a mapping
such that;
X : U → Int ([0,1]).
where; Int ([0,1]) - all closed sub-intervals
of [0,1].

The set of all interval-valued fuzzy sets on U is
denoted by F (U).

 If,
ˆ ~
X F (U ), x U
L U
x ( x)
ˆ [ ˆ
x ( x), ˆ
x ( x)] T hedegree of membership-
of an element xto X

L
ˆ
x ( x) lower degree of membership x toX
U
ˆ
x ( x) upper degree of membership x toX

L U
0 ˆ
x ( x) ˆ
x ( x) 1.

ˆ ˆ ~
 Let X , Y F ( U ). Then,
Union ˆ ˆ
of X and Y , denoted by,
ˆ
X  Y ˆ is given by -
( x ) sup [ ( x) , ( x )]
X  Yˆ
ˆ
ˆ
x ˆ
y
L L
[ sup ( ( x), ( x)),
ˆ
x ˆ
y
U U
sup ( ( x), ˆ
y ( x ) ) ].
ˆ
x

 ˆ ˆ ~
Let X , Y F ( U ). Then,
ˆ ˆ
Intersecti on of X and Y , denoted by,
ˆ
X  Y ˆ is given by -

( x ) inf [ ( x) , ( x)]
X  Yˆ
ˆ
ˆ
x ˆ
y
[ inf ( L ( x), L ( x)),
ˆ
x ˆ
y
inf ( U ( x ) , U ( x ) ) ].
xˆ ˆ
y

 ˆ ˆ ~
Let X, Y F ( U ). Then,
comlement of ˆ
X denoted ˆ c,
by X
and is given by -

(x) 1- ( x) .
ˆ
Xc
ˆ
x
[1 - U ( x ) , 1 - L ( x ) ) ].
ˆ
x ˆ
y

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

 Definition 1.8[5]: The complement of a
interval valued fuzzy soft set (F,A) is,

(F,A)C = (FC,¬A),
where ∀α ∈ A ,¬α = not α .

FC: ¬A F ( U ).

FC(β ) = (F ( ¬β ))C , ∀β ∈ ¬A

 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)=〈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 want.

 Example 2.4:
In example 2.3,
(G,A)C = {
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]〉
}.

2. Application –
in
medical diagnosis.

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.
 We construct another I-V fuzzy soft set (F1,S)
~
over P, F1:S→ F ( P).

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

The membership values are calculated by,
T1
( pi , d k ) [a, b]
L L
a inf { Q ( pi , e j ) R1 (e j , d k , )},
j

U
b sup { U
Q ( pi , e j ) R1 (e j , d k , )}
j

T2
( pi , d k ) [ x, y ]
L L
x inf { Q ( pi , e j ) R2 (e j , d k , )},
j

U
y sup { U
Q ( pi , e j ) R2 (e j , d k , )}
j

The membership values are calculated by,
S T1 ( pi , d j ) p q
L L
p { T1 ( pi , d j ) T1 ( p j , d i )}
j
U U
q { 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

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.

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

6. Find Sk= maxj { ST1 (pi , dj) ─ ST2 (pi,┐dj)}.

Then we conclude that the patient pi is
suffering from the disease dk.

 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.

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) }.

 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 ]

 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 ]

 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] 〉},

 IVFSS - (F1,S) is a parameterized family
={ F1(e1), F1(e2), F1(e3), F1(e4) }.

gives a collection of approximate
description of the patient-symptoms in the
hospital.

 (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]

 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.

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]

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.

References
1. Chetia.B, Das.P.K, An Application of Interval-
Valued Fuzzy Soft Sets in Medical
Diagnosis, Int. J. Contemp. Math. Sciences, Vol.
5, 2010, no. 38, 1887 - 1894

2. De S.K, Biswas R, and Roy A.R, An application
of intuitionistic fuzzy sets in medical
diagnosis, Fuzzy Sets and
Systems,117(2001), 209-213.

3. Maji PK, Biswas R and Roy A.R, Fuzzy Soft
Sets, The Journal of Fuzzy Mathematics

4. Molodtsov D, Soft Set Theory-First
Results, Computers and Mathematics with
Application, 37(1999), 19-31.

5. Roy MK, Biswas R, I-V fuzzy relations and
Sanchez’s approach for medical
diagnosis, Fuzzy Sets and
Systems,47(1992),35-38.

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