October 2007 Volume 10 Number 4 - Educational Technology ...

we can see that
S( X ) − S(
Y ) S(
Y ) − S(
X )
=
,
2
2
S( X ) − S(
Y ) S(
Y ) − S(
Y )
1−
= 1−
2
2
and M(X, Y) = M(Y, X). Q. E. D.
Let à and B ~ be two vague sets in the universe of discourse U, U = {u1, u2, …, un}, where
à = [ t ~ ( u ), 1−
f ~ ( u )] / u1 + [ t ( ), 1 ( )]
A 1 A 1
~ u − f 2 ~ u / u2 + … + [ t ( ), 1 ( )]
A
A 2
~ un
− f ~ u
A
A
n / un,
and
B ~ = t ( u ), 1−
f ( u )] / u1 + t ( u ), 1−
f ( u )] / u2 + … + t ( u ), 1−
f ( u )] / un.
[ ~
B 1 ~
B
1
[ ~ 2 ~
B
B
2
[ ~
B
n ~ n
B
Let ~ ( i)
A u V = [ ) ( ~
A i u t , 1 – ) ( ~
A i u f ] be the vague membership value of ui in the vague set Ã, and let ~ ( )
B i u V =
[ ~ ( )
B i u t , 1 – ) ( ~
B
i u f ] be the vague membership value of ui in the vague set B ~ . By applying Eq. (5), we can see
that the score ( ~ ( ))
A i u V S of the vague membership value ) ( ~
A i u V can be evaluated as follows:
( ~ ( ))
A i u V S = ) ( ~
A i u t – ) ( ~
A i u f ,
and the score ( )) V S of the vague membership value ) ( V can be evaluated as follows:
( ~
B i u
( )) V S = ) ( t – ), ( f
( ~
B i u
~
B i u
~
B i u
~
B i u
where 1 ≤ i ≤ n. Then, the degree of similarity H(Ã, B ~ ) between the vague sets à and B ~ can be evaluated by the
function H,
n ~ ~ 1
H ( A,
B)
= ∑ M ( V ~ ( u ), V u i ~ ( ))
B i
n i=
1
A
1 ( ~ ( )) ( ~ ( ))
∑ 1
,
= 1
2 ⎟ ⎟
n ⎛ S V u − S V u ⎞
= ⎜ A i B i
−
(7)
n i ⎜
⎝
⎠
where H(Ã, B)∈[0, 1]. The larger the value of H(Ã, B ~ ), the higher the similarity between the vague sets à and B ~ .
Let à and B ~ be two vague sets in the universe of discourse U, U = {u1, u2, …, un}, where
à = [ t ~ ( u ), 1−
f ~ ( u )] / u1 + [ t ( ), 1 ( )]
A 1 A 1
~ u − f 2 ~ u / u2 + … + [ t ( ), 1 ( )]
A
A 2
~ un
− f ~ u
A
A
n / un,
and
B ~ = [ t ~ ( u ), 1−
f ~ ( u )] / u1 + [ t ( ), 1 ( )]
B 1
B
1
~ u − f 2 ~ u / u2 + … + [ ( ), 1 ( )]
B
B
2
~ un
f
B
~ un
B
The proposed similarity measure between vague sets has the following properties:
Property 3: Two vague sets à and B ~ are identical if and only if H(Ã, B ~ ) = 1.
Proof:
(i) If à and B ~ are identical, then
t − / un.
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