October 2007 Volume 10 Number 4 - Educational Technology ...
October 2007 Volume 10 Number 4 - Educational Technology ...
October 2007 Volume 10 Number 4 - Educational Technology ...
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t ( u ), 1−<br />
f ( u )] = t ( u ), 1−<br />
f ( u )], where 1 ≤ i ≤ n.<br />
[ ~ ~<br />
A i A i<br />
[ ~<br />
B i ~<br />
B<br />
i<br />
( ~ A i u ( ~<br />
B i u<br />
That is, t ( ) = ) ( t , f ) = f ), and 1 ≤ i ≤ n. Because<br />
and<br />
~<br />
A i u<br />
~<br />
B i u<br />
( )) V S = ) (<br />
( ~<br />
A i u<br />
t – ) ( f<br />
~<br />
A i u<br />
~ A i u<br />
( ~ ( ))<br />
B i u V S = ) ( ~<br />
B i u t – ) ( ~ B i u f = ) ( ~<br />
A i u t – ) ( ~ A i u f = )). ( ( ~ A i u V S<br />
Therefore, we can see that<br />
H(Ã, B ~ ) ∑ ⎟ = ⎟<br />
n 1 ⎛ S(<br />
V ~ ( ui<br />
)) − S(<br />
V ~ ( ui<br />
)) ⎞<br />
= ⎜ A<br />
B<br />
1−<br />
n ⎜ i 1 ⎝<br />
2<br />
⎠<br />
(ii) If H(Ã, B ~ ) = 1, then<br />
= ∑ ⎟ = ⎟<br />
n 1 ⎛ S(<br />
V ~ ( ui<br />
)) − S(<br />
V ~ ( ui<br />
)) ⎞<br />
⎜ A<br />
A<br />
1−<br />
n ⎜ i 1 ⎝<br />
2<br />
⎠<br />
= 1.<br />
H(Ã, B ~ ) ∑ ⎟ = ⎟<br />
n 1 ⎛ S(<br />
V ~ ( ui<br />
)) − S(<br />
V ~ ( ui<br />
)) ⎞<br />
= ⎜ A<br />
B<br />
1−<br />
= 1.<br />
n ⎜ i 1 ⎝<br />
2<br />
⎠<br />
It implies that ( )) V S = )), ( V S where 1 ≤ i ≤ n. Because )) ( V S =<br />
( ~<br />
A i u<br />
( ( ~<br />
B i u V ( ~<br />
B i u<br />
( ~<br />
B i u ( ~ A i u<br />
( ~ B i u<br />
( ~ B i u<br />
( )) V S and S )) = t ) – f ), where 1 ≤ i ≤ n, we can see that<br />
( ~<br />
B i u<br />
t ( ) = )<br />
~<br />
A i u<br />
t and f ) = ) ( f (i.e., 1 – ) ( f = 1 – ) ( f ),<br />
~ B i u<br />
~ A i u<br />
~ B i u<br />
where 1 ≤ i ≤ n. Therefore, the vague sets à and B ~ are identical. Q. E. D.<br />
Property 4: H(Ã, B ~ ) = H( B ~ , Ã).<br />
Proof:<br />
Because<br />
H(Ã, B ~ ) ∑ ⎟ = ⎟<br />
n 1 ⎛ S(<br />
V ~ ( ui<br />
)) − S(<br />
V ~ ( ui<br />
)) ⎞<br />
= ⎜ A<br />
B<br />
1−<br />
n ⎜ i 1 ⎝<br />
2<br />
⎠<br />
and<br />
H( B ~ , Ã) ∑ ⎟ = ⎟<br />
n 1 ⎛ S(<br />
V ~ ( ui<br />
) − S(<br />
V ~ ( ui<br />
)) ⎞<br />
= ⎜ B<br />
A<br />
1−<br />
,<br />
n ⎜ i 1 ⎝<br />
2<br />
⎠<br />
and because<br />
∑<br />
=<br />
⎟ ⎟⎟<br />
⎛ ⎞<br />
1 n S(<br />
V~<br />
( u −<br />
⎜<br />
i))<br />
S(<br />
V~<br />
( ui<br />
))<br />
⎜<br />
1−<br />
A B = ∑ n i 1<br />
2<br />
⎝<br />
⎠<br />
= ⎟ ⎟⎟<br />
⎛ ⎞<br />
1 n S(<br />
V~<br />
( u −<br />
⎜<br />
i)<br />
S(<br />
V~<br />
( ui<br />
))<br />
⎜<br />
1−<br />
B A ,<br />
n i 1<br />
2<br />
⎝<br />
⎠<br />
we can see that H(Ã, B ~ ) = H( B ~ , Ã). Q. E. D.<br />
( ~ A i u<br />
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