<s>
In	O
mathematics	O
,	O
vague	B-General_Concept
sets	I-General_Concept
are	O
an	O
extension	O
of	O
fuzzy	O
sets	O
.	O
</s>
<s>
In	O
a	O
fuzzy	O
set	O
,	O
each	O
object	O
is	O
assigned	O
a	O
single	B-Algorithm
value	I-Algorithm
in	O
the	O
interval	O
 [ 0 , 1 ] 	O
reflecting	O
its	O
grade	O
of	O
membership	O
.	O
</s>
<s>
This	O
single	B-Algorithm
value	I-Algorithm
does	O
not	O
allow	O
a	O
separation	O
of	O
evidence	O
for	O
membership	O
and	O
evidence	O
against	O
membership	O
.	O
</s>
<s>
proposed	O
the	O
notion	O
of	O
vague	B-General_Concept
sets	I-General_Concept
,	O
where	O
each	O
object	O
is	O
characterized	O
by	O
two	O
different	O
membership	O
functions	O
:	O
a	O
true	O
membership	O
function	O
and	O
a	O
false	O
membership	O
function	O
.	O
</s>
<s>
The	O
vague	B-General_Concept
set	I-General_Concept
degenerates	O
to	O
a	O
fuzzy	O
set	O
,	O
if	O
for	O
all	O
x	O
.	O
</s>
