<s>
In	O
mathematics	O
,	O
the	O
membership	B-General_Concept
function	I-General_Concept
of	O
a	O
fuzzy	O
set	O
is	O
a	O
generalization	O
of	O
the	O
indicator	O
function	O
for	O
classical	O
sets	O
.	O
</s>
<s>
Membership	B-General_Concept
functions	I-General_Concept
were	O
introduced	O
by	O
Aliasker	O
Zadeh	O
in	O
the	O
first	O
paper	O
on	O
fuzzy	O
sets	O
(	O
1965	O
)	O
.	O
</s>
<s>
Aliasker	O
Zadeh	O
,	O
in	O
his	O
theory	O
of	O
fuzzy	O
sets	O
,	O
proposed	O
using	O
a	O
membership	B-General_Concept
function	I-General_Concept
(	O
with	O
a	O
range	B-Algorithm
covering	O
the	O
interval	O
(	O
0	O
,	O
1	O
)	O
)	O
operating	O
on	O
the	O
domain	O
of	O
all	O
possible	O
values	O
.	O
</s>
<s>
For	O
any	O
set	O
,	O
a	O
membership	B-General_Concept
function	I-General_Concept
on	O
is	O
any	O
function	O
from	O
to	O
the	O
real	O
unit	O
interval	O
.	O
</s>
<s>
Membership	B-General_Concept
functions	I-General_Concept
represent	O
fuzzy	O
subsets	O
of	O
.	O
</s>
<s>
The	O
membership	B-General_Concept
function	I-General_Concept
which	O
represents	O
a	O
fuzzy	O
set	O
is	O
usually	O
denoted	O
by	O
For	O
an	O
element	O
of	O
,	O
the	O
value	O
is	O
called	O
the	O
membership	O
degree	O
of	O
in	O
the	O
fuzzy	O
set	O
The	O
membership	O
degree	O
quantifies	O
the	O
grade	O
of	O
membership	O
of	O
the	O
element	O
to	O
the	O
fuzzy	O
set	O
The	O
value	O
0	O
means	O
that	O
is	O
not	O
a	O
member	O
of	O
the	O
fuzzy	O
set	O
;	O
the	O
value	O
1	O
means	O
that	O
is	O
fully	O
a	O
member	O
of	O
the	O
fuzzy	O
set	O
.	O
</s>
<s>
Sometimes	O
,	O
a	O
more	O
general	O
definition	O
is	O
used	O
,	O
where	O
membership	B-General_Concept
functions	I-General_Concept
take	O
values	O
in	O
an	O
arbitrary	O
fixed	O
algebra	O
or	O
structure	O
;	O
usually	O
it	O
is	O
required	O
that	O
be	O
at	O
least	O
a	O
poset	O
or	O
lattice	O
.	O
</s>
<s>
The	O
usual	O
membership	B-General_Concept
functions	I-General_Concept
with	O
values	O
in	O
 [ 0 , 1 ] 	O
are	O
then	O
called	O
[0,1]-valued	O
membership	B-General_Concept
functions	I-General_Concept
.	O
</s>
<s>
One	O
application	O
of	O
membership	B-General_Concept
functions	I-General_Concept
is	O
as	O
capacities	O
in	O
decision	O
theory	O
.	O
</s>
