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By analogy with operation of algebraic product (multiplication), exponentiation
operation is defined. If for FN
~
membership function is
~
μ
, then for FN
its
~
membership function is
γ
μ
~
.
Let us give examples of the frequently used FN, or to be exact, membership
functions which define these FN (fig. 1.4).
(
)
L
−
R
1.
Membership function of
-type FN (fig. 1.4.a)
⎧
1
2
1
−
2
x
,
0
≤
x
≤
;
⎪
⎨
()
()
2
2
L
x
=
1
−
x
,
0
≤
x
≤
1
R
x
=
1
(
)
2
⎪
⎩
2
x
−
1
,
<
x
≤
1
2
⎧
0
x
≤
a
−
a
;
x
>
a
+
a
;
1
L
2
R
⎪
2
⎛
a
−
x
⎞
⎪
⎜
⎜
⎝
⎟
⎟
⎠
1
−
1
,
a
−
a
<
x
≤
a
;
1
L
⎪
a
L
⎪
⎪
1
a
<
x
≤
a
;
()
1
2
μ
x
=
⎨
2
~
⎛
−
⎞
A
x
a
a
⎪
⎜
⎜
⎝
⎟
⎟
⎠
1
−
2
2
,
a
<
x
≤
a
+
R
;
2
2
⎪
a
2
R
⎪
2
⎛
x
−
a
⎞
a
⎪
2
⎜
⎜
⎝
2
−
1
⎟
⎟
⎠
,
a
+
R
<
x
≤
a
+
a
.
⎪
2
2
R
a
2
⎩
R
2.
Membership function of
T
-type FN (FN of trapezoidal type)
()
()
0
≤
x
≤
1
,
(Fig. 1.4.b)
L
x
=
R
x
=
1
−
x
a
−
x
a
−
x
⎧
1
−
1
,
0
<
1
≤
1
a
>
0
⎪
L
a
a
⎪
⎪
L
L
x
−
a
x
−
a
()
1
−
2
,
0
<
2
≤
1
a
>
0
μ
x
=
⎨
~
R
A
a
a
⎪
⎪
⎩
R
R
1
a
≤
x
≤
a
;
1
2
⎪
0
x
<
a
−
a
or
x
>
a
+
a
.
1
L
2
R
3.
Membership function of a normal triangular number or normal FN of
triangular type (Fig. 1.4.c)
()
()
() (
)
L
x
=
R
x
=
1
−
x
,
0
≤
x
≤
1
μ
x
=
a
,
a
,
a
~
1
L
R
A
