3.12. EXERCISES AND PROJECTS 131
Find formulas for all the functions involved and also plot them.
- LetA(x)=sinπx,B(x)=x^2 ,A^0 (x)=1−A(x),andB^0 (x)=1−B(x)
forx∈[0,1].Showthat
A(x)∧A^0 (x)=(A(x)∧A^0 (x)∧B(x))∨(A(x)∧A^0 (x)∧B^0 (x))
Find formulas for all the functions involved and also plot them.
- If⊕is an averaging operator andfis a continuous, strictly increasing
functionf:[0,1]→[0,1]satisfyingf(0) = 0andf(1) = 1, show that
x~y=f−^1 (f(x)⊕f(y))
is also an averaging operator.
- Show that an OWA operator satisfies the three basic properties associated
with an averaging operator.
- Show that the function
t(x)=max
Ω
min
μ
x−a
b−a
,
c−x
c−b
∂
, 0
æ
gives the triangular set determined by the points(a,0),(b,1),(c,0)for
a<b<c.
- Show that the function
t(x)=max
Ω
αmin
μ
x−a
b−a
, 1 ,
d−x
d−c
∂
, 0
æ
gives the trapezoidal set determined by the points(a,0),(b,α),(c,α),
(d,0)fora<b<c<dand 0 <α≤ 1.
- LetA[a,b,c]denote the triangular set determined by the points(a,0),
(b,1),(c,0). Using the fuzzy sets
0
0.2
0.4
0.6
0.8
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A 1 [2, 4 ,6]andA 2 [4, 6 ,8]
and
