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1234
respectively, gives a fuzzy set
C
defined by
1234
P
xb b b b
;, ,,
P
xc c c c
;, , ,
,
B
C
where
c
min
a
,
b
,
c
O
a
1
O
b
,
1
1
1
2
1
2
1
2
(7.5)
c
O
a
1
O
b
,
c
max
a
,
b
.
3
2
3
2
3
4
4
4
From the above description one can see that for
aa
and
bb
, trapezoidal
2
3
2
3
fuzzy sets
A
and
B
reduce to two triangular fuzzy sets and for
2
fuzzy set
C
represents the final triangular fuzzy set obtained by merging two triangular fuzzy
sets
A
and
B
. Following the same discussion, one can also merge two similar
Gaussian fuzzy sets
cc
3
and
GG
represented by the corresponding membership
1
2
function as
^
`
2
2
P
xc
;,
V
exp
x c
V
,
i
,2.
G
i
i
i
i
Merging of these two fuzzy sets
and
GG
will result in a new fuzzy set
G
represented also by a Gaussian membership function with mean and variance
parameters respectively as
1
2
c
O
c
1
O
c
,
V OV
.
1
O V
3
1
1
1
2
3
2
1
2
2
The parameters
,o
GG
has
the most influence on the cardinality of
G
. Similarly, in the case of trapezoidal or
triangular fuzzy sets the parameters O and O [0,1] determine which of the
fuzzy sets
A
or
B
has the most influence on the cardinality (kernel) of
C
.
O and
O [0,1] determine which of the fuzzy sets
1
2
Mu(x)
Mu(x)
C
1.0
1.0
A
B
A
B
C
x
b2
x
a1 a2
b1
c2
b2
a3
a4
b3 b4
a1
b1
a2
a3
b3
c1
c3
c4
c1
c2
c3
Figure 7.6.
Merging of two fuzzy sets, trapezoidal (left) and triangular (right)
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