Chapter 6
FUZZY FUNCTION
Chi-Yuan Yeh
Kinds of fuzzy function
• Crisp function with fuzzy constraint
• Fuzzy extension function which propagates the fuzziness of independent variable to dependent variable
• Fuzzifying function which blurs the image of a crisp independent variable
2
Function with fuzzy constraint
3
Fuzzy extension function
4
tes
Fuzzy extension function
5
x~
Single fuzzifying function
6
Single fuzzifying function
7
Single fuzzifying function
8
Fuzzy bunch of functions
9
Fuzzy bunch of functions
10
1
Fuzzy bunch of functions
11
Fuzzy bunch of functions
12
Fuzzy bunch of functions
13
Maximizing set
14
Minimizing set
Minimizing set of f is defined as the maximizing set of -f
15
Xx
xxxm
inf(f)sup(f)
)f(sup(f)
f)inf(f)sup(
f)inf())f(()(f
Maximizing set
16
Maximizing set
17
Maximizing set
18
Maximum value in crisp domain
19
Maximum value in crisp domain
20
Maximum value in crisp domain
21
Maximum value in fuzzy domain
22
Maximum value in fuzzy domain
23
Maximum value in fuzzy domain
24
Maximum value in fuzzy domain
25
Maximum value in fuzzy domain
26
Integration of fuzzying function
27
Integration of fuzzying function
28
Integration of fuzzying function
29
Integration of fuzzying function
30
Integration in fuzzy interval
31
y
Integration in fuzzy interval
32
Differentiation of crisp function in fuzzy interval
34
Differentiation of crisp function in fuzzy interval
35
36
0 0.5 x
Differentiation of fuzzifying function
function
37
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