An introduction to fuzzy math
Fuzzy math differs from conventional math primarily in the area
of set theory. For example in a conventional AND statement both
statements must be true for the statement to be true. However, in
fuzzy logic statements are not always true or false, they merely
have varying levels of confidence. The table below exibits the
differences in how an AND statement is applied in conventional and
fuzzy systems. As can clearly be seen the AND statement in a fuzzy
system is merely the minimum confidence value of the two
values.
| conventional |
|
|
fuzzy |
|
|
| A |
B |
A AND B |
Fuzzy A |
Fuzzy B |
A AND B |
1
1
0
0
|
1
0
1
0
|
1
0
0
0
|
.700
.345
.000
.985
|
1.000
.625
.453
.245
|
.700
.345
.000
.245
|
The OR statement is also different in a fuzzy system as opposed
to a conventional system. Once again the table below demonstrates
how an OR statement functions in both a conventional and fuzzy
system. The table demonstrates that the OR statement returns the
maximum value of its two operatives. Thus, the OR is essentially
the opposite of the AND statement, under a fuzzy system.
| conventional |
|
|
fuzzy |
|
|
| A |
B |
A AND B |
Fuzzy A |
Fuzzy B |
A AND B |
1
1
0
0
|
1
0
1
0
|
1
1
1
0
|
.700
.345
.000
.985
|
1.000
.625
.453
.245
|
1.000
.625
.453
.985
|
The concept of the fuzzy set is also key to many applications of
fuzzy logic. A set in conventional mathematics is very simple. A
set is merely a collection of numbers or things. However, in fuzzy
math the definition of a set is slightly different. A fuzzy set is
a list of values, which describes to what degree an object belongs
to a variety of characteristics. The fuzzy set listed below is an
example of how a variety of temperatures might be classified.
| temp. in Farenheit |
very cold |
cold |
cool |
moderate |
warm |
hot |
brain baking |
20
30
40
50
60
70
80
90
100
110
|
.850
.275
.100
.000
.000
.000
.000
.000
.000
.000
|
.150
.450
.400
.300
.000
.000
.000
.000
.000
.000
|
.000
.275
.400
.500
.000
.000
.000
.000
.000
.000
|
.000
.000
.100
.300
.300
.100
.000
.000
.000
.000
|
.000
.000
.000
.000
.500
.800
.400
.100
.000
.000
|
.000
.000
.000
.000
.200
.100
.500
.700
.400
.100
|
.000
.000
.000
.000
.000
.000
.100
.200
.600
.900
|
|