Image Processing Reference
In-Depth Information
In any case, we get the same classical implication table, given below, in the extreme
cases of true (1) and false (0) propositions, in other words when using binary degrees
of truth.
A
B
A
⇒
B
0
0
1
0
1
1
1
0
0
1
1
1
With these few definitions, we can now define the fuzzy equivalents of the major
reasoning modes: modus ponens, modus tollens, syllogism, contraposition. Consider
the example of modus ponens. In classical logic, it is written:
A
B
)
=
∧
(
A
=
⇒
⇒
B.
[8.88]
Its fuzzy equivalent is defined as follows:
- let us consider the rule:
if
X
is
A
then
Y
is
B
;
- and the knowledge:
X
is
A
where
A
is an approximation of
A
;
- we then come to the conclusion:
Y
is
B
where
B
is an approximation of
B
, with the degree:
t
μ
A
⇒
B
(
x, y
)
,μ
A
(
x
)
.
μ
B
(
y
) = sup
x
[8.89]
We can now model and handle fuzzy rule systems. For example, let us consider
the rule:
IF
(
x
is
A
AND
y
is
B
)
THEN
z
is
C
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