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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