Image Processing Reference
In-Depth Information
has a degree of truth based on a t-conorm
T
:
μ
A
∨
B
(
x, y
)=
T
μ
A
(
x
)
,μ
B
(
y
)
,
and a negation has a degree of truth defined by a fuzzy complementation
c
:
¬
A
(
x
)=
c
μ
A
(
x
)
.
μ
S
In the case of variables with values in a product space, i.e.
X
with values in
and
V
Y
with values in
, conjunction is interpreted as a cartesian product. The degree of
truth of:
X
is
A
and
Y
is
B
is then written:
μ
A
×
B
(
x, y
)=
t
μ
A
(
x
)
,μ
B
(
y
)
.
Now let us consider the implication. In classical logic, we have:
A
=
⇒
(
B
)
⇐⇒
(
B
or non-
A
)
,
[8.85]
and therefore the implication is expressed based on a disjunction and a negation. By
using the same equivalence in the fuzzy case, a fuzzy implication is defined based on a
t-conorm (disjunction) and a complementation (negation). Let
A
and
B
be non-fuzzy
sets. The degree to which
A
implies
B
is defined by:
Imp(
A, B
)=
T
c
(
A
)
,B
[8.86]
where
T
is a t-conorm and
c
is a complementation.
In the case where
A
and
B
are fuzzy, we have:
T
c
μ
A
(
x
)
,μ
B
(
x
)
.
[8.87]
Imp(
A, B
)=inf
x
The following table sums up the major fuzzy implications used in other works for
fuzzy reasoning:
−
T
(
x, y
)=max(
x, y
)
max(1
a, b
)
Kleene-Diene
T
(
x, y
)=min(1
,x
+
y
) min(1
,
1
−
a
+
b
)
Lukasiewicz
T
(
x, y
)=
x
+
y
−
xy
1
−
a
+
ab
Reichenbach
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