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has a degree of truth based on a t-conorm T :
μ A B ( x, y )= T μ A ( x ) 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 ) 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 ) 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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