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that 6 is a complement number of 4, without explicitly mentioning 10 and +.
Such simplification, however, may cause some confusion about the “comple-
ment” concept. Choosing (9) as the complement set of
A
in the Z-system is
an example of such confusion.
For two sets,
A
and
B
,wehave
as
and
U
as
E
; that is,
U
=
A
B
.
However, we are still unable to say that
A
is the complement set of
B
with
respect to
U
under
∪
∪
∪
, because there are many
A
that satisfy
U
=
A
∪
B
for
agiven
B
. To make a unique
A
, we need to add
A
. This leads to
the complement set definition given by (3). For a characteristic function by
(5), the complement set definition is given by (8). It says that
µ
¬A
(
a
)isthe
complement of
µ
A
(
a
) with respect to the constant function
µ
U
(
a
) = 1 under
the operation in (7), subject to 0 = min
{µ
A
(
a
)
,µ
¬A
(
a
)
}
for its uniqueness. For
the binary valued function by (5), the complement
µ
¬A
(
a
) by (8) is equivalent
to the direct expression given by (9). Therefore, in such a case, either (8) or
(9) can be used as the definition of the complement
µ
¬A
(
a
).
For a real valued case by (10), the complement
µ
¬A
(
a
) by (8) is no longer
equivalent to that given by (9). In this case, the correct way is to choose
(8) to define the complement
µ
¬A
(
a
). However, except for those
µ
A
(
a
)inthe
degenerated case by (5), there is no solution for
µ
¬A
(
a
) by (8) for a real-valued
µ
A
(
a
) by (10), i.e., the complement set does not exist.
∩
B
=
∅
4.2 Avoiding a Controversial Definition in Zadeh's
Complement Set
Unfortunately, Zadeh mistakenly selected (9) as the definition of the comple-
ment
µ
¬A
(
a
) in the Z-system, though the
µ
¬A
(
a
) given by (9) is the comple-
ment of
µ
A
(
a
) with respect to
µ
T
(
a
) = 1 under +. In fact, the operation +
has been excluded from (7).
More precisely, 1
µ
A
(
a
) can be regarded as the “mirror” or “conjugate”
function with respect to
µ
T
(
a
) = 1. To distinguish it from
µ
¬A
(
a
), we denote
this conjugate function as
−
µ
ΞA
(
a
)=1
−
µ
A
(
a
)
.
(21)
The confusion of this concept with the commonly adopted complement con-
cept discussed in Sect. 4.1 has caused bad consequences at least in two aspects.
First, it produces unnecessary mistakes in applications of the Z-system.
Though it maybe clear to certain senior fuzzy researchers that the “comple-
ment” concept is different from the commonly adopted complement concept,
it may be not clear to many new comers or those who simply apply the
Z-system for practical uses. In their applications, a classical logical reasoning
problem is extended into a fuzzy logic problem via simply turning a binary
valued characteristic function into a fuzzy membership. This type of practice
may cause mistakes and lead to unsuccessful applications, unless the original
logical problem does not involve logical negation either directly or indirectly
